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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.
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.
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 (\(\alpha\)) 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:
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 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:
Note: You will learn how to calculate the test statistic for each type of test in the following chapters.
There are two main approaches used for hypothesis tests:
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 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.
The following steps are used for a hypothesis test:
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:
You will learn the steps for both types in the following pages.
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Statistics By Jim
Making statistics intuitive
By Jim Frost 10 Comments
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 differences between your sample data and the null hypothesis.
When your test statistic indicates a sufficiently large incompatibility with the null hypothesis, you can reject the null and state that your results are statistically significant—your data support the notion that the sample effect exists in the population . To use a test statistic to evaluate statistical significance, you either compare it to a critical value or use it to calculate the p-value .
Statisticians named the hypothesis tests after the test statistics because they’re the quantity that the tests actually evaluate. For example, t-tests assess t-values, F-tests evaluate F-values, and chi-square tests use, you guessed it, chi-square values.
In this post, learn about test statistics, how to calculate them, interpret them, and evaluate statistical significance using the critical value and p-value methods.
Each test statistic has its own formula. I present several common test statistics examples below. To see worked examples for each one, click the links to my more detailed articles.
T-value for 1-sample t-test | Take the sample mean, subtract the hypothesized mean, and divide by the . | |
T-value for 2-sample t-test | Take one sample mean, subtract the other, and divide by the pooled standard deviation. | |
F-value for F-tests and ANOVA | Calculate the ratio of two . | |
Chi-squared value (χ ) for a Chi-squared test | Sum the squared differences between observed and expected values divided by the expected values. |
In the formulas above, it’s helpful to understand the null condition and the test statistic value that occurs when your sample data match that condition exactly. Also, it’s worthwhile knowing what causes the test statistics to move further away from the null value, potentially becoming significant. Test statistics are statistically significant when they exceed a critical value.
All these test statistics are ratios, which helps you understand their null values.
When a t-value equals 0, it indicates that your sample data match the null hypothesis exactly.
For a 1-sample t-test, when the sample mean equals the hypothesized mean, the numerator is zero, which causes the entire t-value ratio to equal zero. As the sample mean moves away from the hypothesized mean in either the positive or negative direction, the test statistic moves away from zero in the same direction.
A similar case exists for 2-sample t-tests. When the two sample means are equal, the numerator is zero, and the entire test statistic ratio is zero. As the two sample means become increasingly different, the absolute value of the numerator increases, and the t-value becomes more positive or negative.
Related post : How T-tests Work
When an F-value equals 1, it indicates that the two variances in the numerator and denominator are equal, matching the null hypothesis.
As the numerator and denominator become less and less similar, the F-value moves away from one in either direction.
Related post : The F-test in ANOVA
When a chi-squared value equals 0, it indicates that the observed values always match the expected values. This condition causes the numerator to equal zero, making the chi-squared value equal zero.
As the observed values progressively fail to match the expected values, the numerator increases, causing the test statistic to rise from zero.
Related post : How a Chi-Squared Test Works
You’ll never see a test statistic that equals the null value precisely in practice. However, trivial differences been sample values and the null value are not uncommon.
Test statistics are unitless. This fact can make them difficult to interpret on their own. You know they evaluate how well your data agree with the null hypothesis. If your test statistic is extreme enough, your data are so incompatible with the null hypothesis that you can reject it and conclude that your results are statistically significant. But how does that translate to specific values of your test statistic? Where do you draw the line?
For instance, t-values of zero match the null value. But how far from zero should your t-value be to be statistically significant? Is 1 enough? 2? 3? If your t-value is 2, what does it mean anyway? In this case, we know that the sample mean doesn’t equal the null value, but how exceptional is it? To complicate matters, the dividing line changes depending on your sample size and other study design issues.
Similar types of questions apply to the other test statistics too.
To interpret individual values of a test statistic, we need to place them in a larger context. Towards this end, let me introduce you to sampling distributions for test statistics!
Performing a hypothesis test on a sample produces a single test statistic. Now, imagine you carry out the following process:
This process produces the distribution of test statistic values that occurs when the effect does not exist in the population (i.e., the null hypothesis is true). Statisticians refer to this type of distribution as a sampling distribution, a kind of probability distribution.
Why would we need this type of distribution?
It provides the larger context required for interpreting a test statistic. More specifically, it allows us to compare our study’s single test statistic to values likely to occur when the null is true. We can quantify our sample statistic’s rareness while assuming the effect does not exist in the population. Now that’s helpful!
Fortunately, we don’t need to collect many random samples to create this distribution! Statisticians have developed formulas allowing us to estimate sampling distributions for test statistics using the sample data.
To evaluate your data’s compatibility with the null hypothesis, place your study’s test statistic in the distribution.
Related post : Understanding Probability Distributions
Suppose our t-test produces a t-value of two. That’s our test statistic. Let’s see where it fits in.
The sampling distribution below shows a t-distribution with 20 degrees of freedom, equating to a 1-sample t-test with a sample size of 21. The distribution centers on zero because it assumes the null hypothesis is correct. When the null is true, your analysis is most likely to obtain a t-value near zero and less likely to produce t-values further from zero in either direction.
The sampling distribution indicates that our test statistic is somewhat rare when we assume the null hypothesis is correct. However, the chances of observing t-values from -2 to +2 are not totally inconceivable. We need a way to quantify the likelihood.
From this point, we need to use the sampling distributions’ ability to calculate probabilities for test statistics.
Related post : Sampling Distributions Explained
The significance level uses critical values to define how far the test statistic must be from the null value to reject the null hypothesis. When the test statistic exceeds a critical value, the results are statistically significant.
The percentage of the area beneath the sampling distribution curve that is shaded represents the probability that the test statistic will fall in those regions when the null is true. Consequently, to depict a significance level of 0.05, I’ll shade 5% of the sampling distribution furthest away from the null value.
The two shaded areas are equidistant from the null value in the center. Each region has a likelihood of 0.025, which sums to our significance level of 0.05. These shaded areas are the critical regions for a two-tailed hypothesis test. Let’s return to our example t-value of 2.
Related post : What are Critical Values?
In this example, the critical values are -2.086 and +2.086. Our test statistic of 2 is not statistically significant because it does not exceed the critical value.
Other hypothesis tests have their own test statistics and sampling distributions, but their processes for critical values are generally similar.
Learn how to find critical values for test statistics using tables:
Related post : Understanding Significance Levels
P-values are the probability of observing an effect at least as extreme as your sample’s effect if you assume no effect exists in the population.
Test statistics represent effect sizes in hypothesis tests because they denote the difference between your sample effect and no effect —the null hypothesis. Consequently, you use the test statistic to calculate the p-value for your hypothesis test.
The above p-value definition is a bit tortuous. Fortunately, it’s much easier to understand how test statistics and p-values work together using a sampling distribution graph.
Let’s use our hypothetical test statistic t-value of 2 for this example. However, because I’m displaying the results of a two-tailed test, I need to use t-values of +2 and -2 to cover both tails.
Related post : One-tailed vs. Two-Tailed Hypothesis Tests
The graph below displays the probability of t-values less than -2 and greater than +2 using the area under the curve. This graph is specific to our t-test design (1-sample t-test with N = 21).
The sampling distribution indicates that each of the two shaded regions has a probability of 0.02963—for a total of 0.05926. That’s the p-value! The graph shows that the test statistic falls within these areas almost 6% of the time when the null hypothesis is true in the population.
While this likelihood seems small, it’s not low enough to justify rejecting the null under the standard significance level of 0.05. P-value results are always consistent with the critical value method. Learn more about using test statistics to find p values .
While test statistics are a crucial part of hypothesis testing, you’ll probably let your statistical software calculate the p-value for the test. However, understanding test statistics will boost your comprehension of what a hypothesis test actually assesses.
Related post : Interpreting P-values
July 5, 2024 at 8:21 am
“As the observed values progressively fail to match the observed values, the numerator increases, causing the test statistic to rise from zero”.
Sir, this sentence is written in the Chi-squared Test heading. There the observed value is written twice. I think the second one to be replaced with ‘expected values’.
July 5, 2024 at 4:10 pm
Thanks so much, Dr. Raj. You’re correct about the typo and I’ve made the correction.
May 9, 2024 at 1:40 am
Thank you very much (great page on one and two-tailed tests)!
May 6, 2024 at 12:17 pm
I would like to ask a question. If only positive numbers are the possible values in a sample (e.g. absolute values without 0), is it meaningful to test if the sample is significantly different from zero (using for example a one sample t-test or a Wilcoxon signed-rank test) or can I assume that if given a large enough sample, the result will by definition be significant (even if a small or very variable sample results in a non-significant hypothesis test).
Thank you very much,
May 6, 2024 at 4:35 pm
If you’re talking about the raw values you’re assessing using a one-sample t-test, it doesn’t make sense to compare them to zero given your description of the data. You know that the mean can’t possibly equal zero. The mean must be some positive value. Yes, in this scenario, if you have a large enough sample size, you should get statistically significant results. So, that t-test isn’t tell you anything that you don’t already know!
However, you should be aware of several things. The 1-sample test can compare your sample mean to values other than zero. Typically, you’ll need to specify the value of the null hypothesis for your software. This value is the comparison value. The test determines whether your sample data provide enough evidence to conclude that the population mean does not equal the null hypothesis value you specify. You’ll need to specify the value because there is no obvious default value to use. Every 1-sample t-test has its subject-area context with a value that makes sense for its null hypothesis value and it is frequently not zero.
I suspect that you’re getting tripped up with the fact that t-tests use a t-value of zero for its null hypothesis value. That doesn’t mean your 1-sample t-test is comparing your sample mean to zero. The test converts your data to a single t-value and compares the t-value to zero. But your actual null hypothesis value can be something else. It’s just converting your sample to a standardized value to use for testing. So, while the t-test compares your sample’s t-value to zero, you can actually compare your sample mean to any value you specify. You need to use a value that makes sense for your subject area.
I hope that makes sense!
May 8, 2024 at 8:37 am
Thank you very much Jim, this helps a lot! Actually, the value I would like to compare my sample to is zero, but I just couldn’t find the right way to test it apparently (it’s about EEG data). The original data was a sample of numbers between -1 and +1, with the question if they are significantly different from zero in either direction (in which case a one sample t-test makes sense I guess, since the sample mean can in fact be zero). However, since a sample mean of 0 can also occur if half of the sample differs in the negative, and the other half in the positive direction, I also wanted to test if there is a divergence from 0 in ‘absolute’ terms – that’s how the absolute valued numbers came about (I know that absolute values can also be zero, but in this specific case, they were all positive numbers) And a special thanks for the last paragraph – I will definitely keep in mind, it is a potential point of confusion.
May 8, 2024 at 8:33 pm
You can use a 1-sample t test for both cases but you’ll need to set them up slightly different. To detect a positive or negative difference from zero, use a 2-tailed test. For the case with absolute values, use a one-tailed test with a critical region in the positive end. To learn more, read about One- and Two-Tailed Tests Explained . Use zero for the comparison value in both cases.
February 12, 2024 at 1:00 am
Very helpful and well articulated! Thanks Jim 🙂
September 18, 2023 at 10:01 am
Thank you for brief explanation.
July 25, 2022 at 8:32 am
the content was helpful to me. thank you
The bottom line.
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 a data-generating process. The word "population" will be used for both of these cases in the following descriptions.
In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine 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. 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.
If an individual 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 is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is 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 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 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 statisticians 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.”
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.
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.
Hypothesis testing refers to a statistical process that helps researchers 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. 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.
Sage. " Introduction to Hypothesis Testing ," Page 4.
Elder Research. " Who Invented the Null Hypothesis? "
Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."
Five step hypothesis testing procedure.
In the remaining lessons, we will use the following five step hypothesis testing procedure. This is slightly different from the five step procedure that we used when conducting randomization tests.
When using a standard normal distribution (i.e., z distribution), the test statistic is the standardized value that is the boundary of the p-value. Recall the formula for a z score: \(z=\frac{x-\overline x}{s}\). The formula for a test statistic will be similar. When conducting a hypothesis test the sampling distribution will be centered on the null parameter and the standard deviation is known as the standard error.
This formula puts our observed sample statistic on a standard scale (e.g., z distribution). A z score tells us where a score lies on a normal distribution in standard deviation units. The test statistic tells us where our sample statistic falls on the sampling distribution in standard error units.
Research question: Is the mean body temperature in the population different from 98.6° Fahrenheit?
Research question: Is there a positive correlation in the population between the price of an ink jet printer and how many pages per minute (ppm) it prints?
Research question: Is the proportion of NFL overtime coin tosses that are won different from 0.50?
StatKey was used to construct a randomization distribution:
From the given StatKey output, the randomization distribution is approximately normal.
\(H_0\colon p=0.50\)
\(H_a\colon p \ne 0.50\)
\(test\;statistic=\dfrac{sample\;statistic-null\;parameter}{standard\;error}\)
The sample statistic is the proportion in the original sample, 0.561. The null parameter is 0.50. And, the standard error is 0.024.
\(test\;statistic=\dfrac{0.561-0.50}{0.024}=\dfrac{0.061}{0.024}=2.542\)
The p value will be the area on the z distribution that is more extreme than the test statistic of 2.542, in the direction of the alternative hypothesis. This is a two-tailed test:
The p value is the area in the left and right tails combined: \(p=0.0055110+0.0055110=0.011022\)
The p value (0.011022) is less than the standard 0.05 alpha level, therefore we reject the null hypothesis.
There is evidence that the proportion of all NFL overtime coin tosses that are won is different from 0.50
Research question : Are more than 50% of all World Campus STAT 200 students women?
Data were collected from a representative sample of 501 World Campus STAT 200 students. In that sample, 284 students were women and 217 were not women.
StatKey was used to construct a sampling distribution using randomization methods:
Because this randomization distribution is approximately normal, we can find the p value by computing a standardized test statistic and using the z distribution.
The assumption here is that the sampling distribution is approximately normal. From the given StatKey output, the randomization distribution is approximately normal.
\(H_0\colon p=0.50\) \(H_a\colon p>0.50\)
\(test\;statistic=\dfrac{sample\;statistic-hypothesized\;parameter}{standard\;error}\)
The sample statistic is \(\widehat p = 284/501 = 0.567\).
The hypothesized parameter is the value from the hypotheses: \(p_0=0.50\).
The standard error on the randomization distribution above is 0.022.
\(test\;statistic=\dfrac{0.567-0.50}{0.022}=3.045\)
We can find the p value by constructing a standard normal distribution and finding the area under the curve that is more extreme than our observed test statistic of 3.045, in the direction of the alternative hypothesis. In other words, \(P(z>3.045)\):
Our p value is 0.0011634
Our p value is less than or equal to the standard 0.05 alpha level, therefore we reject the null hypothesis.
There is evidence that the proportion of all World Campus STAT 200 students who are women is greater than 0.50.
Research question: Is the mean quiz score different from 14 in the population?
\(H_0\colon \mu = 14\)
\(H_a\colon \mu \ne 14\)
The sample statistic is the mean in the original sample, 13.746 points. The null parameter is 14 points. And, the standard error, 0.142, can be found on the StatKey output.
\(test\;statistic=\dfrac{13.746-14}{0.142}=\dfrac{-0.254}{0.142}=-1.789\)
The p value will be the area on the z distribution that is more extreme than the test statistic of -1.789, in the direction of the alternative hypothesis:
This was a two-tailed test. The p value is the area in the left and right tails combined: \(p=0.0368074+0.0368074=0.0736148\)
The p value (0.0736148) is greater than the standard 0.05 alpha level, therefore we fail to reject the null hypothesis.
There is not enough evidence to state that the mean quiz score in the population is different from 14 points.
Research question: Do the mean commute times in Atlanta and St. Louis differ in the population?
From the given StatKey output, the randomization distribution is approximately normal.
\(H_0: \mu_1-\mu_2=0\)
\(H_a: \mu_1 - \mu_2 \ne 0\)
\(test\;statistic=\dfrac{sample\;statistic - null \; parameter}{standard \;error}\)
The observed sample statistic is \(\overline x _1 - \overline x _2 = 7.14\). The null parameter is 0. And, the standard error, from the StatKey output, is 1.136.
\(test\;statistic=\dfrac{7.14-0}{1.136}=6.285\)
The p value will be the area on the z distribution that is more extreme than the test statistic of 6.285, in the direction of the alternative hypothesis:
This was a two-tailed test. The area in the two tailed combined is 0.000000. Theoretically, the p value cannot be 0 because there is always some chance that a Type I error was committed. This p value would be written as p < 0.001.
The p value is smaller than the standard 0.05 alpha level, therefore we reject the null hypothesis.
There is evidence that the mean commute times in Atlanta and St. Louis are different in the population.
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Hypothesis testing is the act of statistically evaluating a belief or theory. Hypothesis testing is the process of testing your theory using data from the real world obtained either through observation or experiments. Hypothesis testing is the step-by-step process of analyzing empirical data to check if it differs from the expected numbers if the belief or theory you started with was true.
This article walks you through the hypothesis testing concept and lists the process of hypothesis testing step by step.
To illustrate the concept and show you the hypothesis testing process with a example, we evaluate a belief that the companies in the Russell 3000 grow at a rate greater than 10% per year.
Here is a list of subtopics if you want to jump ahead:
Structuring the hypothesis test: the null and alternate hypothesis, the null hypothesis.
Computing the test statistic.
A hypothesis test and a criminal trial: similarities, the sampling distribution, reject region in hypothesis testing, some facts on the null hypothesis, some facts on the alternate hypothesis.
If you already know the concept of hypothesis testing concept and you only need to follow the step-by-step process outlined below.
List of Topics
A hypothesis test starts with a hypothesis that you want to test. It is designed as a statement or belief that you are examining. This statement or belief is termed the null hypothesis. The null hypothesis is what the hypothesis test is evaluating.
The opposite of the null hypothesis is called an alternate hypothesis. We are not examining the alternate hypothesis. Instead, the alternate hypothesis is what remains if the null hypothesis is rejected after being examined.
We will talk more about designing the null and alternate hypotheses later. Remember that we place what we want to prove in the alternate hypothesis. And we put the opposite of what we want to prove in the null hypothesis.
To continue our example, we will place what we believe to be true (mean growth rate is great than 10%) in the alternate hypothesis. And the opposite of the alternate hypothesis (mean growth rate is less than or equal to 10%) in the null hypothesis. Accordingly, we will have the following null and alternate hypotheses for our example: Ho: Mean growth rate <= 10% Ha: Mean growth rate > 10%
If we reject the null hypothesis, we will be concluding that the alternate hypothesis stands. On the other hand, if the evidence does not provide evidence to reject the null hypothesis, we can only conclude that we cannot reject the null hypothesis. In other words, we have not proven the alternate hypothesis. We conclude that we cannot reject the null hypothesis and therefore make no claim to have proven the alternate hypothesis or our starting theory or belief!
In hypothesis testing, the evidence required is gathered from a sample of the relevant population. Then, the parameter of interest from the sample is computed and referred to as the test statistic. This test statistic informs us about the null hypothesis.
Even if the null hypothesis is true, the test statistic is unlikely to be exactly equal to the parameter of interest of the true population because we are basing our test statistic on a sample of the population! A sample is only an unbiased estimator and not the actual population parameter. However, if the null hypothesis is true, the test statistic is likely to be close to the null hypothesis value, and likely agree with the null hypothesis. How close should it be? Or how far away from the null hypothesis value should the test statistic be before we can conclude that the null hypothesis is not true and “can be rejected”?
This is where the significance level comes into play. The significance level is the level of certainty required to reject the null hypothesis. The most commonly used significance levels are 1%, 5%, or 10% in practice. The significance level should be determined by the type of errors we are willing to tolerate (type 1 or type 2 errors).
We will use a 5% level of significance in our example today.
Significance level helps us determine the point beyond which we say that the null hypothesis is not true and “can be rejected”!
Best practice dictates that the critical value must be set up at the design stage and before the hypothesis test is done. The critical value is based on two factors. 1) the sampling distribution and 2) significance levels.
The sampling distribution is a distribution of sample values we can expect if the null hypothesis were true. Theoretically, the sample distribution is the distribution we would get if we took all possible samples that covered the entire population. The reason the sample distribution is central to hypothesis testing is that the mean of the sample distribution will equal the mean of the true population. So we use the sample distribution to evaluate the sample test statistic and check if our data agree with the null hypothesis.
If our null hypothesis is true, the test statistic will lie close to the middle of the sampling distribution. However, if our null hypothesis is NOT true, the test statistic will likely be closer to the tails of the sampling distribution.
To make a firm decision, we need a point beyond which we say that the null hypothesis is not true. That point is referred to as the critical value. The region beyond the critical value is referred to as the critical region or the reject region. If the test statistic falls in this region, we reject the null hypothesis. We conclude that the alternate hypothesis is true.
In our example, we are looking for a 5% confidence level. Therefore the critical value and reject region will be computed using a 5% confidence level. The critical value and reject region can be computed using the Z table, Microsoft Excel or another software program.
In Microsoft Excel we use the =NORM.S.INV(0.95) for a single tail critical value of 1.645 as the z value. We can use the Z table to arrive at the same value too.
Once we have the critical value, we run the experiment or gather sample data. Then, we analyze the sample data and compute the sample parameter of interest.
In our example, we randomly sample __ companies of the Russell 3000. We compute the average growth rates of the sample. We then compute the test statistic using this formula.
We compare the sample parameter of interest with the critical value/critical region. We are essentially checking if the test statistic falls in the reject region.
We are ready to conclude the hypothesis test only when we have the sample parameter of interest and the critical value at hand. We check if the parameter of interest falls in the critical regions identified in the earlier step.
In our example, we can see that the test statistic falls in the reject region.
If the parameter of interest falls in the critical regions, we reject the null hypothesis. Only when we reject the null hypothesis can we conclude that we believe the alternate hypothesis!
In our example, we can conclude that we reject the null hypothesis as the test statistic falls in the reject region. Because we reject the null hypothesis, we can say we believe the alternate hypothesis is true. And we conclude that the growth rate of companies of the Russell 3000 is greater than 10% per year!
A hypothesis test is often compared to and explained as a criminal trial. In a criminal trial, we start with the belief “innocent until proven guilty.” Similarly, in hypothesis testing, we assume that the null hypothesis is true. Therefore, we need to present data to disprove the null hypothesis. That is why we say that hypothesis testing is a trial of the null hypothesis. It is not the alternate hypothesis we are testing! The null hypothesis is similar to the criminal defendant. The data scientist is similar to the prosecutor. It is the prosecutor’s job to prove that the criminal is guilty. The prosecutor or the data scientist/researcher examines the data to present evidence that the null hypothesis is not true. Only if the researcher presents data to prove the null hypothesis is not true, can we conclude that that alternate hypothesis is true. If we do not have evidence to prove the criminal is guilty, he escapes conviction. It does not mean he is truly innocent. It only means that he was not found guilty. Similarly, if we do not have evidence to reject the null hypothesis we can only conclude that we cannot reject the null hypothesis.
We are looking for evidence that the null hypothesis is not true and “can be rejected”. This evidence is provided by a sample. How should this sample be gathered? How large should the sample be to provide this evidence? The sample must be carefully selected to be representative of the true population of interest. A random sample is best. Other sampling methods include cluster sampling, cluster sampling, stratified sampling, convenience sampling, etc. Each has its advantages and disadvantages, which we will not go into here.
Selecting the sample size is important in hypothesis testing. The sample size chosen impacts the risk of Type I and Type 2 errors. The sample size also directly determines the confidence levels and the power of the test. The sample size formula can be resorted to arrive at the sample size.
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Published on July 17, 2020 by Rebecca Bevans . Revised on June 22, 2023.
The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test.
The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.
What exactly is a test statistic, types of test statistics, interpreting test statistics, reporting test statistics, other interesting articles, frequently asked questions about test statistics.
A test statistic describes how closely the distribution of your data matches the distribution predicted under the null hypothesis of the statistical test you are using.
The distribution of data is how often each observation occurs, and can be described by its central tendency and variation around that central tendency. Different statistical tests predict different types of distributions, so it’s important to choose the right statistical test for your hypothesis.
The test statistic summarizes your observed data into a single number using the central tendency, variation, sample size, and number of predictor variables in your statistical model.
Generally, the test statistic is calculated as the pattern in your data (i.e., the correlation between variables or difference between groups) divided by the variance in the data (i.e., the standard deviation ).
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Below is a summary of the most common test statistics, their hypotheses, and the types of statistical tests that use them.
Different statistical tests will have slightly different ways of calculating these test statistics, but the underlying hypotheses and interpretations of the test statistic stay the same.
Test statistic | Null and alternative hypotheses | Statistical tests that use it |
---|---|---|
value | The means of two groups are equal The means of two groups are not equal | test |
value | The means of two groups are equal The means of two groups are not equal | test |
value | The variation among two or more groups is greater than or equal to the variation between the groups The variation among two or more groups is smaller than the variation between the groups | |
-value | Two samples are independent Two samples are not independent (i.e., they are correlated) | correlation tests |
In practice, you will almost always calculate your test statistic using a statistical program (R, SPSS, Excel, etc.), which will also calculate the p value of the test statistic. However, formulas to calculate these statistics by hand can be found online.
The t value of the regression test is 2.36 – this is your test statistic.
For any combination of sample sizes and number of predictor variables, a statistical test will produce a predicted distribution for the test statistic. This shows the most likely range of values that will occur if your data follows the null hypothesis of the statistical test.
The more extreme your test statistic – the further to the edge of the range of predicted test values it is – the less likely it is that your data could have been generated under the null hypothesis of that statistical test.
The agreement between your calculated test statistic and the predicted values is described by the p value . The smaller the p value, the less likely your test statistic is to have occurred under the null hypothesis of the statistical test.
Because the test statistic is generated from your observed data, this ultimately means that the smaller the p value, the less likely it is that your data could have occurred if the null hypothesis was true.
Test statistics can be reported in the results section of your research paper along with the sample size, p value of the test, and any characteristics of your data that will help to put these results into context.
Whether or not you need to report the test statistic depends on the type of test you are reporting.
Which statistics to report | |
---|---|
Correlation and regression tests | or regression coefficient for each predictor variable value for each predictor |
Tests of difference between groups | value for the test statistic |
By surveying a random subset of 100 trees over 25 years we found a statistically significant ( p < 0.01) positive correlation between temperature and flowering dates ( R 2 = 0.36, SD = 0.057).
In our comparison of mouse diet A and mouse diet B, we found that the lifespan on diet A ( M = 2.1 years; SD = 0.12) was significantly shorter than the lifespan on diet B ( M = 2.6 years; SD = 0.1), with an average difference of 6 months ( t (80) = -12.75; p < 0.01).
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.
Methodology
Research bias
A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.
The formula for the test statistic depends on the statistical test being used.
Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).
The test statistic you use will be determined by the statistical test.
You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.
The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.
For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.
Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .
When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
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Table of contents.
Hypothesis testing is one of the major methods in the grand scheme of statistics. It assists in deriving data-driven decisions, evaluating theories, and inferring meaningful conclusions from data. But what is hypothesis testing, and why is it so important in statistics? For those who want to learn and enhance their skills, the IIM Nagpur PG Certificate Programme in Data Science for Business Excellence and Innovation offers the perfect platform for learning from professionals, gaining hands-on experience, and moving up the value chain.
Hypothesis testing is a means to make decisions about population properties based on the information derived from a sample. It simply provides a basis for the testing of assumptions or hypotheses made about a certain parameter in a population. Statement of the whole process involves a null hypothesis denoted by H0 and an alternative hypothesis denoted by H1, followed by the collection of data and subsequent analysis to see whether it favours the rejection of the null hypothesis in favour of the alternative.
In other words, hypothesis testing gives the answers to such questions as:
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The hypothesis testing process is quite logical and it contains several critical steps:
Within hypothesis testing, there are varieties that best fit different sorts of data and research questions. The main kinds include:
Let us see how it works in practice with some examples.
A company would like to know whether the new layout of a website, version B, provides more user engagement than the old layout, version A. The null hypothesis is that both versions have no difference in engagement. The alternative hypothesis is that the new layout leads to higher engagement. Performing an A/B test and examining the data will enable the company to either accept the new layout or not.
A new medication in healthcare is tested in a clinical trial. Researchers want to establish if it can reduce blood pressure better than a previously existing one. The null hypothesis, H0, assumes no difference in blood pressure reduction between the new and the existing medications. The alternative hypothesis, H1, is that the new medication is more potent. The researchers can make an informed inference after administering different groups with the medications, measuring changes in blood pressure, and comparing such changes using a t-test.
A manufacturer wants to be certain that the true mean diameter for bolts manufactured by a machine is 5 mm. The null hypothesis is that the mean diameter is 5 mm. The alternative hypothesis is that the mean diameter is not 5 mm. The manufacturer conducts a hypothesis test based on a sample of bolts, measuring their diameters.
Hypothesis testing is not merely an exercise alma mater; rather, it is something much needed for making a business decision. It may support enhancing or optimizing business strategies, operations, and customer satisfaction by rigorous testing of assumptions based on data-driven decisions. For instance, this would include areas such as product development, market research, or quality assurance.
This nuance in business matters a lot to decisions. It is useful in finding chances for innovation and optimizing marketing strategies to assure product quality. Hypothesis testing is used by businesses to test new ideas and concepts before full-scale implementation to avoid loss and risk, hence maximizing the chances of return maximization.
For instance, in the retail sector, one can use hypothesis testing to determine if the promotional campaign is effective. Comparing the sales data before and after a promotion will help the business learn if the promotion worked. In manufacturing, hypothesis testing can be used to ensure that production processes are within specified limits, to maintain the quality and consistency of the product.
Moreover, hypothesis testing is at the core of any financial analysis and forecasting . It helps the financial analyst measure investment performance, identify anomalies in financial data, and graph the future trends. It provides a firm statistical method to assist an informed decision that turns on empirical evidence rather than gut feelings or assumptions.
Let us consider in greater detail some of the important hypothesis testing types enumerated earlier, and when and how they are used.
The Z-test is very useful in large samples, n > 30, with known population variances. It is frequently used in quality control and survey analysis. For example, suppose a company wants to compare the average customer satisfaction score from a large sample with the national average. A Z-test will tell whether their customer service passes or fails the national standard.
The T-test is useful when sample sizes are small and the population variance is unknown. There are three variants of the T-test, which makes it versatile in application:
Chi-square tests are used in categorical data. They are aimed at establishing whether the relationship between the two variables is significant. Chi-square tests may be applied in market research to establish whether some relationships exist, say between customer demographics and buying behavior. For example, it would tell if age groups have varying preferences for a given product or not.
ANOVA is used to compare three or more group means. The application of this statistical tool is thus very appropriate, especially in multiple treatment experiments. For instance, to establish which type of crop variety yields the highest under similar conditions, an agricultural scientist will apply the ANOVA to compare the yields of different crop varieties.
The F-test is carried out between the two variances to check if they are significantly different. One of the areas this test is commonly applied includes quality control processes, especially when it comes to comparing the variability of product measurements from different machines.
While hypothesis testing is an invaluable tool, there are some practical considerations which have to be borne in mind for effective and reliable results.
Hypothesis testing is one of the critical parts of a statistical analysis and hence sets a clearly defined framework within which data-driven decisions can be made. Once you have grasped how the process and types of hypothesis testing work, you will be able to apply such methods to real-world problems in ways that can drive business excellence and innovation.
It isn’t a theoretical concept but a practical tool that enables better decision-making across many domains. Be it healthcare, marketing, or manufacturing, it is understanding hypothesis testing that, with the application of this understanding, leads toward more informed and data-driven decision-making to propel the organization in the proper direction.
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5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). Null Hypothesis. The statement that there is not a difference in the population (s), denoted as H 0.
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.
Developing a hypothesis (with example) 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.
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
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.
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.
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.
A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators. In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population. The test considers two hypotheses: the ...
Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.
This statistics video tutorial provides a basic introduction into hypothesis testing. It provides examples and practice problems that explains how to state ...
Hypothesis tests # Formal hypothesis testing is perhaps the most prominent and widely-employed form of statistical analysis. It is sometimes seen as the most rigorous and definitive part of a statistical analysis, but it is also the source of many statistical controversies. The currently-prevalent approach to hypothesis testing dates to developments that took place between 1925 and 1940 ...
What is hypothesis testing in statistics with example? Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to ...
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.
Test statistics represent effect sizes in hypothesis tests because they denote the difference between your sample effect and no effect —the null hypothesis. Consequently, you use the test statistic to calculate the p-value for your hypothesis test. The above p-value definition is a bit tortuous.
Hypothesis 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 ...
Here, we'll be using the formula below for the general form of the test statistic. Determine the p-value. The p-value is the area under the standard normal distribution that is more extreme than the test statistic in the direction of the alternative hypothesis. Make a decision. If \(p \leq \alpha\) reject the null hypothesis.
Choose the sample size. Determine the statistical technique. Set up the critical values to identify the reject region and non-reject region. Collect the data sample and compute sample parameters & Test statistic. Compare sample/test statistic with critical value/reject or non-reject region. Make your conclusion clear.
Hypothesis testing. In hypothesis testing, two mutually exclusive statements about a parameter or population (hypotheses) are evaluated to decide which statement is best supported by sample data. Parameters and statistics. In statistics, a parameter is a description of a population, while a statistic describes a small portion of a population ...
Example 1: Weight of Turtles. A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles. Here is how to write the null and alternative hypotheses for this scenario: H0: μ = 300 (the true mean weight is equal to ...
Test statistic example. To test your hypothesis about temperature and flowering dates, you perform a regression test. The regression test generates: a regression coefficient of 0.36. a t value comparing that coefficient to the predicted range of regression coefficients under the null hypothesis of no relationship.
When writing the conclusion of a hypothesis test, we typically include: Whether we reject or fail to reject the null hypothesis. The significance level. A short explanation in the context of the hypothesis test. For example, we would write: We reject the null hypothesis at the 5% significance level.
State the Hypotheses: The hypotheses state the null hypothesis, H0, and the alternative hypothesis, H1. The null hypothesis normally represents a no-effect or no-difference statement, while the alternative represents what is to be proved. Select the Level of Significance (α): Probability of rejecting the null hypothesis when it is true. Common ...
hypothesis A proposed explanation for a fairly narrow set of phenomena, usually based on prior experience, scientific background knowledge, preliminary observations, and logic. To learn more, visit Science at multiple levels .