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## Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

- State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a or H 1 ).
- Collect data in a way designed to test the hypothesis.
- Perform an appropriate statistical test .
- Decide whether to reject or fail to reject your null hypothesis.
- Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

## Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

- H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

- an estimate of the difference in average height between the two groups.
- a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

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The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

- Normal distribution
- Descriptive statistics
- Measures of central tendency
- Correlation coefficient

Methodology

- Cluster sampling
- Stratified sampling
- Types of interviews
- Cohort study
- Thematic analysis

Research bias

- Implicit bias
- Cognitive bias
- Survivorship bias
- Availability heuristic
- Nonresponse bias
- Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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## 7.1: Basics of Hypothesis Testing

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- Kathryn Kozak
- Coconino Community College

To understand the process of a hypothesis tests, you need to first have an understanding of what a hypothesis is, which is an educated guess about a parameter. Once you have the hypothesis, you collect data and use the data to make a determination to see if there is enough evidence to show that the hypothesis is true. However, in hypothesis testing you actually assume something else is true, and then you look at your data to see how likely it is to get an event that your data demonstrates with that assumption. If the event is very unusual, then you might think that your assumption is actually false. If you are able to say this assumption is false, then your hypothesis must be true. This is known as a proof by contradiction. You assume the opposite of your hypothesis is true and show that it can’t be true. If this happens, then your hypothesis must be true. All hypothesis tests go through the same process. Once you have the process down, then the concept is much easier. It is easier to see the process by looking at an example. Concepts that are needed will be detailed in this example.

Example \(\PageIndex{1}\) basics of hypothesis testing

Suppose a manufacturer of the XJ35 battery claims the mean life of the battery is 500 days with a standard deviation of 25 days. You are the buyer of this battery and you think this claim is inflated. You would like to test your belief because without a good reason you can’t get out of your contract.

What do you do?

Well first, you should know what you are trying to measure. Define the random variable.

Let x = life of a XJ35 battery

Now you are not just trying to find different x values. You are trying to find what the true mean is. Since you are trying to find it, it must be unknown. You don’t think it is 500 days. If you did, you wouldn’t be doing any testing. The true mean, \(\mu\), is unknown. That means you should define that too.

Let \(\mu\)= mean life of a XJ35 battery

You may want to collect a sample. What kind of sample?

You could ask the manufacturers to give you batteries, but there is a chance that there could be some bias in the batteries they pick. To reduce the chance of bias, it is best to take a random sample.

How big should the sample be?

A sample of size 30 or more means that you can use the central limit theorem. Pick a sample of size 30.

Example \(\PageIndex{1}\) contains the data for the sample you collected:

Now what should you do? Looking at the data set, you see some of the times are above 500 and some are below. But looking at all of the numbers is too difficult. It might be helpful to calculate the mean for this sample.

The sample mean is \(\overline{x} = 490\) days. Looking at the sample mean, one might think that you are right. However, the standard deviation and the sample size also plays a role, so maybe you are wrong.

Before going any farther, it is time to formalize a few definitions.

You have a guess that the mean life of a battery is less than 500 days. This is opposed to what the manufacturer claims. There really are two hypotheses, which are just guesses here – the one that the manufacturer claims and the one that you believe. It is helpful to have names for them.

Definition \(\PageIndex{1}\)

Null Hypothesis : historical value, claim, or product specification. The symbol used is \(H_{o}\).

Definition \(\PageIndex{2}\)

Alternate Hypothesis : what you want to prove. This is what you want to accept as true when you reject the null hypothesis. There are two symbols that are commonly used for the alternative hypothesis: \(H_{A}\) or \(H_{I}\). The symbol \(H_{A}\) will be used in this book.

In general, the hypotheses look something like this:

\(H_{o} : \mu=\mu_{o}\)

\(H_{A} : \mu<\mu_{o}\)

where \(\mu_{o}\) just represents the value that the claim says the population mean is actually equal to.

Also, \(H_{A}\) can be less than, greater than, or not equal to.

For this problem:

\(H_{o} : \mu=500\) days, since the manufacturer says the mean life of a battery is 500 days.

\(H_{A} : \mu<500\) days, since you believe that the mean life of the battery is less than 500 days.

Now back to the mean. You have a sample mean of 490 days. Is this small enough to believe that you are right and the manufacturer is wrong? How small does it have to be?

If you calculated a sample mean of 235, you would definitely believe the population mean is less than 500. But even if you had a sample mean of 435 you would probably believe that the true mean was less than 500. What about 475? Or 483? There is some point where you would stop being so sure that the population mean is less than 500. That point separates the values of where you are sure or pretty sure that the mean is less than 500 from the area where you are not so sure. How do you find that point?

Well it depends on how much error you want to make. Of course you don’t want to make any errors, but unfortunately that is unavoidable in statistics. You need to figure out how much error you made with your sample. Take the sample mean, and find the probability of getting another sample mean less than it, assuming for the moment that the manufacturer is right. The idea behind this is that you want to know what is the chance that you could have come up with your sample mean even if the population mean really is 500 days.

You want to find \(P\left(\overline{x}<490 | H_{o} \text { is true }\right)=P(\overline{x}<490 | \mu=500)\)

To compute this probability, you need to know how the sample mean is distributed. Since the sample size is at least 30, then you know the sample mean is approximately normally distributed. Remember \(\mu_{\overline{x}}=\mu\) and \(\sigma_{\overline{x}}=\dfrac{\sigma}{\sqrt{n}}\)

A picture is always useful.

Before calculating the probability, it is useful to see how many standard deviations away from the mean the sample mean is. Using the formula for the z-score from chapter 6, you find

\(z=\dfrac{\overline{x}-\mu_{o}}{\sigma / \sqrt{n}}=\dfrac{490-500}{25 / \sqrt{30}}=-2.19\)

This sample mean is more than two standard deviations away from the mean. That seems pretty far, but you should look at the probability too.

On TI-83/84:

\(P(\overline{x}<490 | \mu=500)=\text { normalcdf }(-1 E 99,490,500,25 \div \sqrt{30}) \approx 0.0142\)

\(P(\overline{x}<490 \mu=500)=\text { pnorm }(490,500,25 / \operatorname{sqrt}(30)) \approx 0.0142\)

There is a 1.42% chance that you could find a sample mean less than 490 when the population mean is 500 days. This is really small, so the chances are that the assumption that the population mean is 500 days is wrong, and you can reject the manufacturer’s claim. But how do you quantify really small? Is 5% or 10% or 15% really small? How do you decide?

Before you answer that question, a couple more definitions are needed.

Definition \(\PageIndex{3}\)

Test Statistic : \(z=\dfrac{\overline{x}-\mu_{o}}{\sigma / \sqrt{n}}\) since it is calculated as part of the testing of the hypothesis.

Definition \(\PageIndex{4}\)

p – value : probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true. It is the probability that was calculated above.

Now, how small is small enough? To answer that, you really want to know the types of errors you can make.

There are actually only two errors that can be made. The first error is if you say that \(H_{o}\) is false, when in fact it is true. This means you reject \(H_{o}\) when \(H_{o}\) was true. The second error is if you say that \(H_{o}\) is true, when in fact it is false. This means you fail to reject \(H_{o}\) when \(H_{o}\) is false. The following table organizes this for you:

Type of errors:

Definition \(\PageIndex{5}\)

Type I Error is rejecting \(H_{o}\) when \(H_{o}\) is true, and

Definition \(\PageIndex{6}\)

Type II Error is failing to reject \(H_{o}\) when \(H_{o}\) is false.

Since these are the errors, then one can define the probabilities attached to each error.

Definition \(\PageIndex{7}\)

\(\alpha\) = P(type I error) = P(rejecting \(H_{o} / H_{o}\) is true)

Definition \(\PageIndex{8}\)

\(\beta\) = P(type II error) = P(failing to reject \(H_{o} / H_{o}\) is false)

\(\alpha\) is also called the level of significance .

Another common concept that is used is Power = \(1-\beta \).

Now there is a relationship between \(\alpha\) and \(\beta\). They are not complements of each other. How are they related?

If \(\alpha\) increases that means the chances of making a type I error will increase. It is more likely that a type I error will occur. It makes sense that you are less likely to make type II errors, only because you will be rejecting \(H_{o}\) more often. You will be failing to reject \(H_{o}\) less, and therefore, the chance of making a type II error will decrease. Thus, as \(\alpha\) increases, \(\beta\) will decrease, and vice versa. That makes them seem like complements, but they aren’t complements. What gives? Consider one more factor – sample size.

Consider if you have a larger sample that is representative of the population, then it makes sense that you have more accuracy then with a smaller sample. Think of it this way, which would you trust more, a sample mean of 490 if you had a sample size of 35 or sample size of 350 (assuming a representative sample)? Of course the 350 because there are more data points and so more accuracy. If you are more accurate, then there is less chance that you will make any error. By increasing the sample size of a representative sample, you decrease both \(\alpha\) and \(\beta\).

Summary of all of this:

- For a certain sample size, n , if \(\alpha\) increases, \(\beta\) decreases.
- For a certain level of significance, \(\alpha\), if n increases, \(\beta\) decreases.

Now how do you find \(\alpha\) and \(\beta\)? Well \(\alpha\) is actually chosen. There are only three values that are usually picked for \(\alpha\): 0.01, 0.05, and 0.10. \(\beta\) is very difficult to find, so usually it isn’t found. If you want to make sure it is small you take as large of a sample as you can afford provided it is a representative sample. This is one use of the Power. You want \(\beta\) to be small and the Power of the test is large. The Power word sounds good.

Which pick of \(\alpha\) do you pick? Well that depends on what you are working on. Remember in this example you are the buyer who is trying to get out of a contract to buy these batteries. If you create a type I error, you said that the batteries are bad when they aren’t, most likely the manufacturer will sue you. You want to avoid this. You might pick \(\alpha\) to be 0.01. This way you have a small chance of making a type I error. Of course this means you have more of a chance of making a type II error. No big deal right? What if the batteries are used in pacemakers and you tell the person that their pacemaker’s batteries are good for 500 days when they actually last less, that might be bad. If you make a type II error, you say that the batteries do last 500 days when they last less, then you have the possibility of killing someone. You certainly do not want to do this. In this case you might want to pick \(\alpha\) as 0.10. If both errors are equally bad, then pick \(\alpha\) as 0.05.

The above discussion is why the choice of \(\alpha\) depends on what you are researching. As the researcher, you are the one that needs to decide what \(\alpha\) level to use based on your analysis of the consequences of making each error is.

If a type I error is really bad, then pick \(\alpha\) = 0.01.

If a type II error is really bad, then pick \(\alpha\) = 0.10

If neither error is bad, or both are equally bad, then pick \(\alpha\) = 0.05

The main thing is to always pick the \(\alpha\) before you collect the data and start the test.

The above discussion was long, but it is really important information. If you don’t know what the errors of the test are about, then there really is no point in making conclusions with the tests. Make sure you understand what the two errors are and what the probabilities are for them.

Now it is time to go back to the example and put this all together. This is the basic structure of testing a hypothesis, usually called a hypothesis test. Since this one has a test statistic involving z, it is also called a z-test. And since there is only one sample, it is usually called a one-sample z-test.

Example \(\PageIndex{2}\) battery example revisited

- State the random variable and the parameter in words.
- State the null and alternative hypothesis and the level of significance.
- A random sample of size n is taken.
- The population standard derivation is known.
- The sample size is at least 30 or the population of the random variable is normally distributed.
- Find the sample statistic, test statistic, and p-value.
- Interpretation

1. x = life of battery

\(\mu\) = mean life of a XJ35 battery

2. \(H_{o} : \mu=500\) days

\(H_{A} : \mu<500\) days

\(\alpha = 0.10\) (from above discussion about consequences)

3. Every hypothesis has some assumptions that be met to make sure that the results of the hypothesis are valid. The assumptions are different for each test. This test has the following assumptions.

- This occurred in this example, since it was stated that a random sample of 30 battery lives were taken.
- This is true, since it was given in the problem.
- The sample size was 30, so this condition is met.

4. The test statistic depends on how many samples there are, what parameter you are testing, and assumptions that need to be checked. In this case, there is one sample and you are testing the mean. The assumptions were checked above.

Sample statistic:

\(\overline{x} = 490\)

Test statistic:

Using TI-83/84:

\(P(\overline{x}<490 | \mu=500)=\text { normalcdf }(-1 \mathrm{E} 99,490,500,25 / \sqrt{30}) \approx 0.0142\)

\(P(\overline{x}<490 | \mu=500)=\operatorname{pnorm}(490,500,25 / \operatorname{sqrt}(30)) \approx 0.0142\)

5. Now what? Well, this p-value is 0.0142. This is a lot smaller than the amount of error you would accept in the problem -\(\alpha\) = 0.10. That means that finding a sample mean less than 490 days is unusual to happen if \(H_{o}\) is true. This should make you think that \(H_{o}\) is not true. You should reject \(H_{o}\).

In fact, in general:

Reject \(H_{o}\) if the p-value < \(\alpha\) and

Fail to reject \(H_{o}\) if the p-value \(\geq \alpha\).

6. Since you rejected \(H_{o}\), what does this mean in the real world? That is what goes in the interpretation. Since you rejected the claim by the manufacturer that the mean life of the batteries is 500 days, then you now can believe that your hypothesis was correct. In other words, there is enough evidence to show that the mean life of the battery is less than 500 days.

Now that you know that the batteries last less than 500 days, should you cancel the contract? Statistically, there is evidence that the batteries do not last as long as the manufacturer says they should. However, based on this sample there are only ten days less on average that the batteries last. There may not be practical significance in this case. Ten days do not seem like a large difference. In reality, if the batteries are used in pacemakers, then you would probably tell the patient to have the batteries replaced every year. You have a large buffer whether the batteries last 490 days or 500 days. It seems that it might not be worth it to break the contract over ten days. What if the 10 days was practically significant? Are there any other things you should consider? You might look at the business relationship with the manufacturer. You might also look at how much it would cost to find a new manufacturer. These are also questions to consider before making any changes. What this discussion should show you is that just because a hypothesis has statistical significance does not mean it has practical significance. The hypothesis test is just one part of a research process. There are other pieces that you need to consider.

That’s it. That is what a hypothesis test looks like. All hypothesis tests are done with the same six steps. Those general six steps are outlined below.

- State the random variable and the parameter in words. This is where you are defining what the unknowns are in this problem. x = random variable \(\mu\) = mean of random variable, if the parameter of interest is the mean. There are other parameters you can test, and you would use the appropriate symbol for that parameter.
- State the null and alternative hypotheses and the level of significance \(H_{o} : \mu=\mu_{o}\), where \(\mu_{o}\) is the known mean \(H_{A} : \mu<\mu_{o}\) \(H_{A} : \mu>\mu_{o}\), use the appropriate one for your problem \(H_{A} : \mu \neq \mu_{o}\) Also, state your \(\alpha\) level here.
- State and check the assumptions for a hypothesis test. Each hypothesis test has its own assumptions. They will be stated when the different hypothesis tests are discussed.
- Find the sample statistic, test statistic, and p-value. This depends on what parameter you are working with, how many samples, and the assumptions of the test. The p-value depends on your \(H_{A}\). If you are doing the \(H_{A}\) with the less than, then it is a left-tailed test, and you find the probability of being in that left tail. If you are doing the \(H_{A}\) with the greater than, then it is a right-tailed test, and you find the probability of being in the right tail. If you are doing the \(H_{A}\) with the not equal to, then you are doing a two-tail test, and you find the probability of being in both tails. Because of symmetry, you could find the probability in one tail and double this value to find the probability in both tails.
- Conclusion This is where you write reject \(H_{o}\) or fail to reject \(H_{o}\). The rule is: if the p-value < \(\alpha\), then reject \(H_{o}\). If the p-value \(\geq \alpha\), then fail to reject \(H_{o}\).
- Interpretation This is where you interpret in real world terms the conclusion to the test. The conclusion for a hypothesis test is that you either have enough evidence to show \(H_{A}\) is true, or you do not have enough evidence to show \(H_{A}\) is true.

Sorry, one more concept about the conclusion and interpretation. First, the conclusion is that you reject \(H_{o}\) or you fail to reject \(H_{o}\). Why was it said like this? It is because you never accept the null hypothesis. If you wanted to accept the null hypothesis, then why do the test in the first place? In the interpretation, you either have enough evidence to show \(H_{A}\) is true, or you do not have enough evidence to show \(H_{A}\) is true. You wouldn’t want to go to all this work and then find out you wanted to accept the claim. Why go through the trouble? You always want to show that the alternative hypothesis is true. Sometimes you can do that and sometimes you can’t. It doesn’t mean you proved the null hypothesis; it just means you can’t prove the alternative hypothesis. Here is an example to demonstrate this.

Example \(\PageIndex{3}\) conclusion in hypothesis tests

In the U.S. court system a jury trial could be set up as a hypothesis test. To really help you see how this works, let’s use OJ Simpson as an example. In the court system, a person is presumed innocent until he/she is proven guilty, and this is your null hypothesis. OJ Simpson was a football player in the 1970s. In 1994 his ex-wife and her friend were killed. OJ Simpson was accused of the crime, and in 1995 the case was tried. The prosecutors wanted to prove OJ was guilty of killing his wife and her friend, and that is the alternative hypothesis

\(H_{0}\): OJ is innocent of killing his wife and her friend

\(H_{A}\): OJ is guilty of killing his wife and her friend

In this case, a verdict of not guilty was given. That does not mean that he is innocent of this crime. It means there was not enough evidence to prove he was guilty. Many people believe that OJ was guilty of this crime, but the jury did not feel that the evidence presented was enough to show there was guilt. The verdict in a jury trial is always guilty or not guilty!

The same is true in a hypothesis test. There is either enough or not enough evidence to show that alternative hypothesis. It is not that you proved the null hypothesis true.

When identifying hypothesis, it is important to state your random variable and the appropriate parameter you want to make a decision about. If count something, then the random variable is the number of whatever you counted. The parameter is the proportion of what you counted. If the random variable is something you measured, then the parameter is the mean of what you measured. (Note: there are other parameters you can calculate, and some analysis of those will be presented in later chapters.)

Example \(\PageIndex{4}\) stating hypotheses

Identify the hypotheses necessary to test the following statements:

- The average salary of a teacher is more than $30,000.
- The proportion of students who like math is less than 10%.
- The average age of students in this class differs from 21.

a. x = salary of teacher

\(\mu\) = mean salary of teacher

The guess is that \(\mu>\$ 30,000\) and that is the alternative hypothesis.

The null hypothesis has the same parameter and number with an equal sign.

\(\begin{array}{l}{H_{0} : \mu=\$ 30,000} \\ {H_{A} : \mu>\$ 30,000}\end{array}\)

b. x = number od students who like math

p = proportion of students who like math

The guess is that p < 0.10 and that is the alternative hypothesis.

\(\begin{array}{l}{H_{0} : p=0.10} \\ {H_{A} : p<0.10}\end{array}\)

c. x = age of students in this class

\(\mu\) = mean age of students in this class

The guess is that \(\mu \neq 21\) and that is the alternative hypothesis.

\(\begin{array}{c}{H_{0} : \mu=21} \\ {H_{A} : \mu \neq 21}\end{array}\)

Example \(\PageIndex{5}\) Stating Type I and II Errors and Picking Level of Significance

- The plant-breeding department at a major university developed a new hybrid raspberry plant called YumYum Berry. Based on research data, the claim is made that from the time shoots are planted 90 days on average are required to obtain the first berry with a standard deviation of 9.2 days. A corporation that is interested in marketing the product tests 60 shoots by planting them and recording the number of days before each plant produces its first berry. The sample mean is 92.3 days. The corporation wants to know if the mean number of days is more than the 90 days claimed. State the type I and type II errors in terms of this problem, consequences of each error, and state which level of significance to use.
- A concern was raised in Australia that the percentage of deaths of Aboriginal prisoners was higher than the percent of deaths of non-indigenous prisoners, which is 0.27%. State the type I and type II errors in terms of this problem, consequences of each error, and state which level of significance to use.

a. x = time to first berry for YumYum Berry plant

\(\mu\) = mean time to first berry for YumYum Berry plant

\(\begin{array}{l}{H_{0} : \mu=90} \\ {H_{A} : \mu>90}\end{array}\)

Type I Error: If the corporation does a type I error, then they will say that the plants take longer to produce than 90 days when they don’t. They probably will not want to market the plants if they think they will take longer. They will not market them even though in reality the plants do produce in 90 days. They may have loss of future earnings, but that is all.

Type II error: The corporation do not say that the plants take longer then 90 days to produce when they do take longer. Most likely they will market the plants. The plants will take longer, and so customers might get upset and then the company would get a bad reputation. This would be really bad for the company.

Level of significance: It appears that the corporation would not want to make a type II error. Pick a 10% level of significance, \(\alpha = 0.10\).

b. x = number of Aboriginal prisoners who have died

p = proportion of Aboriginal prisoners who have died

\(\begin{array}{l}{H_{o} : p=0.27 \%} \\ {H_{A} : p>0.27 \%}\end{array}\)

Type I error: Rejecting that the proportion of Aboriginal prisoners who died was 0.27%, when in fact it was 0.27%. This would mean you would say there is a problem when there isn’t one. You could anger the Aboriginal community, and spend time and energy researching something that isn’t a problem.

Type II error: Failing to reject that the proportion of Aboriginal prisoners who died was 0.27%, when in fact it is higher than 0.27%. This would mean that you wouldn’t think there was a problem with Aboriginal prisoners dying when there really is a problem. You risk causing deaths when there could be a way to avoid them.

Level of significance: It appears that both errors may be issues in this case. You wouldn’t want to anger the Aboriginal community when there isn’t an issue, and you wouldn’t want people to die when there may be a way to stop it. It may be best to pick a 5% level of significance, \(\alpha = 0.05\).

Hypothesis testing is really easy if you follow the same recipe every time. The only differences in the various problems are the assumptions of the test and the test statistic you calculate so you can find the p-value. Do the same steps, in the same order, with the same words, every time and these problems become very easy.

Exercise \(\PageIndex{1}\)

For the problems in this section, a question is being asked. This is to help you understand what the hypotheses are. You are not to run any hypothesis tests and come up with any conclusions in this section.

- Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a given time period and found that 11% of all lenses had defects of some type. Looking at the type of defects, they found in a three-month time period that out of 34,641 defective lenses, 5865 were due to scratches. Are there more defects from scratches than from all other causes? State the random variable, population parameter, and hypotheses.
- According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Alaska had 321 complaints of identity theft out of 1,432 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? State the random variable, population parameter, and hypotheses.
- The Kyoto Protocol was signed in 1997, and required countries to start reducing their carbon emissions. The protocol became enforceable in February 2005. In 2004, the mean CO2 emission was 4.87 metric tons per capita. Is there enough evidence to show that the mean CO2 emission is lower in 2010 than in 2004? State the random variable, population parameter, and hypotheses.
- The FDA regulates that fish that is consumed is allowed to contain 1.0 mg/kg of mercury. In Florida, bass fish were collected in 53 different lakes to measure the amount of mercury in the fish. The data for the average amount of mercury in each lake is in Example \(\PageIndex{5}\) ("Multi-disciplinary niser activity," 2013). Do the data provide enough evidence to show that the fish in Florida lakes has more mercury than the allowable amount? State the random variable, population parameter, and hypotheses.
- Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a given time period and found that 11% of all lenses had defects of some type. Looking at the type of defects, they found in a three-month time period that out of 34,641 defective lenses, 5865 were due to scratches. Are there more defects from scratches than from all other causes? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the manufacturer, and the appropriate alpha level to use. State why you picked this alpha level.
- According to the February 2008 Federal Trade Commission report on consumer fraud and identity theft, 23% of all complaints in 2007 were for identity theft. In that year, Alaska had 321 complaints of identity theft out of 1,432 consumer complaints ("Consumer fraud and," 2008). Does this data provide enough evidence to show that Alaska had a lower proportion of identity theft than 23%? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the state of Arizona, and the appropriate alpha level to use. State why you picked this alpha level.
- The Kyoto Protocol was signed in 1997, and required countries to start reducing their carbon emissions. The protocol became enforceable in February 2005. In 2004, the mean CO2 emission was 4.87 metric tons per capita. Is there enough evidence to show that the mean CO2 emission is lower in 2010 than in 2004? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the agency overseeing the protocol, and the appropriate alpha level to use. State why you picked this alpha level.
- The FDA regulates that fish that is consumed is allowed to contain 1.0 mg/kg of mercury. In Florida, bass fish were collected in 53 different lakes to measure the amount of mercury in the fish. The data for the average amount of mercury in each lake is in Example \(\PageIndex{5}\) ("Multi-disciplinary niser activity," 2013). Do the data provide enough evidence to show that the fish in Florida lakes has more mercury than the allowable amount? State the type I and type II errors in this case, consequences of each error type for this situation from the perspective of the FDA, and the appropriate alpha level to use. State why you picked this alpha level.

1. \(H_{o} : p=0.11, H_{A} : p>0.11\)

3. \(H_{o} : \mu=4.87 \text { metric tons per capita, } H_{A} : \mu<4.87 \text { metric tons per capita }\)

5. See solutions

7. See solutions

- How it works

## Hypothesis Testing – A Complete Guide with Examples

Published by Alvin Nicolas at August 14th, 2021 , Revised On October 26, 2023

In statistics, hypothesis testing is a critical tool. It allows us to make informed decisions about populations based on sample data. Whether you are a researcher trying to prove a scientific point, a marketer analysing A/B test results, or a manufacturer ensuring quality control, hypothesis testing plays a pivotal role. This guide aims to introduce you to the concept and walk you through real-world examples.

## What is a Hypothesis and a Hypothesis Testing?

A hypothesis is considered a belief or assumption that has to be accepted, rejected, proved or disproved. In contrast, a research hypothesis is a research question for a researcher that has to be proven correct or incorrect through investigation.

## What is Hypothesis Testing?

Hypothesis testing is a scientific method used for making a decision and drawing conclusions by using a statistical approach. It is used to suggest new ideas by testing theories to know whether or not the sample data supports research. A research hypothesis is a predictive statement that has to be tested using scientific methods that join an independent variable to a dependent variable.

Example: The academic performance of student A is better than student B

## Characteristics of the Hypothesis to be Tested

A hypothesis should be:

- Clear and precise
- Capable of being tested
- Able to relate to a variable
- Stated in simple terms
- Consistent with known facts
- Limited in scope and specific
- Tested in a limited timeframe
- Explain the facts in detail

## What is a Null Hypothesis and Alternative Hypothesis?

A null hypothesis is a hypothesis when there is no significant relationship between the dependent and the participants’ independent variables .

In simple words, it’s a hypothesis that has been put forth but hasn’t been proved as yet. A researcher aims to disprove the theory. The abbreviation “Ho” is used to denote a null hypothesis.

If you want to compare two methods and assume that both methods are equally good, this assumption is considered the null hypothesis.

Example: In an automobile trial, you feel that the new vehicle’s mileage is similar to the previous model of the car, on average. You can write it as: Ho: there is no difference between the mileage of both vehicles. If your findings don’t support your hypothesis and you get opposite results, this outcome will be considered an alternative hypothesis.

If you assume that one method is better than another method, then it’s considered an alternative hypothesis. The alternative hypothesis is the theory that a researcher seeks to prove and is typically denoted by H1 or HA.

If you support a null hypothesis, it means you’re not supporting the alternative hypothesis. Similarly, if you reject a null hypothesis, it means you are recommending the alternative hypothesis.

Example: In an automobile trial, you feel that the new vehicle’s mileage is better than the previous model of the vehicle. You can write it as; Ha: the two vehicles have different mileage. On average/ the fuel consumption of the new vehicle model is better than the previous model.

If a null hypothesis is rejected during the hypothesis test, even if it’s true, then it is considered as a type-I error. On the other hand, if you don’t dismiss a hypothesis, even if it’s false because you could not identify its falseness, it’s considered a type-II error.

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## How to Conduct Hypothesis Testing?

Here is a step-by-step guide on how to conduct hypothesis testing.

## Step 1: State the Null and Alternative Hypothesis

Once you develop a research hypothesis, it’s important to state it is as a Null hypothesis (Ho) and an Alternative hypothesis (Ha) to test it statistically.

A null hypothesis is a preferred choice as it provides the opportunity to test the theory. In contrast, you can accept the alternative hypothesis when the null hypothesis has been rejected.

Example: You want to identify a relationship between obesity of men and women and the modern living style. You develop a hypothesis that women, on average, gain weight quickly compared to men. Then you write it as: Ho: Women, on average, don’t gain weight quickly compared to men. Ha: Women, on average, gain weight quickly compared to men.

## Step 2: Data Collection

Hypothesis testing follows the statistical method, and statistics are all about data. It’s challenging to gather complete information about a specific population you want to study. You need to gather the data obtained through a large number of samples from a specific population.

Example: Suppose you want to test the difference in the rate of obesity between men and women. You should include an equal number of men and women in your sample. Then investigate various aspects such as their lifestyle, eating patterns and profession, and any other variables that may influence average weight. You should also determine your study’s scope, whether it applies to a specific group of population or worldwide population. You can use available information from various places, countries, and regions.

## Step 3: Select Appropriate Statistical Test

There are many types of statistical tests , but we discuss the most two common types below, such as One-sided and two-sided tests.

Note: Your choice of the type of test depends on the purpose of your study

## One-sided Test

In the one-sided test, the values of rejecting a null hypothesis are located in one tail of the probability distribution. The set of values is less or higher than the critical value of the test. It is also called a one-tailed test of significance.

Example: If you want to test that all mangoes in a basket are ripe. You can write it as: Ho: All mangoes in the basket, on average, are ripe. If you find all ripe mangoes in the basket, the null hypothesis you developed will be true.

## Two-sided Test

In the two-sided test, the values of rejecting a null hypothesis are located on both tails of the probability distribution. The set of values is less or higher than the first critical value of the test and higher than the second critical value test. It is also called a two-tailed test of significance.

Example: Nothing can be explicitly said whether all mangoes are ripe in the basket. If you reject the null hypothesis (Ho: All mangoes in the basket, on average, are ripe), then it means all mangoes in the basket are not likely to be ripe. A few mangoes could be raw as well.

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## Step 4: Select the Level of Significance

When you reject a null hypothesis, even if it’s true during a statistical hypothesis, it is considered the significance level . It is the probability of a type one error. The significance should be as minimum as possible to avoid the type-I error, which is considered severe and should be avoided.

If the significance level is minimum, then it prevents the researchers from false claims.

The significance level is denoted by P, and it has given the value of 0.05 (P=0.05)

If the P-Value is less than 0.05, then the difference will be significant. If the P-value is higher than 0.05, then the difference is non-significant.

Example: Suppose you apply a one-sided test to test whether women gain weight quickly compared to men. You get to know about the average weight between men and women and the factors promoting weight gain.

## Step 5: Find out Whether the Null Hypothesis is Rejected or Supported

After conducting a statistical test, you should identify whether your null hypothesis is rejected or accepted based on the test results. It would help if you observed the P-value for this.

Example: If you find the P-value of your test is less than 0.5/5%, then you need to reject your null hypothesis (Ho: Women, on average, don’t gain weight quickly compared to men). On the other hand, if a null hypothesis is rejected, then it means the alternative hypothesis might be true (Ha: Women, on average, gain weight quickly compared to men. If you find your test’s P-value is above 0.5/5%, then it means your null hypothesis is true.

## Step 6: Present the Outcomes of your Study

The final step is to present the outcomes of your study . You need to ensure whether you have met the objectives of your research or not.

In the discussion section and conclusion , you can present your findings by using supporting evidence and conclude whether your null hypothesis was rejected or supported.

In the result section, you can summarise your study’s outcomes, including the average difference and P-value of the two groups.

If we talk about the findings, our study your results will be as follows:

Example: In the study of identifying whether women gain weight quickly compared to men, we found the P-value is less than 0.5. Hence, we can reject the null hypothesis (Ho: Women, on average, don’t gain weight quickly than men) and conclude that women may likely gain weight quickly than men.

Did you know in your academic paper you should not mention whether you have accepted or rejected the null hypothesis?

Always remember that you either conclude to reject Ho in favor of Haor do not reject Ho . It would help if you never rejected Ha or even accept Ha .

Suppose your null hypothesis is rejected in the hypothesis testing. If you conclude reject Ho in favor of Haor do not reject Ho, then it doesn’t mean that the null hypothesis is true. It only means that there is a lack of evidence against Ho in favour of Ha. If your null hypothesis is not true, then the alternative hypothesis is likely to be true.

Example: We found that the P-value is less than 0.5. Hence, we can conclude reject Ho in favour of Ha (Ho: Women, on average, don’t gain weight quickly than men) reject Ho in favour of Ha. However, rejected in favour of Ha means (Ha: women may likely to gain weight quickly than men)

## Frequently Asked Questions

What are the 3 types of hypothesis test.

The 3 types of hypothesis tests are:

- One-Sample Test : Compare sample data to a known population value.
- Two-Sample Test : Compare means between two sample groups.
- ANOVA : Analyze variance among multiple groups to determine significant differences.

## What is a hypothesis?

A hypothesis is a proposed explanation or prediction about a phenomenon, often based on observations. It serves as a starting point for research or experimentation, providing a testable statement that can either be supported or refuted through data and analysis. In essence, it’s an educated guess that drives scientific inquiry.

## What are null hypothesis?

A null hypothesis (often denoted as H0) suggests that there is no effect or difference in a study or experiment. It represents a default position or status quo. Statistical tests evaluate data to determine if there’s enough evidence to reject this null hypothesis.

## What is the probability value?

The probability value, or p-value, is a measure used in statistics to determine the significance of an observed effect. It indicates the probability of obtaining the observed results, or more extreme, if the null hypothesis were true. A small p-value (typically <0.05) suggests evidence against the null hypothesis, warranting its rejection.

## What is p value?

The p-value is a fundamental concept in statistical hypothesis testing. It represents the probability of observing a test statistic as extreme, or more so, than the one calculated from sample data, assuming the null hypothesis is true. A low p-value suggests evidence against the null, possibly justifying its rejection.

## What is a t test?

A t-test is a statistical test used to compare the means of two groups. It determines if observed differences between the groups are statistically significant or if they likely occurred by chance. Commonly applied in research, there are different t-tests, including independent, paired, and one-sample, tailored to various data scenarios.

## When to reject null hypothesis?

Reject the null hypothesis when the test statistic falls into a predefined rejection region or when the p-value is less than the chosen significance level (commonly 0.05). This suggests that the observed data is unlikely under the null hypothesis, indicating evidence for the alternative hypothesis. Always consider the study’s context.

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Baffled by the concept of reliability and validity? Reliability refers to the consistency of measurement. Validity refers to the accuracy of measurement.

What are the different research strategies you can use in your dissertation? Here are some guidelines to help you choose a research strategy that would make your research more credible.

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## How to Test Statistical Hypotheses

This lesson describes a general procedure that can be used to test statistical hypotheses.

## How to Conduct Hypothesis Tests

All hypothesis tests are conducted the same way. The researcher states a hypothesis to be tested, formulates an analysis plan, analyzes sample data according to the plan, and accepts or rejects the null hypothesis, based on results of the analysis.

- State the hypotheses. Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis . The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa.
- Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
- Test method. Typically, the test method involves a test statistic and a sampling distribution . Computed from sample data, the test statistic might be a mean score, proportion, difference between means, difference between proportions, z-score, t statistic, chi-square, etc. Given a test statistic and its sampling distribution, a researcher can assess probabilities associated with the test statistic. If the test statistic probability is less than the significance level, the null hypothesis is rejected.

Test statistic = (Statistic - Parameter) / (Standard deviation of statistic)

Test statistic = (Statistic - Parameter) / (Standard error of statistic)

- P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic, assuming the null hypothesis is true.
- Interpret the results. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

## Applications of the General Hypothesis Testing Procedure

The next few lessons show how to apply the general hypothesis testing procedure to different kinds of statistical problems.

- Proportions
- Difference between proportions
- Regression slope
- Difference between means
- Difference between matched pairs
- Goodness of fit
- Homogeneity
- Independence

At this point, don't worry if the general procedure for testing hypotheses seems a little bit unclear. The procedure will be clearer as you see it applied in the next few lessons.

## Test Your Understanding

In hypothesis testing, which of the following statements is always true?

I. The P-value is greater than the significance level. II. The P-value is computed from the significance level. III. The P-value is the parameter in the null hypothesis. IV. The P-value is a test statistic. V. The P-value is a probability.

(A) I only (B) II only (C) III only (D) IV only (E) V only

The correct answer is (E). The P-value is the probability of observing a sample statistic as extreme as the test statistic. It can be greater than the significance level, but it can also be smaller than the significance level. It is not computed from the significance level, it is not the parameter in the null hypothesis, and it is not a test statistic.

Statistics Made Easy

## Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about a population parameter .

For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

## The Two Types of Statistical Hypotheses

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.

## Hypothesis Tests

A hypothesis test consists of five steps:

1. State the hypotheses.

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

2. Determine a significance level to use for the hypothesis.

Decide on a significance level. Common choices are .01, .05, and .1.

3. Find the test statistic.

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

4. Reject or fail to reject the null hypothesis.

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

5. Interpret the results.

Interpret the results of the hypothesis test in the context of the question being asked.

## The Two Types of Decision Errors

There are two types of decision errors that one can make when doing a hypothesis test:

Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.

Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.

## One-Tailed and Two-Tailed Tests

A statistical hypothesis can be one-tailed or two-tailed.

A one-tailed hypothesis involves making a “greater than” or “less than ” statement.

For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.

For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.

Related: What is a Directional Hypothesis?

## Types of Hypothesis Tests

There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.

The following tutorials provide an explanation of the most common types of hypothesis tests:

Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test

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## Keyboard Shortcuts

Hypothesis testing.

Key Topics:

- Basic approach
- Null and alternative hypothesis
- Decision making and the p -value
- Z-test & Nonparametric alternative

## Basic approach to hypothesis testing

- State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
- Specify the null and alternative hypotheses in terms of the parameters of the model.
- Invent a test statistic that will tend to be different under the null and alternative hypotheses.
- Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
- Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
- Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.
- \(\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}\)
- general form is: (estimate - value we are testing)/(st.dev of the estimate)
- z-statistic follows N(0,1) distribution
- 2 × the area above |z|, area above z,or area below z, or
- compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
- Choose the acceptable level of Alpha = 0.05, we conclude …. ?

## Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

- either our initial assumption is correct and we experienced an unusual event or,
- our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

## Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

- H 0 : Defendant is not guilty.
- H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

- It can one-sided and two-sided.
- In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

- Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

- If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
- If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

## Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

## Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

- α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

- β is the probability of making a Type II error

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

- the effect of the sample size,
- the correctness of the underlying assumptions about the population,
- statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

## One sample t -test

- Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .
- t-statistic: \(\frac{\bar{X}-\mu_0}{s / \sqrt{n}}\) where s is a sample st.dev.
- t-statistic follows t -distribution with df = n - 1
- Alpha = 0.05, we conclude ….

## Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

## Hypothesis Testing

When you conduct a piece of quantitative research, you are inevitably attempting to answer a research question or hypothesis that you have set. One method of evaluating this research question is via a process called hypothesis testing , which is sometimes also referred to as significance testing . Since there are many facets to hypothesis testing, we start with the example we refer to throughout this guide.

## An example of a lecturer's dilemma

Two statistics lecturers, Sarah and Mike, think that they use the best method to teach their students. Each lecturer has 50 statistics students who are studying a graduate degree in management. In Sarah's class, students have to attend one lecture and one seminar class every week, whilst in Mike's class students only have to attend one lecture. Sarah thinks that seminars, in addition to lectures, are an important teaching method in statistics, whilst Mike believes that lectures are sufficient by themselves and thinks that students are better off solving problems by themselves in their own time. This is the first year that Sarah has given seminars, but since they take up a lot of her time, she wants to make sure that she is not wasting her time and that seminars improve her students' performance.

## The research hypothesis

The first step in hypothesis testing is to set a research hypothesis. In Sarah and Mike's study, the aim is to examine the effect that two different teaching methods – providing both lectures and seminar classes (Sarah), and providing lectures by themselves (Mike) – had on the performance of Sarah's 50 students and Mike's 50 students. More specifically, they want to determine whether performance is different between the two different teaching methods. Whilst Mike is skeptical about the effectiveness of seminars, Sarah clearly believes that giving seminars in addition to lectures helps her students do better than those in Mike's class. This leads to the following research hypothesis:

Before moving onto the second step of the hypothesis testing process, we need to take you on a brief detour to explain why you need to run hypothesis testing at all. This is explained next.

## Sample to population

If you have measured individuals (or any other type of "object") in a study and want to understand differences (or any other type of effect), you can simply summarize the data you have collected. For example, if Sarah and Mike wanted to know which teaching method was the best, they could simply compare the performance achieved by the two groups of students – the group of students that took lectures and seminar classes, and the group of students that took lectures by themselves – and conclude that the best method was the teaching method which resulted in the highest performance. However, this is generally of only limited appeal because the conclusions could only apply to students in this study. However, if those students were representative of all statistics students on a graduate management degree, the study would have wider appeal.

In statistics terminology, the students in the study are the sample and the larger group they represent (i.e., all statistics students on a graduate management degree) is called the population . Given that the sample of statistics students in the study are representative of a larger population of statistics students, you can use hypothesis testing to understand whether any differences or effects discovered in the study exist in the population. In layman's terms, hypothesis testing is used to establish whether a research hypothesis extends beyond those individuals examined in a single study.

Another example could be taking a sample of 200 breast cancer sufferers in order to test a new drug that is designed to eradicate this type of cancer. As much as you are interested in helping these specific 200 cancer sufferers, your real goal is to establish that the drug works in the population (i.e., all breast cancer sufferers).

As such, by taking a hypothesis testing approach, Sarah and Mike want to generalize their results to a population rather than just the students in their sample. However, in order to use hypothesis testing, you need to re-state your research hypothesis as a null and alternative hypothesis. Before you can do this, it is best to consider the process/structure involved in hypothesis testing and what you are measuring. This structure is presented on the next page .

## 6 Steps to Evaluate the Effectiveness of Statistical Hypothesis Testing

You know what is tragic? Having the potential to complete the research study but not doing the correct hypothesis testing. Quite often, researchers think the most challenging aspect of research is standardization of experiments, data analysis or writing the thesis! But in all honesty, creating an effective research hypothesis is the most crucial step in designing and executing a research study. An effective research hypothesis will provide researchers the correct basic structure for building the research question and objectives.

In this article, we will discuss how to formulate and identify an effective research hypothesis testing to benefit researchers in designing their research work.

Table of Contents

## What Is Research Hypothesis Testing?

Hypothesis testing is a systematic procedure derived from the research question and decides if the results of a research study support a certain theory which can be applicable to the population. Moreover, it is a statistical test used to determine whether the hypothesis assumed by the sample data stands true to the entire population.

The purpose of testing the hypothesis is to make an inference about the population of interest on the basis of random sample taken from that population. Furthermore, it is the assumption which is tested to determine the relationship between two data sets.

## Types of Statistical Hypothesis Testing

## Source: https://www.youtube.com/c/365DataScience

1. there are two types of hypothesis in statistics, a. null hypothesis.

This is the assumption that the event will not occur or there is no relation between the compared variables. A null hypothesis has no relation with the study’s outcome unless it is rejected. Null hypothesis uses H0 as its symbol.

## b. Alternate Hypothesis

The alternate hypothesis is the logical opposite of the null hypothesis. Furthermore, the acceptance of the alternative hypothesis follows the rejection of the null hypothesis. It uses H1 or Ha as its symbol

Hypothesis Testing Example: A sanitizer manufacturer company claims that its product kills 98% of germs on average. To put this company’s claim to test, create null and alternate hypothesis H0 (Null Hypothesis): Average = 98% H1/Ha (Alternate Hypothesis): The average is less than 98%

## 2. Depending on the population distribution, you can categorize the statistical hypothesis into two types.

A. simple hypothesis.

A simple hypothesis specifies an exact value for the parameter.

## b. Composite Hypothesis

A composite hypothesis specifies a range of values.

Hypothesis Testing Example: A company claims to have achieved 1000 units as their average sales for this quarter. (Simple Hypothesis) The company claims to achieve the sales in the range of 900 to 100o units. (Composite Hypothesis).

## 3. Based on the type of statistical testing, the hypothesis in statistics is of two types.

A. one-tailed.

One-Tailed test or directional test considers a critical region of data which would result in rejection of the null hypothesis if the test sample falls in that data region. Therefore, accepting the alternate hypothesis. Furthermore, the critical distribution area in this test is one-sided which means the test sample is either greater or lesser than a specific value.

## b. Two-Tailed

Two-Tailed test or nondirectional test is designed to show if the sample mean is significantly greater than and significantly less than the mean population. Here, the critical distribution area is two-sided. If the sample falls within the range, the alternate hypothesis is accepted and the null hypothesis is rejected.

Statistical Hypothesis Testing Example: Suppose H0: mean = 100 and H1: mean is not equal to 100 According to the H1, the mean can be greater than or less than 100. (Two-Tailed test) Similarly, if H0: mean >= 100, then H1: mean < 100 Here the mean is less than 100. (One-Tailed test)

## Steps in Statistical Hypothesis Testing

Step 1: develop initial research hypothesis.

Research hypothesis is developed from research question. It is the prediction that you want to investigate. Moreover, an initial research hypothesis is important for restating the null and alternate hypothesis, to test the research question mathematically.

## Step 2: State the null and alternate hypothesis based on your research hypothesis

Usually, the alternate hypothesis is your initial hypothesis that predicts relationship between variables. However, the null hypothesis is a prediction of no relationship between the variables you are interested in.

## Step 3: Perform sampling and collection of data for statistical testing

It is important to perform sampling and collect data in way that assists the formulated research hypothesis. You will have to perform a statistical testing to validate your data and make statistical inferences about the population of your interest.

## Step 4: Perform statistical testing based on the type of data you collected

There are various statistical tests available. Based on the comparison of within group variance and between group variance, you can carry out the statistical tests for the research study. If the between group variance is large enough and there is little or no overlap between groups, then the statistical test will show low p-value. (Difference between the groups is not a chance event).

Alternatively, if the within group variance is high compared to between group variance, then the statistical test shows a high p-value. (Difference between the groups is a chance event).

## Step 5: Based on the statistical outcome, reject or fail to reject your null hypothesis

In most cases, you will use p-value generated from your statistical test to guide your decision. You will consider a predetermined level of significance of 0.05 for rejecting your null hypothesis , i.e. there is less than 5% chance of getting the results wherein the null hypothesis is true.

## Step 6: Present your final results of hypothesis testing

You will present the results of your hypothesis in the results and discussion section of the research paper . In results section, you provide a brief summary of the data and a summary of the results of your statistical test. Meanwhile, in discussion, you can mention whether your results support your initial hypothesis.

Note that we never reject or fail to reject the alternate hypothesis. This is because the testing of hypothesis is not designed to prove or disprove anything. However, it is designed to test if a result is spuriously occurred, or by chance. Thus, statistical hypothesis testing becomes a crucial statistical tool to mathematically define the outcome of a research question.

Have you ever used hypothesis testing as a means of statistically analyzing your research data? How was your experience? Do write to us or comment below.

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## A Beginner’s Guide to Hypothesis Testing in Business

- 30 Mar 2021

Becoming a more data-driven decision-maker can bring several benefits to your organization, enabling you to identify new opportunities to pursue and threats to abate. Rather than allowing subjective thinking to guide your business strategy, backing your decisions with data can empower your company to become more innovative and, ultimately, profitable.

If you’re new to data-driven decision-making, you might be wondering how data translates into business strategy. The answer lies in generating a hypothesis and verifying or rejecting it based on what various forms of data tell you.

Below is a look at hypothesis testing and the role it plays in helping businesses become more data-driven.

Access your free e-book today.

## What Is Hypothesis Testing?

To understand what hypothesis testing is, it’s important first to understand what a hypothesis is.

A hypothesis or hypothesis statement seeks to explain why something has happened, or what might happen, under certain conditions. It can also be used to understand how different variables relate to each other. Hypotheses are often written as if-then statements; for example, “If this happens, then this will happen.”

Hypothesis testing , then, is a statistical means of testing an assumption stated in a hypothesis. While the specific methodology leveraged depends on the nature of the hypothesis and data available, hypothesis testing typically uses sample data to extrapolate insights about a larger population.

## Hypothesis Testing in Business

When it comes to data-driven decision-making, there’s a certain amount of risk that can mislead a professional. This could be due to flawed thinking or observations, incomplete or inaccurate data , or the presence of unknown variables. The danger in this is that, if major strategic decisions are made based on flawed insights, it can lead to wasted resources, missed opportunities, and catastrophic outcomes.

The real value of hypothesis testing in business is that it allows professionals to test their theories and assumptions before putting them into action. This essentially allows an organization to verify its analysis is correct before committing resources to implement a broader strategy.

As one example, consider a company that wishes to launch a new marketing campaign to revitalize sales during a slow period. Doing so could be an incredibly expensive endeavor, depending on the campaign’s size and complexity. The company, therefore, may wish to test the campaign on a smaller scale to understand how it will perform.

In this example, the hypothesis that’s being tested would fall along the lines of: “If the company launches a new marketing campaign, then it will translate into an increase in sales.” It may even be possible to quantify how much of a lift in sales the company expects to see from the effort. Pending the results of the pilot campaign, the business would then know whether it makes sense to roll it out more broadly.

Related: 9 Fundamental Data Science Skills for Business Professionals

## Key Considerations for Hypothesis Testing

1. alternative hypothesis and null hypothesis.

In hypothesis testing, the hypothesis that’s being tested is known as the alternative hypothesis . Often, it’s expressed as a correlation or statistical relationship between variables. The null hypothesis , on the other hand, is a statement that’s meant to show there’s no statistical relationship between the variables being tested. It’s typically the exact opposite of whatever is stated in the alternative hypothesis.

For example, consider a company’s leadership team that historically and reliably sees $12 million in monthly revenue. They want to understand if reducing the price of their services will attract more customers and, in turn, increase revenue.

In this case, the alternative hypothesis may take the form of a statement such as: “If we reduce the price of our flagship service by five percent, then we’ll see an increase in sales and realize revenues greater than $12 million in the next month.”

The null hypothesis, on the other hand, would indicate that revenues wouldn’t increase from the base of $12 million, or might even decrease.

Check out the video below about the difference between an alternative and a null hypothesis, and subscribe to our YouTube channel for more explainer content.

## 2. Significance Level and P-Value

Statistically speaking, if you were to run the same scenario 100 times, you’d likely receive somewhat different results each time. If you were to plot these results in a distribution plot, you’d see the most likely outcome is at the tallest point in the graph, with less likely outcomes falling to the right and left of that point.

With this in mind, imagine you’ve completed your hypothesis test and have your results, which indicate there may be a correlation between the variables you were testing. To understand your results' significance, you’ll need to identify a p-value for the test, which helps note how confident you are in the test results.

In statistics, the p-value depicts the probability that, assuming the null hypothesis is correct, you might still observe results that are at least as extreme as the results of your hypothesis test. The smaller the p-value, the more likely the alternative hypothesis is correct, and the greater the significance of your results.

## 3. One-Sided vs. Two-Sided Testing

When it’s time to test your hypothesis, it’s important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests , or one-tailed and two-tailed tests, respectively.

Typically, you’d leverage a one-sided test when you have a strong conviction about the direction of change you expect to see due to your hypothesis test. You’d leverage a two-sided test when you’re less confident in the direction of change.

## 4. Sampling

To perform hypothesis testing in the first place, you need to collect a sample of data to be analyzed. Depending on the question you’re seeking to answer or investigate, you might collect samples through surveys, observational studies, or experiments.

A survey involves asking a series of questions to a random population sample and recording self-reported responses.

Observational studies involve a researcher observing a sample population and collecting data as it occurs naturally, without intervention.

Finally, an experiment involves dividing a sample into multiple groups, one of which acts as the control group. For each non-control group, the variable being studied is manipulated to determine how the data collected differs from that of the control group.

## Learn How to Perform Hypothesis Testing

Hypothesis testing is a complex process involving different moving pieces that can allow an organization to effectively leverage its data and inform strategic decisions.

If you’re interested in better understanding hypothesis testing and the role it can play within your organization, one option is to complete a course that focuses on the process. Doing so can lay the statistical and analytical foundation you need to succeed.

Do you want to learn more about hypothesis testing? Explore Business Analytics —one of our online business essentials courses —and download our Beginner’s Guide to Data & Analytics .

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- Fundamental Analysis

## Hypothesis to Be Tested: Definition and 4 Steps for Testing with Example

## What Is Hypothesis Testing?

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population, or from a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

## Key Takeaways

- Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
- The test provides evidence concerning the plausibility of the hypothesis, given the data.
- Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
- The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

## How Hypothesis Testing Works

In hypothesis testing, an analyst tests a statistical sample, with the goal of providing evidence on the plausibility of the null hypothesis.

Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis (e.g., the population mean return is not equal to zero). Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

## 4 Steps of Hypothesis Testing

All hypotheses are tested using a four-step process:

- The first step is for the analyst to state the hypotheses.
- The second step is to formulate an analysis plan, which outlines how the data will be evaluated.
- The third step is to carry out the plan and analyze the sample data.
- The final step is to analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

## Real-World Example of Hypothesis Testing

If, for example, a person wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct.

Mathematically, the null hypothesis would be represented as Ho: P = 0.5. The alternative hypothesis would be denoted as "Ha" and be identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is then tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If, on the other hand, there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

Some staticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

## What is Hypothesis Testing?

Hypothesis testing refers to a process used by analysts to assess the plausibility of a hypothesis by using sample data. In hypothesis testing, statisticians formulate two hypotheses: the null hypothesis and the alternative hypothesis. A null hypothesis determines there is no difference between two groups or conditions, while the alternative hypothesis determines that there is a difference. Researchers evaluate the statistical significance of the test based on the probability that the null hypothesis is true.

## What are the Four Key Steps Involved in Hypothesis Testing?

Hypothesis testing begins with an analyst stating two hypotheses, with only one that can be right. The analyst then formulates an analysis plan, which outlines how the data will be evaluated. Next, they move to the testing phase and analyze the sample data. Finally, the analyst analyzes the results and either rejects the null hypothesis or states that the null hypothesis is plausible, given the data.

## What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

## What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

## The Bottom Line

Hypothesis testing refers to a statistical process that helps researchers and/or analysts determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. There are different types of hypothesis testing, each with their own set of rules and procedures. However, all hypothesis testing methods have the same four step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. Hypothesis testing plays a vital part of the scientific process, helping to test assumptions and make better data-based decisions.

Sage. " Introduction to Hypothesis Testing. " Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples. "

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- How to Write a Strong Hypothesis | Guide & Examples

## How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

## Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

## Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

## Prevent plagiarism, run a free check.

Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

## Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

## Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

## Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

- The relevant variables
- The specific group being studied
- The predicted outcome of the experiment or analysis

## Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

## Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

## Cite this Scribbr article

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McCombes, S. (2022, May 06). How to Write a Strong Hypothesis | Guide & Examples. Scribbr. Retrieved 22 February 2024, from https://www.scribbr.co.uk/research-methods/hypothesis-writing/

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## Shona McCombes

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## Unit 12: Significance tests (hypothesis testing)

About this unit, the idea of significance tests.

- Simple hypothesis testing (Opens a modal)
- Idea behind hypothesis testing (Opens a modal)
- Examples of null and alternative hypotheses (Opens a modal)
- P-values and significance tests (Opens a modal)
- Comparing P-values to different significance levels (Opens a modal)
- Estimating a P-value from a simulation (Opens a modal)
- Using P-values to make conclusions (Opens a modal)
- Simple hypothesis testing Get 3 of 4 questions to level up!
- Writing null and alternative hypotheses Get 3 of 4 questions to level up!
- Estimating P-values from simulations Get 3 of 4 questions to level up!

## Error probabilities and power

- Introduction to Type I and Type II errors (Opens a modal)
- Type 1 errors (Opens a modal)
- Examples identifying Type I and Type II errors (Opens a modal)
- Introduction to power in significance tests (Opens a modal)
- Examples thinking about power in significance tests (Opens a modal)
- Consequences of errors and significance (Opens a modal)
- Type I vs Type II error Get 3 of 4 questions to level up!
- Error probabilities and power Get 3 of 4 questions to level up!

## Tests about a population proportion

- Constructing hypotheses for a significance test about a proportion (Opens a modal)
- Conditions for a z test about a proportion (Opens a modal)
- Reference: Conditions for inference on a proportion (Opens a modal)
- Calculating a z statistic in a test about a proportion (Opens a modal)
- Calculating a P-value given a z statistic (Opens a modal)
- Making conclusions in a test about a proportion (Opens a modal)
- Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
- Conditions for a z test about a proportion Get 3 of 4 questions to level up!
- Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
- Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
- Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

## Tests about a population mean

- Writing hypotheses for a significance test about a mean (Opens a modal)
- Conditions for a t test about a mean (Opens a modal)
- Reference: Conditions for inference on a mean (Opens a modal)
- When to use z or t statistics in significance tests (Opens a modal)
- Example calculating t statistic for a test about a mean (Opens a modal)
- Using TI calculator for P-value from t statistic (Opens a modal)
- Using a table to estimate P-value from t statistic (Opens a modal)
- Comparing P-value from t statistic to significance level (Opens a modal)
- Free response example: Significance test for a mean (Opens a modal)
- Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
- Conditions for a t test about a mean Get 3 of 4 questions to level up!
- Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
- Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
- Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

## More significance testing videos

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## How to Test a Hypothesis

Last Updated: July 4, 2023 Fact Checked

This article was co-authored by Meredith Juncker, PhD . Meredith Juncker is a PhD candidate in Biochemistry and Molecular Biology at Louisiana State University Health Sciences Center. Her studies are focused on proteins and neurodegenerative diseases. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 40,473 times.

Testing a hypothesis is an important part of the scientific method. It allows you to evaluate the validity of an educated guess. In a typical process, you’ll form a hypothesis based on evidence that you’ve gathered, and then test that hypothesis through experiments. As you gather more and more data, you’ll be able to see if your original hypothesis was correct. If there were flaws in your first guess, you can revise your hypothesis to better match what you’ve learned from your data.

## Asking a Question and Researching

- For example, a question could be something like, “Which brand of stain remover will remove stains from fabrics most effectively?” [2] X Research source

- For the stain-remover experiment, you could dirty 4 types of fabric (e.g. cotton, linen, wool, polyester) each with 4 different types of stains (e.g. red wine, grass, mud and dirt, grease), and then test the top four or five brands of stain remover (e.g. Mr. Clean, Tide, Shout, Clorox) to see which removes the largest number of stains.

- In the case of the stain-remover experiment, you’d need to purchase a bottle of each of the major stain-remover brands and dirty a variety of fabrics with a variety of stains.
- Then, test out each type of detergent on each of the stained fabrics. (If you live at your parents’ house, you’ll need to get permission to use the laundry room for most of a day.)

## Making and Challenging Your Hypothesis

- For instance, if you’ve run some loads of laundry (maybe testing which brand of stain remover works best on removing different stains from linen), you can use your results to take a stab at a hypothesis.
- A good working hypothesis would look like: “If fabrics are stained with common household items, Tide stain remover will remove the stains most effectively.”

- In our example, since you only tested 1 type of fabric (linen), you’ll need to repeat the laundry test with the other 3 fabrics (cotton, wool, polyester) and note which stain remover most effectively eliminates the stains.

- While it can be tempting to only accept data that supports your hypothesis, it’s not scientific or ethical.
- You must accept all the data and watch for whatever patterns appear, even if it proves your hypothesis to be likely false.
- Note that significant results don’t mean your hypothesis is proven, but rather that based on the data you collected, the differences you observed are likely not due to chance.

## Revising Your Hypothesis

- For example, if you began the experiment thinking that Tide would have the most effective stain remover, but you’ve noticed that Tide does a poor job of removing stains from red wine and mud, you likely need to change your working assumptions.

- So, if Tide turns out to be ineffective at removing certain types of stains, your early working hypothesis will have been incorrect.

- A final, tested hypothesis would look like: “Shout is the most effective stain remover for removing a variety of household stains from a variety of common fabrics.”

## Expert Q&A

- Deductive (or “top-down”) reasoning won’t help you much in testing out a scientific hypothesis. Your hypothesis needs to be grounded in the tests you’ve run and the data that you’ve gathered. Thanks Helpful 0 Not Helpful 0
- Depending on the type of hypothesis you’re testing, you may also need a control group. For example, if you’re testing how effective a drug is, you need to have a group on a placebo. Thanks Helpful 0 Not Helpful 0
- Remember that a null hypothesis (when the control and tested variable are the same) is different from an alternative hypothesis (when the control and tested variable are different). Thanks Helpful 0 Not Helpful 0

## You Might Also Like

- ↑ https://grammar.yourdictionary.com/for-students-and-parents/how-create-hypothesis.html
- ↑ https://www.education.com/science-fair/article/best-way-stains-out-of-clothes/
- ↑ https://www.grammarly.com/blog/how-to-write-a-hypothesis/
- ↑ https://www.sciencebuddies.org/science-fair-projects/science-fair/steps-of-the-scientific-method
- ↑ https://online.stat.psu.edu/stat502_fa21/lesson/1/1.1
- ↑ https://pressbooks.pub/scientificinquiryinsocialwork/chapter/6-3-inductive-and-deductive-reasoning/
- ↑ https://online.stat.psu.edu/stat100/lesson/10/10.1
- ↑ https://www.khanacademy.org/science/biology/intro-to-biology/science-of-biology/a/the-science-of-biology

## About this article

If you need to test a hypothesis, first come up with a question you’d like to answer, then develop an experiment to answer your question. As you set up your experiment, create a statement about what you think is happening. This is your hypothesis. As you perform the test, compare your data to your hypothesis. By the end of the experiment, you should be able to conclude whether your hypothesis was true or not. Read on to learn more from our Science co-author about how to set up an experiment! Did this summary help you? Yes No

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## Statistical Analysis: Developing and Testing Hypotheses

Statistical hypothesis testing is sometimes known as confirmatory data analysis. It is a way of drawing inferences from data. In the process, you develop a hypothesis or theory about what you might see in your research. You then test that hypothesis against the data that you collect.

Hypothesis testing is generally used when you want to compare two groups, or compare a group against an idealised position.

## Before You Start: Developing A Research Hypothesis

Before you can do any kind of research in social science fields such as management, you need a research question or hypothesis. Research is generally designed to either answer a research question or consider a research hypothesis . These two are closely linked, and generally one or the other is used, rather than both.

A research question is the question that your research sets out to answer . For example:

Do men and women like ice cream equally?

Do men and women like the same flavours of ice cream?

What are the main problems in the market for ice cream?

How can the market for ice cream be segmented and targeted?

Research hypotheses are statements of what you believe you will find in your research.

These are then tested statistically during the research to see if your belief is correct. Examples include:

Men and women like ice cream to different extents.

Men and women like different flavours of ice cream.

Men are more likely than women to like mint ice cream.

Women are more likely than men to like chocolate ice cream.

Both men and women prefer strawberry to vanilla ice cream.

Relationships vs Differences

Research hypotheses can be expressed in terms of differences between groups, or relationships between variables. However, these are two sides of the same coin: almost any hypothesis could be set out in either way.

For example:

There is a relationship between gender and liking ice cream OR

Men are more likely to like ice cream than women.

## Testing Research Hypotheses

The purpose of statistical hypothesis testing is to use a sample to draw inferences about a population.

Testing research hypotheses requires a number of steps:

Step 1. Define your research hypothesis

The first step in any hypothesis testing is to identify your hypothesis, which you will then go on to test. How you define your hypothesis may affect the type of statistical testing that you do, so it is important to be clear about it. In particular, consider whether you are going to hypothesise simply that there is a relationship or speculate about the direction of the relationship.

Using the examples above:

There is a relationship between gender and liking ice cream is a non-directional hypothesis. You have simply specified that there is a relationship, not whether men or women like ice cream more.

However, men are more likely to like ice cream than women is directional : you have specified which gender is more likely to like ice cream.

Generally, it is better not to specify direction unless you are moderately sure about it.

Step 2. Define the null hypothesis

The null hypothesis is basically a statement of what you are hoping to disprove: the opposite of your ‘guess’ about the relationship. For example, in the hypotheses above, the null hypothesis would be:

Men and women like ice cream equally, or

There is no relationship between gender and ice cream.

This also defines your ‘alternative hypothesis’ which is your ‘test hypothesis’ ( men like ice cream more than women ). Your null hypothesis is generally that there is no difference, because this is the simplest position.

The purpose of hypothesis testing is to disprove the null hypothesis. If you cannot disprove the null hypothesis, you have to assume it is correct.

Step 3. Develop a summary measure that describes your variable of interest for each group you wish to compare

Our page on Simple Statistical Analysis describes several summary measures, including two of the most common, mean and median.

The next step in your hypothesis testing is to develop a summary measure for each of your groups. For example, to test the gender differences in liking for ice cream, you might ask people how much they liked ice cream on a scale of 1 to 5. Alternatively, you might have data about the number of times that ice creams are consumed each week in the summer months.

You then need to produce a summary measure for each group, usually mean and standard deviation. These may be similar for each group, or quite different.

Step 4. Choose a reference distribution and calculate a test statistic

To decide whether there is a genuine difference between the two groups, you have to use a reference distribution against which to measure the values from the two groups.

The most common source of reference distributions is a standard distribution such as the normal distribution or t - distribution. These two are the same, except that the standard deviation of the t -distribution is estimated from the sample, and that of the normal distribution is known. There is more about this in our page on Statistical Distributions .

You then compare the summary data from the two groups by using them to calculate a test statistic. There is a standard formula for every test statistic and reference distribution. The test and reference distribution depend on your data and the purpose of your testing (see below).

The test that you use to compare your groups will depend on how many groups you have, the type of data that you have collected, and how reliable your data are. In general, you would use different tests for comparing two groups than you would for comparing three or more.

Our page Surveys and Survey Design explains that there are two types of answer scale, continuous and categorical. Age, for example, is a continuous scale, although it can also be grouped into categories. You may also find it helpful to read our page on Types of Data .

Gender is a category scale.

For a continuous scale, you can use the mean values of the two groups that you are comparing.

For a category scale, you need to use the median values.

Source: Easterby-Smith, Thorpe and Jackson, Management Research 4th Edition

## One- or Two-Tailed Test

The other thing that you have to decide is whether you use what is known as a ‘one-tailed’ or ‘two-tailed’ test.

This allows you to compare differences between groups in either one or both directions.

In practice, this boils down to whether your research hypothesis is expressed as ‘x is likely to be more than y’, or ‘x is likely to be different from y’. If you are confident of the direction of the distance (that is, you are sure that the only options are that ‘x is likely to be more than y’ or ‘x and y are the same’), then your test will be one-tailed. If not, it will be two-tailed .

If there is any doubt, it is better to use a two-tailed test.

You should only use a one-tailed test when you are certain about the direction of the difference, and it doesn’t matter if you are wrong.

The graph under Step 5 shows a two-tailed test.

If you are not very confident about the quality of the data collected, for example because the inputting was done quickly and cheaply, or because the data have not been checked, then you may prefer to use the median even if the data are continuous to avoid any problems with outliers. This makes the tests more robust, and the results more reliable.

Our page on correlations suggests that you may also want to plot a scattergraph before undertaking any further analysis. This will also help you to identify any outliers or potential problems with the data.

## Calculating the Test Statistic

For each type of test, there is a standard formula for the test statistic. For example, for the t -test, it is:

(M1-M2)/SE(diff)

M1 is the mean of the first group

M2 is the mean of the second group

SE(diff) is the standard error of the difference, which is calculated from the standard deviation and the sample size of each group.

The formula for calculating the standard error of the difference between means is:

- sd 2 = the square of the standard deviation of the source population (i.e., the variance);
- n a = the size of sample A; and
- n b = the size of sample B.

Step 5. Identify Acceptance and Rejection Regions

The final part of the test is to see if your test statistic is significant—in other words, whether you are going to accept or reject your null hypothesis. You need to consider first what level of significance is required. This tells you the probability that you have achieved your result by chance.

Significance (or p-value) is usually required to be either 5% or 1%, meaning that you are 95% or 99% confident that your result was not achieved by chance.

NOTE: the significance level is sometimes expressed as p < 0.05 or p < 0.01.

For more about significance, you may like to read our page on Significance and Confidence Intervals .

The graph below shows a reference distribution (this one could be either the normal or the t- distribution) with the acceptance and rejection regions marked. It also shows the critical values. µ is the mean. For more about this, you may like to read our page on Statistical Distributions .

The critical values are identified from published statistical tables for your reference distribution, which are available for different levels of significance.

If your test statistic falls within either of the two rejection regions (that is, it is greater than the higher critical value, or less than the lower one), you will reject the null hypothesis. You can therefore accept your alternative hypothesis.

Step 6. Draw Conclusions and Inferences

The final step is to draw conclusions.

If your test statistic fell within the rejection region, and you have rejected the null hypothesis, you can therefore conclude that there is a gender difference in liking for ice cream, using the example above.

## Types of Error

There are four possible outcomes from statistical testing (see table):

The groups are different, and you conclude that they are different (correct result)

The groups are different, but you conclude that they are not (Type II error)

The groups are the same, but you conclude that they are different (Type I error)

The groups are the same, and you conclude that they are the same (correct result).

Type I errors are generally considered more important than Type II, because they have the potential to change the status quo.

For example, if you wrongly conclude that a new medical treatment is effective, doctors are likely to move to providing that treatment. Patients may receive the treatment instead of an alternative that could have fewer side effects, and pharmaceutical companies may stop looking for an alternative treatment.

Further Reading from Skills You Need

Data Handling and Algebra Part of The Skills You Need Guide to Numeracy

This eBook covers the basics of data handling, data visualisation, basic statistical analysis and algebra. The book contains plenty of worked examples to improve understanding as well as real-world examples to show you how these concepts are useful.

Whether you want to brush up on your basics, or help your children with their learning, this is the book for you.

There are statistical software packages available that will carry out all these tests for you. However, if you have never studied statistics, and you’re not very confident about what you’re doing, you are probably best off discussing it with a statistician or consulting a detailed statistical textbook.

Poorly executed statistical analysis can invalidate very good research. It is much better to find someone to help you. However, this page will help you to understand your friendly statistician!

Continue to: Significance and Confidence Intervals Statistical Analysis: Types of Data

See also: Understanding Correlations Understanding Statistical Distributions Averages (Mean, Median and Mode)

## Statistics Tutorial

Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.

Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.

## Hypothesis Testing

A hypothesis is a claim about a population parameter .

A hypothesis test is a formal procedure to check if a hypothesis is true or not.

Examples of claims that can be checked:

The average height of people in Denmark is more than 170 cm.

The share of left handed people in Australia is not 10%.

The average income of dentists is less the average income of lawyers.

## The Null and Alternative Hypothesis

Hypothesis testing is based on making two different claims about a population parameter.

The null hypothesis (\(H_{0} \)) and the alternative hypothesis (\(H_{1}\)) are the claims.

The two claims needs to be mutually exclusive , meaning only one of them can be true.

The alternative hypothesis is typically what we are trying to prove.

For example, we want to check the following claim:

"The average height of people in Denmark is more than 170 cm."

In this case, the parameter is the average height of people in Denmark (\(\mu\)).

The null and alternative hypothesis would be:

Null hypothesis : The average height of people in Denmark is 170 cm.

Alternative hypothesis : The average height of people in Denmark is more than 170 cm.

The claims are often expressed with symbols like this:

\(H_{0}\): \(\mu = 170 \: cm \)

\(H_{1}\): \(\mu > 170 \: cm \)

If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

If the data does not support the alternative hypothesis, we keep the null hypothesis.

Note: The alternative hypothesis is also referred to as (\(H_{A} \)).

## The Significance Level

The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.

The significance level is a percentage probability of accidentally making the wrong conclusion.

Typical significance levels are:

- \(\alpha = 0.1\) (10%)
- \(\alpha = 0.05\) (5%)
- \(\alpha = 0.01\) (1%)

A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.

There is no "correct" significance level - it only states the uncertainty of the conclusion.

Note: A 5% significance level means that when we reject a null hypothesis:

We expect to reject a true null hypothesis 5 out of 100 times.

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## The Test Statistic

The test statistic is used to decide the outcome of the hypothesis test.

The test statistic is a standardized value calculated from the sample.

Standardization means converting a statistic to a well known probability distribution .

The type of probability distribution depends on the type of test.

Common examples are:

- Standard Normal Distribution (Z): used for Testing Population Proportions
- Student's T-Distribution (T): used for Testing Population Means

Note: You will learn how to calculate the test statistic for each type of test in the following chapters.

## The Critical Value and P-Value Approach

There are two main approaches used for hypothesis tests:

- The critical value approach compares the test statistic with the critical value of the significance level.
- The p-value approach compares the p-value of the test statistic and with the significance level.

## The Critical Value Approach

The critical value approach checks if the test statistic is in the rejection region .

The rejection region is an area of probability in the tails of the distribution.

The size of the rejection region is decided by the significance level (\(\alpha\)).

The value that separates the rejection region from the rest is called the critical value .

Here is a graphical illustration:

If the test statistic is inside this rejection region, the null hypothesis is rejected .

For example, if the test statistic is 2.3 and the critical value is 2 for a significance level (\(\alpha = 0.05\)):

We reject the null hypothesis (\(H_{0} \)) at 0.05 significance level (\(\alpha\))

## The P-Value Approach

The p-value approach checks if the p-value of the test statistic is smaller than the significance level (\(\alpha\)).

The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.

If the p-value is smaller than the significance level, the null hypothesis is rejected .

The p-value directly tells us the lowest significance level where we can reject the null hypothesis.

For example, if the p-value is 0.03:

We reject the null hypothesis (\(H_{0} \)) at a 0.05 significance level (\(\alpha\))

We keep the null hypothesis (\(H_{0}\)) at a 0.01 significance level (\(\alpha\))

Note: The two approaches are only different in how they present the conclusion.

## Steps for a Hypothesis Test

The following steps are used for a hypothesis test:

- Check the conditions
- Define the claims
- Decide the significance level
- Calculate the test statistic

One condition is that the sample is randomly selected from the population.

The other conditions depends on what type of parameter you are testing the hypothesis for.

Common parameters to test hypotheses are:

- Proportions (for qualitative data)
- Mean values (for numerical data)

You will learn the steps for both types in the following pages.

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## How to Write a Great Hypothesis

Hypothesis Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk, "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

- The Scientific Method

## Hypothesis Format

Falsifiability of a hypothesis, operational definitions, types of hypotheses, hypotheses examples.

- Collecting Data

## Frequently Asked Questions

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study.

One hypothesis example would be a study designed to look at the relationship between sleep deprivation and test performance might have a hypothesis that states: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

## The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

- Forming a question
- Performing background research
- Creating a hypothesis
- Designing an experiment
- Collecting data
- Analyzing the results
- Drawing conclusions
- Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. It is only at this point that researchers begin to develop a testable hypothesis. Unless you are creating an exploratory study, your hypothesis should always explain what you expect to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore a number of factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment do not support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk wisdom that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

## Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

- Is your hypothesis based on your research on a topic?
- Can your hypothesis be tested?
- Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the journal articles you read . Many authors will suggest questions that still need to be explored.

To form a hypothesis, you should take these steps:

- Collect as many observations about a topic or problem as you can.
- Evaluate these observations and look for possible causes of the problem.
- Create a list of possible explanations that you might want to explore.
- After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method , falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that if something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in a number of different ways. One of the basic principles of any type of scientific research is that the results must be replicable. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. How would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

In order to measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming other people. In this situation, the researcher might utilize a simulated task to measure aggressiveness.

## Hypothesis Checklist

- Does your hypothesis focus on something that you can actually test?
- Does your hypothesis include both an independent and dependent variable?
- Can you manipulate the variables?
- Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

- Simple hypothesis : This type of hypothesis suggests that there is a relationship between one independent variable and one dependent variable.
- Complex hypothesis : This type of hypothesis suggests a relationship between three or more variables, such as two independent variables and a dependent variable.
- Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
- Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
- Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative sample of the population and then generalizes the findings to the larger group.
- Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the dependent variable if you change the independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

## A few examples of simple hypotheses:

- "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
- Complex hypothesis: "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."
- "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."

## Examples of a complex hypothesis include:

- "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
- "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

## Examples of a null hypothesis include:

- "Children who receive a new reading intervention will have scores different than students who do not receive the intervention."
- "There will be no difference in scores on a memory recall task between children and adults."

## Examples of an alternative hypothesis:

- "Children who receive a new reading intervention will perform better than students who did not receive the intervention."
- "Adults will perform better on a memory task than children."

## Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

## Descriptive Research Methods

Descriptive research such as case studies , naturalistic observations , and surveys are often used when it would be impossible or difficult to conduct an experiment . These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a correlational study can then be used to look at how the variables are related. This type of research method might be used to investigate a hypothesis that is difficult to test experimentally.

## Experimental Research Methods

Experimental methods are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually cause another to change.

## A Word From Verywell

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Some examples of how to write a hypothesis include:

- "Staying up late will lead to worse test performance the next day."
- "People who consume one apple each day will visit the doctor fewer times each year."
- "Breaking study sessions up into three 20-minute sessions will lead to better test results than a single 60-minute study session."

The four parts of a hypothesis are:

- The research question
- The independent variable (IV)
- The dependent variable (DV)
- The proposed relationship between the IV and DV

Castillo M. The scientific method: a need for something better? . AJNR Am J Neuroradiol. 2013;34(9):1669-71. doi:10.3174/ajnr.A3401

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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Learn how to test your research hypothesis using a formal procedure for investigating ideas about the world using statistics. Follow the 5 main steps of hypothesis testing: state your null and alternate hypothesis, collect data, perform a statistical test, decide whether to reject or fail to reject your null hypothesis, and present your results and discussion. See examples of how to apply hypothesis testing in different fields and contexts.

Test Statistic: z = x¯¯¯ −μo σ/ n−−√ since it is calculated as part of the testing of the hypothesis. Definition 7.1.4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.

Need help with a homework problem? Check out our tutoring page! What is a Hypothesis? Andreas Cellarius hypothesis, showing the planetary motions. A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example: A new medicine you think might work.

Explain the facts in detail What is a Null Hypothesis and Alternative Hypothesis? A null hypothesis is a hypothesis when there is no significant relationship between the dependent and the participants' independent variables . In simple words, it's a hypothesis that has been put forth but hasn't been proved as yet.

The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data). Based on the available evidence (data), deciding whether to reject or not reject the initial assumption. Every hypothesis test — regardless of the population parameter involved — requires the above three steps. Example S.3.1

Test statistic = (Statistic - Parameter) / (Standard error of statistic) where Parameter is the value appearing in the null hypothesis, and Statistic is the point estimate of Parameter. As part of the analysis, you may need to compute the standard deviation or standard error of the statistic.

A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level. Common choices are .01, .05, and .1. 3.

Basic approach to hypothesis testing. State a model describing the relationship between the explanatory variables and the outcome variable (s) in the population and the nature of the variability. State all of your assumptions. Specify the null and alternative hypotheses in terms of the parameters of the model.

However, in order to use hypothesis testing, you need to re-state your research hypothesis as a null and alternative hypothesis. Before you can do this, it is best to consider the process/structure involved in hypothesis testing and what you are measuring. This structure is presented on the next page. Understand the structure of hypothesis ...

Step 3: Perform sampling and collection of data for statistical testing. It is important to perform sampling and collect data in way that assists the formulated research hypothesis. You will have to perform a statistical testing to validate your data and make statistical inferences about the population of your interest.

3. One-Sided vs. Two-Sided Testing. When it's time to test your hypothesis, it's important to leverage the correct testing method. The two most common hypothesis testing methods are one-sided and two-sided tests, or one-tailed and two-tailed tests, respectively. Typically, you'd leverage a one-sided test when you have a strong conviction ...

All hypotheses are tested using a four-step process: The first step is for the analyst to state the hypotheses. The second step is to formulate an analysis plan, which outlines how the data will be...

Step 1: Ask a question Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project. Example: Research question Do students who attend more lectures get better exam results? Step 2: Do some preliminary research

Start Unit test. Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The most common way to test a hypothesis is to create an experiment. A good experiment uses test subjects or creates conditions where you can see if your hypothesis seems to be true by evaluating a broad range of data (test results).

Testing Research Hypotheses The purpose of statistical hypothesis testing is to use a sample to draw inferences about a population. Testing research hypotheses requires a number of steps: Step 1. Define your research hypothesis

A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger ...

One-tailed hypothesis tests are also known as directional and one-sided tests because you can test for effects in only one direction. When you perform a one-tailed test, the entire significance level percentage goes into the extreme end of one tail of the distribution. In the examples below, I use an alpha of 5%.

From there, you can begin experiments to test your hypothesis; in this example, you might set aside two plants, water one but not the other, and then record the results to see the differences. The language of hypotheses always discusses variables, or the elements that you're testing.

Hypothesis testing is based on making two different claims about a population parameter. The null hypothesis ( H 0) and the alternative hypothesis ( H 1) are the claims. The two claims needs to be mutually exclusive, meaning only one of them can be true. The alternative hypothesis is typically what we are trying to prove.

Simple hypothesis: This type of hypothesis suggests that there is a relationship between one independent variable and one dependent variable.; Complex hypothesis: This type of hypothesis suggests a relationship between three or more variables, such as two independent variables and a dependent variable.; Null hypothesis: This hypothesis suggests no relationship exists between two or more variables.

For instance, bootstrap hypothesis testing involves constructing a confidence interval for the test statistic and determining if the null value (the value under the null hypothesis) is within the ...

Math 243 Hwk Sheet on Ch.5 Hypothesis Testing Name: Maximum Points: 5 pts Show all your work and interpret the results for full credit. Please make sure to round the decimals to 4 places. Percentages to 2 decimal places. Explain the answer in the context of the given question meaning write a complete sentence after you find the answer.