hypothesis testing 2 population means

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Inference for Comparing 2 Population Means (HT for 2 Means, independent samples)

More of the good stuff! We will need to know how to label the null and alternative hypothesis, calculate the test statistic, and then reach our conclusion using the critical value method or the p-value method.

The Test Statistic for a Test of 2 Means from Independent Samples:

[latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}}[/latex]

What the different symbols mean:

[latex]n_1[/latex] is the sample size for the first group

[latex]n_2[/latex] is the sample size for the second group

[latex]df[/latex], the degrees of freedom, is the smaller of [latex]n_1 - 1[/latex] and [latex]n_2 - 1[/latex]

[latex]\mu_1[/latex] is the population mean from the first group

[latex]\mu_2[/latex] is the population mean from the second group

[latex]\bar{x_1}[/latex] is the sample mean for the first group

[latex]\bar{x_2}[/latex] is the sample mean for the second group

[latex]s_1[/latex] is the sample standard deviation for the first group

[latex]s_2[/latex] is the sample standard deviation for the second group

[latex]\alpha[/latex] is the significance level , usually given within the problem, or if not given, we assume it to be 5% or 0.05

Assumptions when conducting a Test for 2 Means from Independent Samples:

• We do not know the population standard deviations, and we do not assume they are equal
• The two samples or groups are independent
• Both samples are simple random samples
• Both populations are Normally distributed OR both samples are large ([latex]n_1 > 30[/latex] and [latex]n_2 > 30[/latex])

Steps to conduct the Test for 2 Means from Independent Samples:

• Identify all the symbols listed above (all the stuff that will go into the formulas). This includes [latex]n_1[/latex] and [latex]n_2[/latex], [latex]df[/latex], [latex]\mu_1[/latex] and [latex]\mu_2[/latex], [latex]\bar{x_1}[/latex] and [latex]\bar{x_2}[/latex], [latex]s_1[/latex] and [latex]s_2[/latex], and [latex]\alpha[/latex]
• Identify the null and alternative hypotheses
• Calculate the test statistic, [latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}}[/latex]
• Find the critical value(s) OR the p-value OR both
• Apply the Decision Rule
• Write up a conclusion for the test

Example 1: Study on the effectiveness of stents for stroke patients [1]

In this study , researchers randomly assigned stroke patients to two groups: one received the current standard care (control) and the other received a stent surgery in addition to the standard care (stent treatment). If the stents work, the treatment group should have a lower average disability score . Do the results give convincing statistical evidence that the stent treatment reduces the average disability from stroke?

Since we are being asked for convincing statistical evidence, a hypothesis test should be conducted. In this case, we are dealing with averages from two samples or groups (the patients with stent treatment and patients receiving the standard care), so we will conduct a Test of 2 Means.

• [latex]n_1 = 98[/latex] is the sample size for the first group
• [latex]n_2 = 93[/latex] is the sample size for the second group
• [latex]df[/latex], the degrees of freedom, is the smaller of [latex]98 - 1 = 97[/latex] and [latex]93 - 1 = 92[/latex], so [latex]df = 92[/latex]
• [latex]\bar{x_1} = 2.26[/latex] is the sample mean for the first group
• [latex]\bar{x_2} = 3.23[/latex] is the sample mean for the second group
• [latex]s_1 = 1.78[/latex] is the sample standard deviation for the first group
• [latex]s_2 = 1.78[/latex] is the sample standard deviation for the second group
• [latex]\alpha = 0.05[/latex] (we were not told a specific value in the problem, so we are assuming it is 5%)
• One additional assumption we extend from the null hypothesis is that [latex]\mu_1 - \mu_2 = 0[/latex]; this means that in our formula, those variables cancel out
• [latex]H_{0}: \mu_1 = \mu_2[/latex]
• [latex]H_{A}: \mu_1 < \mu_2[/latex]
• [latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}} = \displaystyle \frac{(2.26 - 3.23) - 0)}{\sqrt{\displaystyle \frac{1.78^2}{98} + \displaystyle \frac{1.78^2}{93}}} = -3.76[/latex]
• StatDisk : We can conduct this test using StatDisk. The nice thing about StatDisk is that it will also compute the test statistic. From the main menu above we click on Analysis, Hypothesis Testing, and then Mean Two Independent Samples. From there enter the 0.05 significance, along with the specific values as outlined in the picture below in Step 2. Notice the alternative hypothesis is the [latex]<[/latex] option. Enter the sample size, mean, and standard deviation for each group, and make sure that unequal variances is selected. Now we click on Evaluate. If you check the values, the test statistic is reported in the Step 3 display, as well as the P-Value of 0.00011.

• Applying the Decision Rule: We now compare this to our significance level, which is 0.05. If the p-value is smaller or equal to the alpha level, we have enough evidence for our claim, otherwise we do not. Here, [latex]p-value = 0.00011[/latex], which is definitely smaller than [latex]\alpha = 0.05[/latex], so we have enough evidence for the alternative hypothesis…but what does this mean?
• Conclusion: Because our p-value  of [latex]0.00011[/latex] is less than our [latex]\alpha[/latex] level of [latex]0.05[/latex], we reject [latex]H_{0}[/latex]. We have convincing statistical evidence that the stent treatment reduces the average disability from stroke.

Example 2: Home Run Distances

In 1998, Sammy Sosa and Mark McGwire (2 players in Major League Baseball) were on pace to set a new home run record. At the end of the season McGwire ended up with 70 home runs, and Sosa ended up with 66. The home run distances were recorded and compared (sometimes a player’s home run distance is used to measure their “power”). Do the results give convincing statistical evidence that the home run distances are different from each other? Who would you say “hit the ball farther” in this comparison?

Since we are being asked for convincing statistical evidence, a hypothesis test should be conducted. In this case, we are dealing with averages from two samples or groups (the home run distances), so we will conduct a Test of 2 Means.

• [latex]n_1 = 70[/latex] is the sample size for the first group
• [latex]n_2 = 66[/latex] is the sample size for the second group
• [latex]df[/latex], the degrees of freedom, is the smaller of [latex]70 - 1 = 69[/latex] and [latex]66 - 1 = 65[/latex], so [latex]df = 65[/latex]
• [latex]\bar{x_1} = 418.5[/latex] is the sample mean for the first group
• [latex]\bar{x_2} = 404.8[/latex] is the sample mean for the second group
• [latex]s_1 = 45.5[/latex] is the sample standard deviation for the first group
• [latex]s_2 = 35.7[/latex] is the sample standard deviation for the second group
• [latex]H_{A}: \mu_1 \neq \mu_2[/latex]
• [latex]t = \displaystyle \frac{(\bar{x_1} - \bar{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\displaystyle \frac{s_1^2}{n_1} + \displaystyle \frac{s_2^2}{n_2}}} = \displaystyle \frac{(418.5 - 404.8) - 0)}{\sqrt{\displaystyle \frac{45.5^2}{70} + \displaystyle \frac{35.7^2}{65}}} = 1.95[/latex]
• StatDisk : We can conduct this test using StatDisk. The nice thing about StatDisk is that it will also compute the test statistic. From the main menu above we click on Analysis, Hypothesis Testing, and then Mean Two Independent Samples. From there enter the 0.05 significance, along with the specific values as outlined in the picture below in Step 2. Notice the alternative hypothesis is the [latex]\neq[/latex] option. Enter the sample size, mean, and standard deviation for each group, and make sure that unequal variances is selected. Now we click on Evaluate. If you check the values, the test statistic is reported in the Step 3 display, as well as the P-Value of 0.05221.
• Applying the Decision Rule: We now compare this to our significance level, which is 0.05. If the p-value is smaller or equal to the alpha level, we have enough evidence for our claim, otherwise we do not. Here, [latex]p-value = 0.05221[/latex], which is larger than [latex]\alpha = 0.05[/latex], so we do not have enough evidence for the alternative hypothesis…but what does this mean?
• Conclusion: Because our p-value  of [latex]0.05221[/latex] is larger than our [latex]\alpha[/latex] level of [latex]0.05[/latex], we fail to reject [latex]H_{0}[/latex]. We do not have convincing statistical evidence that the home run distances are different.
• Follow-up commentary: But what does this mean? There actually was a difference, right? If we take McGwire’s average and subtract Sosa’s average we get a difference of 13.7. What this result indicates is that the difference is not statistically significant; it could be due more to random chance than something meaningful. Other factors, such as sample size, could also be a determining factor (with a larger sample size, the difference may have been more meaningful).
• Adapted from the Skew The Script curriculum ( skewthescript.org ), licensed under CC BY-NC-Sa 4.0 ↵

Basic Statistics Copyright © by Allyn Leon is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Hypothesis Test: Difference Between Means

This lesson explains how to conduct a hypothesis test for the difference between two means. The test procedure, called the two-sample t-test , is appropriate when the following conditions are met:

• The sampling method for each sample is simple random sampling .
• The samples are independent .
• Each population is at least 20 times larger than its respective sample .
• The population distribution is normal.
• The population data are symmetric , unimodal , without outliers , and the sample size is 15 or less.
• The population data are slightly skewed , unimodal, without outliers, and the sample size is 16 to 40.
• The sample size is greater than 40, without outliers.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

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.

The table below shows three sets of null and alternative hypotheses. Each makes a statement about the difference d between the mean of one population μ 1 and the mean of another population μ 2 . (In the table, the symbol ≠ means " not equal to ".)

The first set of hypotheses (Set 1) is an example of a two-tailed test , since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests , since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.

When the null hypothesis states that there is no difference between the two population means (i.e., d = 0), the null and alternative hypothesis are often stated in the following form.

H o : μ 1 = μ 2

H a : μ 1 ≠ μ 2

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

• 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. Use the two-sample t-test to determine whether the difference between means found in the sample is significantly different from the hypothesized difference between means.

Analyze Sample Data

Using sample data, find the standard error, degrees of freedom, test statistic, and the P-value associated with the test statistic.

SE = sqrt[ (s 1 2 /n 1 ) + (s 2 2 /n 2 ) ]

DF = (s 1 2 /n 1 + s 2 2 /n 2 ) 2 / { [ (s 1 2 / n 1 ) 2 / (n 1 - 1) ] + [ (s 2 2 / n 2 ) 2 / (n 2 - 1) ] }

t = [ ( x 1 - x 2 ) - d ] / SE

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, having the degrees of freedom computed above. (See sample problems at the end of this lesson for examples of how this is done.)

Interpret 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.

Test Your Understanding

In this section, two sample problems illustrate how to conduct a hypothesis test of a difference between mean scores. The first problem involves a two-tailed test; the second problem, a one-tailed test.

Problem 1: Two-Tailed Test

Within a school district, students were randomly assigned to one of two Math teachers - Mrs. Smith and Mrs. Jones. After the assignment, Mrs. Smith had 30 students, and Mrs. Jones had 25 students.

At the end of the year, each class took the same standardized test. Mrs. Smith's students had an average test score of 78, with a standard deviation of 10; and Mrs. Jones' students had an average test score of 85, with a standard deviation of 15.

Test the hypothesis that Mrs. Smith and Mrs. Jones are equally effective teachers. Use a 0.10 level of significance. (Assume that student performance is approximately normal.)

Solution: The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: μ 1 - μ 2 = 0

Alternative hypothesis: μ 1 - μ 2 ≠ 0

• Formulate an analysis plan . For this analysis, the significance level is 0.10. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

SE = sqrt[(s 1 2 /n 1 ) + (s 2 2 /n 2 )]

SE = sqrt[(10 2 /30) + (15 2 /25] = sqrt(3.33 + 9)

SE = sqrt(12.33) = 3.51

DF = (10 2 /30 + 15 2 /25) 2 / { [ (10 2 / 30) 2 / (29) ] + [ (15 2 / 25) 2 / (24) ] }

DF = (3.33 + 9) 2 / { [ (3.33) 2 / (29) ] + [ (9) 2 / (24) ] } = 152.03 / (0.382 + 3.375) = 152.03/3.757 = 40.47

t = [ ( x 1 - x 2 ) - d ] / SE = [ (78 - 85) - 0 ] / 3.51 = -7/3.51 = -1.99

where s 1 is the standard deviation of sample 1, s 2 is the standard deviation of sample 2, n 1 is the size of sample 1, n 2 is the size of sample 2, x 1 is the mean of sample 1, x 2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test , the P-value is the probability that a t statistic having 40 degrees of freedom is more extreme than -1.99; that is, less than -1.99 or greater than 1.99.

We use the t Distribution Calculator to find P(t < -1.99) is about 0.027.

• If you enter 1.99 as the sample mean in the t Distribution Calculator, you will find the that the P(t ≤ 1.99) is about 0.973. Therefore, P(t > 1.99) is 1 minus 0.973 or 0.027. Thus, the P-value = 0.027 + 0.027 = 0.054.
• Interpret results . Since the P-value (0.054) is less than the significance level (0.10), we cannot accept the null hypothesis.

Note: If you use this approach on an exam, you may also want to mention why this approach is appropriate. Specifically, the approach is appropriate because the sampling method was simple random sampling, the samples were independent, the sample size was much smaller than the population size, and the samples were drawn from a normal population.

Problem 2: One-Tailed Test

The Acme Company has developed a new battery. The engineer in charge claims that the new battery will operate continuously for at least 7 minutes longer than the old battery.

To test the claim, the company selects a simple random sample of 100 new batteries and 100 old batteries. The old batteries run continuously for 190 minutes with a standard deviation of 20 minutes; the new batteries, 200 minutes with a standard deviation of 40 minutes.

Test the engineer's claim that the new batteries run at least 7 minutes longer than the old. Use a 0.05 level of significance. (Assume that there are no outliers in either sample.)

Null hypothesis: μ 1 - μ 2 <= 7

Alternative hypothesis: μ 1 - μ 2 > 7

where μ 1 is battery life for the new battery, and μ 2 is battery life for the old battery.

• Formulate an analysis plan . For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

SE = sqrt[(40 2 /100) + (20 2 /100]

SE = sqrt(16 + 4) = 4.472

DF = (40 2 /100 + 20 2 /100) 2 / { [ (40 2 / 100) 2 / (99) ] + [ (20 2 / 100) 2 / (99) ] }

DF = (20) 2 / { [ (16) 2 / (99) ] + [ (2) 2 / (99) ] } = 400 / (2.586 + 0.162) = 145.56

t = [ ( x 1 - x 2 ) - d ] / SE = [(200 - 190) - 7] / 4.472 = 3/4.472 = 0.67

where s 1 is the standard deviation of sample 1, s 2 is the standard deviation of sample 2, n 1 is the size of sample 1, n 2 is the size of sample 2, x 1 is the mean of sample 1, x 2 is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

Here is the logic of the analysis: Given the alternative hypothesis (μ 1 - μ 2 > 7), we want to know whether the observed difference in sample means is big enough (i.e., sufficiently greater than 7) to cause us to reject the null hypothesis.

Interpret results . Suppose we replicated this study many times with different samples. If the true difference in population means were actually 7, we would expect the observed difference in sample means to be 10 or less in 75% of our samples. And we would expect to find an observed difference to be more than 10 in 25% of our samples Therefore, the P-value in this analysis is 0.25.

Module 10: Hypothesis Testing With Two Samples

Testing for two population means, learning outcomes.

• Conduct a hypothesis test for a difference in two population means with unknown standard deviations and interpret the conclusion in context
• The two independent samples are simple random samples from two distinct populations.
• if the sample sizes are small, the distributions are important (should be normal)
• if the sample sizes are large, the distributions are not important (need not be normal)

Note: The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch t-test. The degrees of freedom formula was developed by Aspin-Welch.

The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, [latex]\displaystyle\overline{{X}}_{{1}}-\overline{{X}}_{{2}} [/latex], and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.

Because we do not know the population standard deviations, we estimate them using the two-sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error , of the difference in sample means, [latex]\displaystyle\overline{{X}}_{{1}}-\overline{{X}}_{{2}} [/latex].

The standard error is: [latex]\displaystyle\sqrt{\frac{(s_1)^2}{n_1}+\frac{(s_2)^2}{n_2}} [/latex]

Recall: Order of Operations

When simplifying mathematical expressions perform the operations in the following order: 1. P arentheses and other Grouping Symbols

• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. E xponents

• Simplify all expressions with exponents.

3. M ultiplication and D ivision

• Perform all multiplication and division in order from left to right. These operations have equal priority.

4. A ddition and S ubtraction

• Perform all addition and subtraction in order from left to right. These operations have equal priority.

For the standard error formula, you would follow the following steps:

First, calculate [latex]\frac{(s_1)^2}{n_1}[/latex] by removing the parentheses by squaring the standard deviation of the first data set and then divide by n of the first data set.

Second, calculate [latex]\frac{(s_2)^2}{n_2}[/latex] by removing the parentheses by squaring the standard deviation of the second data set and then divide by n of the second data set.

Third, add what you got in both previous steps and then take the square root of the sum.

The test statistic ( t -score) is calculated as follows: [latex]t= \dfrac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{\displaystyle\sqrt{\frac{(s_1)^2}{n_1}+\frac{(s_2)^2}{n_2}}} [/latex]

• s 1 and s 2 , the sample standard deviations, are estimates of σ 1 and σ 2 , respectively.
• σ 1 and σ 1 are the unknown population standard deviations.
• [latex]\displaystyle\overline{{x}}_{{1}} [/latex] and [latex]\overline{{x}}_{{2}} [/latex] are the sample means.
• [latex]\mu_1 [/latex] and [latex]\mu_2[/latex] are the population means.

The number of degrees of freedom ( df ) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student’s t -distribution with df as follows:

[latex]\displaystyle{df}=\dfrac{((\dfrac{(s_1)^2}{n_1})+(\dfrac{(s_2)^2}{n_2}))^2}{(\dfrac{1}{n_1-1})(\dfrac{(s_1)^2}{n_1})^2+(\dfrac{1}{n_2-1})(\dfrac{(s_2)^2}{n_2})^2} [/latex]

When both sample sizes n 1 and n 2 are five or larger, the Student’s t approximation is very good. Notice that the sample variances ( s 1 ) 2 and ( s 2 ) 2 are not pooled. (If the question comes up, do not pool the variances.)

Note: It is not necessary to compute this by hand. A calculator or computer easily computes it.

Independent groups

The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in the table below. Each populations has a normal distribution.

Is there a difference in the mean amount of time boys and girls aged seven to 11 play sports each day? Test at the 5% level of significance.

The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, μ g is the population mean for girls and μ b is the population mean for boys. This is a test of two independent groups , two population means .

Random variable: [latex]\displaystyle\overline{{X}}_{{{g}}}-\overline{{X}}_{{b}} [/latex] = difference in the sample mean amount of time girls and boys play sports each day.

[latex]H_0:\mu_g=\mu_b [/latex]; [latex]H_0:\mu_g-\mu_b=0 [/latex]

[latex]H_a:\mu_g\neq\mu_b [/latex]; [latex]H_a:\mu_g-\mu_b\neq{0} [/latex]

The words “ the same ” tell you H 0 has an equal sign. Since there are no other words to indicate H a , assume it says “ is different .” This is a two-tailed test.

Distribution for the test: Use t df where df is calculated using the df formula for independent groups, two population means. Using a calculator, df is approximately 18.8462. Do not pool the variances.

Calculate the p -value using a Student’s t -distribution: p -value = 0.0054

s g = 0.866

So, [latex]\displaystyle\overline{x}_g-\overline{x}_b=2-3.2=-1.2 [/latex]

Half the p -value is below –1.2 and half is above 1.2.

Make a decision: Since α > p -value, reject H 0 . This means you reject μ g = μ b . The means are different.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

• Press STAT .
• Arrow over to TESTS and press 4:2-SampTTest .
• Arrow over to Stats and press ENTER .
• Arrow down and enter 2 for the first sample mean, [latex]0.866[/latex] for Sx1, 9 for n1, 3.2 for the second sample mean, 1 for Sx2, and 16 for n2.
• Arrow down to μ1: and arrow to does not equal μ2.
• Press ENTER .
• Arrow down to Pooled: and No .
• Arrow down to Calculate and press ENTER .
• The p -value is p = 0.0054, the dfs are approximately 18.8462, and the test statistic is –3.14.
• Do the procedure again but instead of Calculate do Draw.

Conclusion: At the 5% level of significance, the sample data show there is sufficient evidence to conclude that the mean number of hours that girls and boys aged seven to 11 play sports per day is different (the mean number of hours boys aged seven to 11 play sports per day is greater than the mean number of hours played by girls OR the mean number of hours girls aged seven to 11 play sports per day is greater than the mean number of hours played by boys).

Two samples are shown in the table. Both have normal distributions. The means for the two populations are thought to be the same. Is there a difference in the means? Test at the 5% level of significance.

The p -value is 0.4125, which is much higher than 0.05, so we decline to reject the null hypothesis. There is not sufficient evidence to conclude that the means of the two populations are not the same.

Note: When the sum of the sample sizes is larger than 30 ( n 1 + n 2 > 30) you can use the normal distribution to approximate the Student’s t .

A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes. College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class. The community group believes that a student who graduates from College A has taken more math classes , on the average. Both populations have a normal distribution. Test at a 1% significance level. Answer the following questions.

• Is this a test of two means or two proportions?
• Are the populations standard deviations known or unknown?
• Which distribution do you use to perform the test?
• What is the random variable?
• What are the null and alternate hypotheses?
• Is this test right-, left-, or two-tailed?
• What is the p -value?
• Do you reject or not reject the null hypothesis?
• Student’s t
• [latex]\displaystyle\overline{{X}}_{{{A}}}-\overline{{X}}_{{B}} [/latex]
• H 0 : μ A ≤ μ B H a : μ A > μ B

• do not reject

Conclusion: At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from College A has taken more math classes, on the average, than a student who graduates from College B.

A study is done to determine if Company A retains its workers longer than Company B. Company A samples 15 workers, and their average time with the company is five years with a standard deviation of 1.2. Company B samples 20 workers, and their average time with the company is 4.5 years with a standard deviation of 0.8. The populations are normally distributed.

• Are the population standard deviations known?
• Conduct an appropriate hypothesis test. At the 5% significance level, what is your conclusion?
• They are unknown.
• The p -value = 0.0878. At the 5% level of significance, there is insufficient evidence to conclude that the workers of Company A stay longer with the company.

A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. Was the professor correct? The randomly selected 30 final exam scores from each group are listed in the two tables below:

Online Class:

Face-to-face Class:

Is the mean of the Final Exam scores of the online class lower than the mean of the Final Exam scores of the face-to-face class? Test at a 5% significance level. Answer the following questions:

• Are the population standard deviations known or unknown?
• What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
• Is this test right, left, or two tailed?
• At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.

(Review the conclusion in Example 2, and write yours in a similar fashion)

Be careful not to mix up the information for Group 1 and Group 2!

First put the data for each group into two lists (such as L1 and L2). Press STAT. Arrow over to TESTS and press 4:2SampTTest. Make sure Data is highlighted and press ENTER. Arrow down and enter L1 for the first list and L2 for the second list. Arrow down to μ 1 : and arrow to ≠ μ 2 (does not equal). Press ENTER. Arrow down to Pooled: No. Press ENTER. Arrow down to Calculate and press ENTER.

• H 0 : μ 1 = μ 2 Null hypothesis: the means of the final exam scores are equal for the online and face-to-face statistics classes.
• H a : μ 1 < μ 2 Alternative hypothesis: the mean of the final exam scores of the online class is less than the mean of the final exam scores of the face-to-face class.
• left-tailed

• reject the null hypothesis
• The professor was correct. The evidence shows that the mean of the final exam scores for the online class is lower than that of the face-to-face class. At the 5% level of significance, from the sample data, there is (is/is not) sufficient evidence to conclude that the mean of the final exam scores for the online class is less than the mean of final exam scores of the face-to-face class.
• Two Population Means with Unknown Standard Deviations. Provided by : OpenStax. Located at : https://openstax.org/books/introductory-statistics/pages/10-1-two-population-means-with-unknown-standard-deviations . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
• Introductory Statistics. Authored by : Barbara Illowsky, Susan Dean. Provided by : OpenStax. Located at : https://openstax.org/books/introductory-statistics/pages/1-introduction . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
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Module 10: Inference for Means

Hypothesis Test for a Difference in Two Population Means (1 of 2)

Learning outcomes.

• Under appropriate conditions, conduct a hypothesis test about a difference between two population means. State a conclusion in context.

Using the Hypothesis Test for a Difference in Two Population Means

The general steps of this hypothesis test are the same as always. As expected, the details of the conditions for use of the test and the test statistic are unique to this test (but similar in many ways to what we have seen before.)

Step 1: Determine the hypotheses.

The hypotheses for a difference in two population means are similar to those for a difference in two population proportions. The null hypothesis, H 0 , is again a statement of “no effect” or “no difference.”

• H 0 : μ 1 – μ 2 = 0, which is the same as H 0 : μ 1 = μ 2

The alternative hypothesis, H a , can be any one of the following.

• H a : μ 1 – μ 2 < 0, which is the same as H a : μ 1 < μ 2
• H a : μ 1 – μ 2 > 0, which is the same as H a : μ 1 > μ 2
• H a : μ 1 – μ 2 ≠ 0, which is the same as H a : μ 1 ≠ μ 2

Step 2: Collect the data.

As usual, how we collect the data determines whether we can use it in the inference procedure. We have our usual two requirements for data collection.

• Samples must be random to remove or minimize bias.
• Samples must be representative of the populations in question.

We use this hypothesis test when the data meets the following conditions.

• The two random samples are independent .
• The variable is normally distributed in both populations . If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the t-distribution. As we discussed in “Hypothesis Test for a Population Mean,” t-procedures are robust even when the variable is not normally distributed in the population. If checking normality in the populations is impossible, then we look at the distribution in the samples. If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the populations, and we use the inference procedure. (Note: This is the same condition we used for the one-sample t-test in “Hypothesis Test for a Population Mean.”)

Step 3: Assess the evidence.

If the conditions are met, then we calculate the t-test statistic. The t-test statistic has a familiar form.

[latex]T=\frac{Observeddifferenceinsamplemeans-Hypothesizeddiferenceinpopulationmeans}{ standarderror}[/latex]

[latex]T=\frac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}}+\frac{s_{2}^{2}}{n_{2}}}[/latex]

Since the null hypothesis assumes there is no difference in the population means, the expression (μ 1 – μ 2 ) is always zero.

As we learned in “Estimating a Population Mean,” the t-distribution depends on the degrees of freedom (df) . In the one-sample and matched-pair cases df = n – 1. For the two-sample t-test, determining the correct df is based on a complicated formula that we do not cover in this course. We will either give the df or use technology to find the df . With the t-test statistic and the degrees of freedom, we can use the appropriate t-model to find the P-value, just as we did in “Hypothesis Test for a Population Mean.” We can even use the same simulation.

Step 4: State a conclusion.

To state a conclusion, we follow what we have done with other hypothesis tests. We compare our P-value to a stated level of significance.

• If the P-value ≤ α, we reject the null hypothesis in favor of the alternative hypothesis.
• If the P-value > α, we fail to reject the null hypothesis. We do not have enough evidence to support the alternative hypothesis.

As always, we state our conclusion in context, usually by referring to the alternative hypothesis.

“Context and Calories”

Does the company you keep impact what you eat? This example comes from an article titled “Impact of Group Settings and Gender on Meals Purchased by College Students” (Allen-O’Donnell, M., T. C. Nowak, K. A. Snyder, and M. D. Cottingham, Journal of Applied Social Psychology 49(9), 2011, onlinelibrary.wiley.com/doi/10.1111/j.1559-1816.2011.00804.x/full) . In this study, researchers examined this issue in the context of gender-related theories in their field. For our purposes, we look at this research more narrowly.

Step 1: Stating the hypotheses.

In the article, the authors make the following hypothesis. “The attempt to appear feminine will be empirically demonstrated by the purchase of fewer calories by women in mixed-gender groups than by women in same-gender groups.” We translate this into a simpler and narrower research question: Do women purchase fewer calories when they eat with men compared to when they eat with women?

Here the two populations are “women eating with women” (population 1) and “women eating with men” (population 2). The variable is the calories in the meal. We test the following hypotheses at the 5% level of significance.

The null hypothesis is always H 0 : μ 1 – μ 2 = 0, which is the same as H 0 : μ 1 = μ 2 .

The alternative hypothesis H a : μ 1 – μ 2 > 0, which is the same as H a : μ 1 > μ 2 .

Here μ 1 represents the mean number of calories ordered by women when they were eating with other women, and μ 2 represents the mean number of calories ordered by women when they were eating with men.

Note: It does not matter which population we label as 1 or 2, but once we decide, we have to stay consistent throughout the hypothesis test. Since we expect the number of calories to be greater for the women eating with other women, the difference is positive if “women eating with women” is population 1. If you prefer to work with positive numbers, choose the group with the larger expected mean as population 1. This is a good general tip.

Step 2: Collect Data.

As usual, there are two major things to keep in mind when considering the collection of data.

• Samples need to be representative of the population in question.
• Samples need to be random in order to remove or minimize bias.

Representative Samples?

The researchers state their hypothesis in terms of “women.” We did the same. But the researchers gathered data by watching people eat at the HUB Rock Café II on the campus of Indiana University of Pennsylvania during the Spring semester of 2006. Almost all of the women in the data set were white undergraduates between the ages of 18 and 24, so there are some definite limitations on the scope of this study. These limitations will affect our conclusion (and the specific definition of the population means in our hypotheses.)

Random Samples?

The observations were collected on February 13, 2006, through February 22, 2006, between 11 a.m. and 7 p.m. We can see that the researchers included both lunch and dinner. They also made observations on all days of the week to ensure that weekly customer patterns did not confound their findings. The authors state that “since the time period for observations and the place where [they] observed students were limited, the sample was a convenience sample.” Despite these limitations, the researchers conducted inference procedures with the data, and the results were published in a reputable journal. We will also conduct inference with this data, but we also include a discussion of the limitations of the study with our conclusion. The authors did this, also.

Do the data meet the conditions for use of a t-test?

The researchers reported the following sample statistics.

• In a sample of 45 women dining with other women, the average number of calories ordered was 850, and the standard deviation was 252.
• In a sample of 27 women dining with men, the average number of calories ordered was 719, and the standard deviation was 322.

One of the samples has fewer than 30 women. We need to make sure the distribution of calories in this sample is not heavily skewed and has no outliers, but we do not have access to a spreadsheet of the actual data. Since the researchers conducted a t-test with this data, we will assume that the conditions are met. This includes the assumption that the samples are independent.

As noted previously, the researchers reported the following sample statistics.

To compute the t-test statistic, make sure sample 1 corresponds to population 1. Here our population 1 is “women eating with other women.” So x 1 = 850, s 1 = 252, n 1 =45, and so on.

[latex]T=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}}+\frac{s_{2}^{2}}{n_{2}}}= \frac{850-719}{\sqrt{\frac{252^{2}}{45}+\frac{322^{2}}{27}}}\approx \frac{131}{72.47}\approx 1.81[/latex]

Using technology, we determined that the degrees of freedom are about 45 for this data. To find the P-value, we use our familiar simulation of the t-distribution. Since the alternative hypothesis is a “greater than” statement, we look for the area to the right of T = 1.81. The P-value is 0.0385.

Generic Conclusion

The hypotheses for this test are H 0 : μ 1 – μ 2 = 0 and H a : μ 1 – μ 2 > 0. Since the P-value is less than the significance level (0.0385 < 0.05), we reject H 0 and accept H a .

Conclusion in context

At Indiana University of Pennsylvania, the mean number of calories ordered by undergraduate women eating with other women is greater than the mean number of calories ordered by undergraduate women eating with men (P-value = 0.0385).

Comment about Conclusions

In the conclusion above, we did not generalize the findings to all women. Since the samples included only undergraduate women at one university, we included this information in our conclusion. But our conclusion is a cautious statement of the findings. The authors see the results more broadly in the context of theories in the field of social psychology. In the context of these theories, they write, “Our findings support the assertion that meal size is a tool for influencing the impressions of others. For traditional-age, predominantly White college women, diminished meal size appears to be an attempt to assert femininity in groups that include men.” This viewpoint is echoed in the following summary of the study for the general public on National Public Radio (npr.org).

• Both men and women appear to choose larger portions when they eat with women, and both men and women choose smaller portions when they eat in the company of men, according to new research published in the Journal of Applied Social Psychology . The study, conducted among a sample of 127 college students, suggests that both men and women are influenced by unconscious scripts about how to behave in each other’s company. And these scripts change the way men and women eat when they eat together and when they eat apart.

Should we be concerned that the findings of this study are generalized in this way? Perhaps. But the authors of the article address this concern by including the following disclaimer with their findings: “While the results of our research are suggestive, they should be replicated with larger, representative samples. Studies should be done not only with primarily White, middle-class college students, but also with students who differ in terms of race/ethnicity, social class, age, sexual orientation, and so forth.” This is an example of good statistical practice. It is often very difficult to select truly random samples from the populations of interest. Researchers therefore discuss the limitations of their sampling design when they discuss their conclusions.

In the following activities, you will have the opportunity to practice parts of the hypothesis test for a difference in two population means. On the next page, the activities focus on the entire process and also incorporate technology.

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Statistics By Jim

Making statistics intuitive

Statistical Hypothesis Testing Overview

By Jim Frost 59 Comments

In this blog post, I explain why you need to use statistical hypothesis testing and help you navigate the essential terminology. 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. If you need to perform hypothesis tests, consider getting my book, Hypothesis Testing: An Intuitive Guide .

Why You Should Perform Statistical Hypothesis Testing

Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. You gain tremendous benefits by working with a sample. In most cases, it is simply impossible to observe the entire population to understand its properties. The only alternative is to collect a random sample and then use statistics to analyze it.

While samples are much more practical and less expensive to work with, there are trade-offs. When you estimate the properties of a population from a sample, the sample statistics are unlikely to equal the actual population value exactly.  For instance, your sample mean is unlikely to equal the population mean. The difference between the sample statistic and the population value is the sample error.

Differences that researchers observe in samples might be due to sampling error rather than representing a true effect at the population level. If sampling error causes the observed difference, the next time someone performs the same experiment the results might be different. Hypothesis testing incorporates estimates of the sampling error to help you make the correct decision. Learn more about Sampling Error .

For example, if you are studying the proportion of defects produced by two manufacturing methods, any difference you observe between the two sample proportions might be sample error rather than a true difference. If the difference does not exist at the population level, you won’t obtain the benefits that you expect based on the sample statistics. That can be a costly mistake!

Let’s cover some basic hypothesis testing terms that you need to know.

Background information : Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Hypothesis Testing

Hypothesis testing is a statistical analysis that uses sample data to assess two mutually exclusive theories about the properties of a population. Statisticians call these theories the null hypothesis and the alternative hypothesis. A hypothesis test assesses your sample statistic and factors in an estimate of the sample error to determine which hypothesis the data support.

When you can reject the null hypothesis, the results are statistically significant, and your data support the theory that an effect exists at the population level.

The effect is the difference between the population value and the null hypothesis value. The effect is also known as population effect or the difference. For example, the mean difference between the health outcome for a treatment group and a control group is the effect.

Typically, you do not know the size of the actual effect. However, you can use a hypothesis test to help you determine whether an effect exists and to estimate its size. Hypothesis tests convert your sample effect into a test statistic, which it evaluates for statistical significance. Learn more about Test Statistics .

An effect can be statistically significant, but that doesn’t necessarily indicate that it is important in a real-world, practical sense. For more information, read my post about Statistical vs. Practical Significance .

Null Hypothesis

The null hypothesis is one of two mutually exclusive theories about the properties of the population in hypothesis testing. Typically, the null hypothesis states that there is no effect (i.e., the effect size equals zero). The null is often signified by H 0 .

In all hypothesis testing, the researchers are testing an effect of some sort. The effect can be the effectiveness of a new vaccination, the durability of a new product, the proportion of defect in a manufacturing process, and so on. There is some benefit or difference that the researchers hope to identify.

However, it’s possible that there is no effect or no difference between the experimental groups. In statistics, we call this lack of an effect the null hypothesis. Therefore, if you can reject the null, you can favor the alternative hypothesis, which states that the effect exists (doesn’t equal zero) at the population level.

You can think of the null as the default theory that requires sufficiently strong evidence against in order to reject it.

For example, in a 2-sample t-test, the null often states that the difference between the two means equals zero.

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Related post : Understanding the Null Hypothesis in More Detail

Alternative Hypothesis

The alternative hypothesis is the other theory about the properties of the population in hypothesis testing. Typically, the alternative hypothesis states that a population parameter does not equal the null hypothesis value. In other words, there is a non-zero effect. If your sample contains sufficient evidence, you can reject the null and favor the alternative hypothesis. The alternative is often identified with H 1 or H A .

For example, in a 2-sample t-test, the alternative often states that the difference between the two means does not equal zero.

You can specify either a one- or two-tailed alternative hypothesis:

If you perform a two-tailed hypothesis test, the alternative states that the population parameter does not equal the null value. For example, when the alternative hypothesis is H A : μ ≠ 0, the test can detect differences both greater than and less than the null value.

A one-tailed alternative has more power to detect an effect but it can test for a difference in only one direction. For example, H A : μ > 0 can only test for differences that are greater than zero.

Related posts : Understanding T-tests and One-Tailed and Two-Tailed Hypothesis Tests Explained

P-values are the probability that you would obtain the effect observed in your sample, or larger, if the null hypothesis is correct. In simpler terms, p-values tell you how strongly your sample data contradict the null. Lower p-values represent stronger evidence against the null. You use P-values in conjunction with the significance level to determine whether your data favor the null or alternative hypothesis.

Related post : Interpreting P-values Correctly

Significance Level (Alpha)

For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.

Use p-values and significance levels together to help you determine which hypothesis the data support. If the p-value is less than your significance level, you can reject the null and conclude that the effect is statistically significant. In other words, the evidence in your sample is strong enough to be able to reject the null hypothesis at the population level.

Related posts : Graphical Approach to Significance Levels and P-values and Conceptual Approach to Understanding Significance Levels

Types of Errors in Hypothesis Testing

Statistical hypothesis tests are not 100% accurate because they use a random sample to draw conclusions about entire populations. There are two types of errors related to drawing an incorrect conclusion.

• False positives: You reject a null that is true. Statisticians call this a Type I error . The Type I error rate equals your significance level or alpha (α).
• False negatives: You fail to reject a null that is false. Statisticians call this a Type II error. Generally, you do not know the Type II error rate. However, it is a larger risk when you have a small sample size , noisy data, or a small effect size. The type II error rate is also known as beta (β).

Statistical power is the probability that a hypothesis test correctly infers that a sample effect exists in the population. In other words, the test correctly rejects a false null hypothesis. Consequently, power is inversely related to a Type II error. Power = 1 – β. Learn more about Power in Statistics .

Related posts : Types of Errors in Hypothesis Testing and Estimating a Good Sample Size for Your Study Using Power Analysis

Which Type of Hypothesis Test is Right for You?

There are many different types of procedures you can use. The correct choice depends on your research goals and the data you collect. Do you need to understand the mean or the differences between means? Or, perhaps you need to assess proportions. You can even use hypothesis testing to determine whether the relationships between variables are statistically significant.

To choose the proper statistical procedure, you’ll need to assess your study objectives and collect the correct type of data . This background research is necessary before you begin a study.

Related Post : Hypothesis Tests for Continuous, Binary, and Count Data

Statistical tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and p-values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.

To see an alternative approach to these traditional hypothesis testing methods, learn about bootstrapping in statistics !

If you want to see examples of hypothesis testing in action, I recommend the following posts that I have written:

• How Effective Are Flu Shots? This example shows how you can use statistics to test proportions.
• Fatality Rates in Star Trek . This example shows how to use hypothesis testing with categorical data.
• Busting Myths About the Battle of the Sexes . A fun example based on a Mythbusters episode that assess continuous data using several different tests.
• Are Yawns Contagious? Another fun example inspired by a Mythbusters episode.

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Reader Interactions

January 14, 2024 at 8:43 am

Hello professor Jim, how are you doing! Pls. What are the properties of a population and their examples? Thanks for your time and understanding.

January 14, 2024 at 12:57 pm

Please read my post about Populations vs. Samples for more information and examples.

Also, please note there is a search bar in the upper-right margin of my website. Use that to search for topics.

July 5, 2023 at 7:05 am

Hello, I have a question as I read your post. You say in p-values section

“P-values are the probability that you would obtain the effect observed in your sample, or larger, if the null hypothesis is correct. In simpler terms, p-values tell you how strongly your sample data contradict the null. Lower p-values represent stronger evidence against the null.”

But according to your definition of effect, the null states that an effect does not exist, correct? So what I assume you want to say is that “P-values are the probability that you would obtain the effect observed in your sample, or larger, if the null hypothesis is **incorrect**.”

July 6, 2023 at 5:18 am

Hi Shrinivas,

The correct definition of p-value is that it is a probability that exists in the context of a true null hypothesis. So, the quotation is correct in stating “if the null hypothesis is correct.”

Essentially, the p-value tells you the likelihood of your observed results (or more extreme) if the null hypothesis is true. It gives you an idea of whether your results are surprising or unusual if there is no effect.

Hence, with sufficiently low p-values, you reject the null hypothesis because it’s telling you that your sample results were unlikely to have occurred if there was no effect in the population.

I hope that helps make it more clear. If not, let me know I’ll attempt to clarify!

May 8, 2023 at 12:47 am

Thanks a lot Ny best regards

May 7, 2023 at 11:15 pm

Hi Jim Can you tell me something about size effect? Thanks

May 8, 2023 at 12:29 am

Here’s a post that I’ve written about Effect Sizes that will hopefully tell you what you need to know. Please read that. Then, if you have any more specific questions about effect sizes, please post them there. Thanks!

January 7, 2023 at 4:19 pm

Hi Jim, I have only read two pages so far but I am really amazed because in few paragraphs you made me clearly understand the concepts of months of courses I received in biostatistics! Thanks so much for this work you have done it helps a lot!

January 10, 2023 at 3:25 pm

Thanks so much!

June 17, 2021 at 1:45 pm

Can you help in the following question: Rocinante36 is priced at ₹7 lakh and has been designed to deliver a mileage of 22 km/litre and a top speed of 140 km/hr. Formulate the null and alternative hypotheses for mileage and top speed to check whether the new models are performing as per the desired design specifications.

April 19, 2021 at 1:51 pm

Its indeed great to read your work statistics.

I have a doubt regarding the one sample t-test. So as per your book on hypothesis testing with reference to page no 45, you have mentioned the difference between “the sample mean and the hypothesised mean is statistically significant”. So as per my understanding it should be quoted like “the difference between the population mean and the hypothesised mean is statistically significant”. The catch here is the hypothesised mean represents the sample mean.

Please help me understand this.

Regards Rajat

April 19, 2021 at 3:46 pm

Thanks for buying my book. I’m so glad it’s been helpful!

The test is performed on the sample but the results apply to the population. Hence, if the difference between the sample mean (observed in your study) and the hypothesized mean is statistically significant, that suggests that population does not equal the hypothesized mean.

For one sample tests, the hypothesized mean is not the sample mean. It is a mean that you want to use for the test value. It usually represents a value that is important to your research. In other words, it’s a value that you pick for some theoretical/practical reasons. You pick it because you want to determine whether the population mean is different from that particular value.

I hope that helps!

November 5, 2020 at 6:24 am

Jim, you are such a magnificent statistician/economist/econometrician/data scientist etc whatever profession. Your work inspires and simplifies the lives of so many researchers around the world. I truly admire you and your work. I will buy a copy of each book you have on statistics or econometrics. Keep doing the good work. Remain ever blessed

November 6, 2020 at 9:47 pm

Hi Renatus,

Thanks so much for you very kind comments. You made my day!! I’m so glad that my website has been helpful. And, thanks so much for supporting my books! 🙂

November 2, 2020 at 9:32 pm

Hi Jim, I hope you are aware of 2019 American Statistical Association’s official statement on Statistical Significance: https://www.tandfonline.com/doi/full/10.1080/00031305.2019.1583913 In case you do not bother reading the full article, may I quote you the core message here: “We conclude, based on our review of the articles in this special issue and the broader literature, that it is time to stop using the term “statistically significant” entirely. Nor should variants such as “significantly different,” “p < 0.05,” and “nonsignificant” survive, whether expressed in words, by asterisks in a table, or in some other way."

With best wishes,

November 3, 2020 at 2:09 am

I’m definitely aware of the debate surrounding how to use p-values most effectively. However, I need to correct you on one point. The link you provide is NOT a statement by the American Statistical Association. It is an editorial by several authors.

There is considerable debate over this issue. There are problems with p-values. However, as the authors state themselves, much of the problem is over people’s mindsets about how to use p-values and their incorrect interpretations about what statistical significance does and does not mean.

If you were to read my website more thoroughly, you’d be aware that I share many of their concerns and I address them in multiple posts. One of the authors’ key points is the need to be thoughtful and conduct thoughtful research and analysis. I emphasize this aspect in multiple posts on this topic. I’ll ask you to read the following three because they all address some of the authors’ concerns and suggestions. But you might run across others to read as well.

Five Tips for Using P-values to Avoid Being Misled How to Interpret P-values Correctly P-values and the Reproducibility of Experimental Results

September 24, 2020 at 11:52 pm

HI Jim, i just want you to know that you made explanation for Statistics so simple! I should say lesser and fewer words that reduce the complexity. All the best! 🙂

September 25, 2020 at 1:03 am

Thanks, Rene! Your kind words mean a lot to me! I’m so glad it has been helpful!

September 23, 2020 at 2:21 am

Honestly, I never understood stats during my entire M.Ed course and was another nightmare for me. But how easily you have explained each concept, I have understood stats way beyond my imagination. Thank you so much for helping ignorant research scholars like us. Looking forward to get hardcopy of your book. Kindly tell is it available through flipkart?

September 24, 2020 at 11:14 pm

I’m so happy to hear that my website has been helpful!

I checked on flipkart and it appears like my books are not available there. I’m never exactly sure where they’re available due to the vagaries of different distribution channels. They are available on Amazon in India.

Introduction to Statistics: An Intuitive Guide (Amazon IN) Hypothesis Testing: An Intuitive Guide (Amazon IN)

July 26, 2020 at 11:57 am

Dear Jim I am a teacher from India . I don’t have any background in statistics, and still I should tell that in a single read I can follow your explanations . I take my entire biostatistics class for botany graduates with your explanations. Thanks a lot. May I know how I can avail your books in India

July 28, 2020 at 12:31 am

Right now my books are only available as ebooks from my website. However, soon I’ll have some exciting news about other ways to obtain it. Stay tuned! I’ll announce it on my email list. If you’re not already on it, you can sign up using the form that is in the right margin of my website.

June 22, 2020 at 2:02 pm

Also can you please let me if this book covers topics like EDA and principal component analysis?

June 22, 2020 at 2:07 pm

This book doesn’t cover principal components analysis. Although, I wouldn’t really classify that as a hypothesis test. In the future, I might write a multivariate analysis book that would cover this and others. But, that’s well down the road.

My Introduction to Statistics covers EDA. That’s the largely graphical look at your data that you often do prior to hypothesis testing. The Introduction book perfectly leads right into the Hypothesis Testing book.

June 22, 2020 at 1:45 pm

Thanks for the detailed explanation. It does clear my doubts. I saw that your book related to hypothesis testing has the topics that I am studying currently. I am looking forward to purchasing it.

Regards, Take Care

June 19, 2020 at 1:03 pm

For this particular article I did not understand a couple of statements and it would great if you could help: 1)”If sample error causes the observed difference, the next time someone performs the same experiment the results might be different.” 2)”If the difference does not exist at the population level, you won’t obtain the benefits that you expect based on the sample statistics.”

I discovered your articles by chance and now I keep coming back to read & understand statistical concepts. These articles are very informative & easy to digest. Thanks for the simplifying things.

June 20, 2020 at 9:53 pm

I’m so happy to hear that you’ve found my website to be helpful!

To answer your questions, keep in mind that a central tenant of inferential statistics is that the random sample that a study drew was only one of an infinite number of possible it could’ve drawn. Each random sample produces different results. Most results will cluster around the population value assuming they used good methodology. However, random sampling error always exists and makes it so that population estimates from a sample almost never exactly equal the correct population value.

So, imagine that we’re studying a medication and comparing the treatment and control groups. Suppose that the medicine is truly not effect and that the population difference between the treatment and control group is zero (i.e., no difference.) Despite the true difference being zero, most sample estimates will show some degree of either a positive or negative effect thanks to random sampling error. So, just because a study has an observed difference does not mean that a difference exists at the population level. So, on to your questions:

1. If the observed difference is just random error, then it makes sense that if you collected another random sample, the difference could change. It could change from negative to positive, positive to negative, more extreme, less extreme, etc. However, if the difference exists at the population level, most random samples drawn from the population will reflect that difference. If the medicine has an effect, most random samples will reflect that fact and not bounce around on both sides of zero as much.

2. This is closely related to the previous answer. If there is no difference at the population level, but say you approve the medicine because of the observed effects in a sample. Even though your random sample showed an effect (which was really random error), that effect doesn’t exist. So, when you start using it on a larger scale, people won’t benefit from the medicine. That’s why it’s important to separate out what is easily explained by random error versus what is not easily explained by it.

I think reading my post about how hypothesis tests work will help clarify this process. Also, in about 24 hours (as I write this), I’ll be releasing my new ebook about Hypothesis Testing!

May 29, 2020 at 5:23 am

Hi Jim, I really enjoy your blog. Can you please link me on your blog where you discuss about Subgroup analysis and how it is done? I need to use non parametric and parametric statistical methods for my work and also do subgroup analysis in order to identify potential groups of patients that may benefit more from using a treatment than other groups.

May 29, 2020 at 2:12 pm

Hi, I don’t have a specific article about subgroup analysis. However, subgroup analysis is just the dividing up of a larger sample into subgroups and then analyzing those subgroups separately. You can use the various analyses I write about on the subgroups.

Alternatively, you can include the subgroups in regression analysis as an indicator variable and include that variable as a main effect and an interaction effect to see how the relationships vary by subgroup without needing to subdivide your data. I write about that approach in my article about comparing regression lines . This approach is my preferred approach when possible.

April 19, 2020 at 7:58 am

sir is confidence interval is a part of estimation?

April 17, 2020 at 3:36 pm

Sir can u plz briefly explain alternatives of hypothesis testing? I m unable to find the answer

April 18, 2020 at 1:22 am

Assuming you want to draw conclusions about populations by using samples (i.e., inferential statistics ), you can use confidence intervals and bootstrap methods as alternatives to the traditional hypothesis testing methods.

March 9, 2020 at 10:01 pm

Hi JIm, could you please help with activities that can best teach concepts of hypothesis testing through simulation, Also, do you have any question set that would enhance students intuition why learning hypothesis testing as a topic in introductory statistics. Thanks.

March 5, 2020 at 3:48 pm

Hi Jim, I’m studying multiple hypothesis testing & was wondering if you had any material that would be relevant. I’m more trying to understand how testing multiple samples simultaneously affects your results & more on the Bonferroni Correction

March 5, 2020 at 4:05 pm

I write about multiple comparisons (aka post hoc tests) in the ANOVA context . I don’t talk about Bonferroni Corrections specifically but I cover related types of corrections. I’m not sure if that exactly addresses what you want to know but is probably the closest I have already written. I hope it helps!

January 14, 2020 at 9:03 pm

Thank you! Have a great day/evening.

January 13, 2020 at 7:10 pm

Any help would be greatly appreciated. What is the difference between The Hypothesis Test and The Statistical Test of Hypothesis?

January 14, 2020 at 11:02 am

They sound like the same thing to me. Unless this is specialized terminology for a particular field or the author was intending something specific, I’d guess they’re one and the same.

April 1, 2019 at 10:00 am

so these are the only two forms of Hypothesis used in statistical testing?

April 1, 2019 at 10:02 am

Are you referring to the null and alternative hypothesis? If so, yes, that’s those are the standard hypotheses in a statistical hypothesis test.

April 1, 2019 at 9:57 am

year very insightful post, thanks for the write up

October 27, 2018 at 11:09 pm

hi there, am upcoming statistician, out of all blogs that i have read, i have found this one more useful as long as my problem is concerned. thanks so much

October 27, 2018 at 11:14 pm

Hi Stano, you’re very welcome! Thanks for your kind words. They mean a lot! I’m happy to hear that my posts were able to help you. I’m sure you will be a fantastic statistician. Best of luck with your studies!

October 26, 2018 at 11:39 am

Dear Jim, thank you very much for your explanations! I have a question. Can I use t-test to compare two samples in case each of them have right bias?

October 26, 2018 at 12:00 pm

Hi Tetyana,

You’re very welcome!

The term “right bias” is not a standard term. Do you by chance mean right skewed distributions? In other words, if you plot the distribution for each group on a histogram they have longer right tails? These are not the symmetrical bell-shape curves of the normal distribution.

If that’s the case, yes you can as long as you exceed a specific sample size within each group. I include a table that contains these sample size requirements in my post about nonparametric vs parametric analyses .

Bias in statistics refers to cases where an estimate of a value is systematically higher or lower than the true value. If this is the case, you might be able to use t-tests, but you’d need to be sure to understand the nature of the bias so you would understand what the results are really indicating.

I hope this helps!

April 2, 2018 at 7:28 am

Simple and upto the point 👍 Thank you so much.

April 2, 2018 at 11:11 am

Hi Kalpana, thanks! And I’m glad it was helpful!

March 26, 2018 at 8:41 am

Am I correct if I say: Alpha – Probability of wrongly rejection of null hypothesis P-value – Probability of wrongly acceptance of null hypothesis

March 28, 2018 at 3:14 pm

You’re correct about alpha. Alpha is the probability of rejecting the null hypothesis when the null is true.

Unfortunately, your definition of the p-value is a bit off. The p-value has a fairly convoluted definition. It is the probability of obtaining the effect observed in a sample, or more extreme, if the null hypothesis is true. The p-value does NOT indicate the probability that either the null or alternative is true or false. Although, those are very common misinterpretations. To learn more, read my post about how to interpret p-values correctly .

March 2, 2018 at 6:10 pm

I recently started reading your blog and it is very helpful to understand each concept of statistical tests in easy way with some good examples. Also, I recommend to other people go through all these blogs which you posted. Specially for those people who have not statistical background and they are facing to many problems while studying statistical analysis.

Thank you for your such good blogs.

March 3, 2018 at 10:12 pm

Hi Amit, I’m so glad that my blog posts have been helpful for you! It means a lot to me that you took the time to write such a nice comment! Also, thanks for recommending by blog to others! I try really hard to write posts about statistics that are easy to understand.

January 17, 2018 at 7:03 am

I recently started reading your blog and I find it very interesting. I am learning statistics by my own, and I generally do many google search to understand the concepts. So this blog is quite helpful for me, as it have most of the content which I am looking for.

January 17, 2018 at 3:56 pm

Hi Shashank, thank you! And, I’m very glad to hear that my blog is helpful!

January 2, 2018 at 2:28 pm

thank u very much sir.

January 2, 2018 at 2:36 pm

You’re very welcome, Hiral!

November 21, 2017 at 12:43 pm

Thank u so much sir….your posts always helps me to be a #statistician

November 21, 2017 at 2:40 pm

Hi Sachin, you’re very welcome! I’m happy that you find my posts to be helpful!

November 19, 2017 at 8:22 pm

great post as usual, but it would be nice to see an example.

November 19, 2017 at 8:27 pm

Thank you! At the end of this post, I have links to four other posts that show examples of hypothesis tests in action. You’ll find what you’re looking for in those posts!

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Lesson 11: tests of the equality of two means, overview section  .

In this lesson, we'll continue our investigation of hypothesis testing. In this case, we'll focus our attention on a hypothesis test for the difference in two population means \(\mu_1-\mu_2\) for two situations:

• a hypothesis test based on the \(t\)-distribution, known as the pooled two-sample \(t\)-test , for \(\mu_1-\mu_2\) when the (unknown) population variances \(\sigma^2_X\) and \(\sigma^2_Y\) are equal
• a hypothesis test based on the \(t\)-distribution, known as Welch's \(t\)-test , for \(\mu_1-\mu_2\) when the (unknown) population variances \(\sigma^2_X\) and \(\sigma^2_Y\) are not equal

Of course, because population variances are generally not known, there is no way of being 100% sure that the population variances are equal or not equal. In order to be able to determine, therefore, which of the two hypothesis tests we should use, we'll need to make some assumptions about the equality of the variances based on our previous knowledge of the populations we're studying.

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• Knowledge Base

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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Business Statistics

3.2 two population means.

In addition to testing claims about the mean of a population, hypothesis testing can be used to compare the equality of two different population means

The procedure for testing hypotheses about two population means is similar to the procedure for a single population mean. The null hypothesis states that there is no difference between two population means \(\mu_1\) and \(\mu_2\) :

\[\begin{equation} H_0:~~\mu_1=\mu_2 \tag{3.6} \end{equation}\]

• The alternative hypothesis can take one of three forms:

\[\begin{align} H_1:&~~\mu_1 \ne \mu_2 &\text{two-tailed test} \\ \\ H_1:&~~\mu_1 < \mu_2 &\text{left-tailed test} \\ \\ H_1:&~~\mu_1 > \mu_2 &\text{right-tailed test} \\ \tag{3.7} \end{align}\]

• To compute a test statistic it is usually assumed:

Both populations are normally distributed

Samples are chosen from the two independent populations

Variances of the two populations are equal, although unknown

• The test statistic follows a Student’s t-distribution with \((n_1+n_2-2)\) degrees of freedom:

\[\begin{align} t&=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{S_p^2 \bigg(\frac{1}{n_1}+\frac{1}{n_2} \bigg)}} \\ \\ S_p^2 &=\frac{S_1^2 (n_1-1)+S_2^2 (n_2-1)}{n_1+n_2-2} \\ \tag{3.8} \end{align}\]

• Here is what each term means:

\[\begin{align} \bar{x}_1&~\text{is the mean of the first sample (chosen from population 1)} \\ \\ \bar{x}_2&~\text{is the mean of the second sample (chosen from population 2)} \\ \\ n_1&~\text{is the size of the first sample} \\ \\ n_2&~\text{is the size of the second sample} \\ \\ S_1^2&~\text{is the variance of the first sample} \\ \\ S_2^2&~\text{is the variance of the second sample} \\ \\ S_p^2&~\text{is the pooled (common) variance from the both samples} \end{align}\]

Example 3.3 A company produces coconut milk. They have both a day shift and a night shift. They would like to know if the day shift and the night shift are equally efficient in processing the coconuts. A study is done by sampling \(9\) shifts during the day and \(16\) shifts during the night. The results of the number of hours required to process \(100\) pounds of coconuts is presented in table 3.1 . Is there a statisticaly significant difference in the average number of hours for each shift to process 100 pounds of coconuts? Significance level is \(5\%\) .

Example 3.4 At \(1\%\) significanve level test the hypothesis that the mean annual revenue of the listed companies is greater than the mean annual revenue of the companies not listed on the stock exchange. Use the data from Excel file. Obtain the p-value in Excel using function =T.TEST() . Perform the same testing by Data Analysis ToolPak.

Two Population Calculator

Confidence interval.

Hypothesis Testing

When computing confidence intervals for two population means, we are interested in the difference between the population means ($ \mu_1 - \mu_2 $). A confidence interval is made up of two parts, the point estimate and the margin of error. The point estimate of the difference between two population means is simply the difference between two sample means ($ \bar{x}_1 - \bar{x}_2 $). The standard error of $ \bar{x}_1 - \bar{x}_2 $, which is used in computing the margin of error, is given by the formula below.

The formula for the margin of error depends on whether the population standard deviations ($\sigma_1$ and $\sigma_2$) are known or unknown. If the population standard deviations are known, then they are used in the formula. If they are unknown, then the sample standard deviations ($s_1$ and $s_2$)are used in their place. To change from $\sigma$ known to $\sigma$ unknown, click on $\boxed{σ}$ and select $\boxed{s}$ in the Two Population Calculator.

While the formulas for the margin of error in the two population case are similar to those in the one population case, the formula for the degrees of freedom is quite a bit more complicated. Although this formula does seem intimidating at first sight, there is a shortcut to get the answer faster. Notice that the terms $\frac{s_1^2}{n_1}$ and $\frac{s_2^2}{n_2}$ each repeat twice. The terms are actually computed previously when finding the margin of error so they don't need to be calculated again.

If the two population variances are assumed to be equal, an alternative formula for computing the degrees of freedom is used. It's simply df = n1 + n2 - 2. This is a simple extension of the formula for the one population case. In the one population case the degrees of freedom is given by df = n - 1. If we add up the degrees of freedom for the two samples we would get df = (n1 - 1) + (n2 - 1) = n1 + n2 - 2. This formula gives a pretty good approximation of the more complicated formula above.

Just like in hypothesis tests about a single population mean, there are lower-tail, upper-tail and two tailed tests. However, the null and alternative are slightly different. First of all, instead of having mu on the left side of the equality, we have $\mu_1 - \mu_2$. On the right side of the equality, we don't have $\mu_0$, the hypothesized value of the population mean. Instead we have $D_0$, the hypothesized difference between the population means. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

Again, hypothesis testing for a single population mean is very similar to hypothesis testing for two population means. For a single population mean, the test statistics is the difference between mu and mu0 dividied by the standard error. For two population means, the test statistic is the difference between $\bar{x}_1 - \bar{x}_2$ and $D_0$ divided by the standard error. The procedure after computing the test statistic is identical to the one population case. That is, you proceed with the p-value approach or critical value approach in the same exact way.

The calculator above computes confidence intervals and hypothesis tests for the difference between two population means. The simpler version of this is confidence intervals and hypothesis tests for a single population mean. For confidence intervals about a single population mean, visit the Confidence Interval Calculator . For hypothesis tests about a single population mean, visit the Hypothesis Testing Calculator .

Z-test for two Means, with Known Population Standard Deviations

Instructions: This calculator conducts a Z-test for two population means (\(\mu_1\) and \(\mu_2\)), with known population standard deviations ( \(\sigma_1\) and \(\sigma_2\)). Please select the null and alternative hypotheses, type the significance level, the sample means, the population standard deviations, the sample sizes, and the results of the z-test will be displayed for you:

The Z-test for Two Means

More about the z-test for two means so you can better use the results delivered by this solver: A z-test for two means is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, we are interested in assessing whether or not it is reasonable to claim that the two population means the population means \(\mu\) 1 and \(\mu\) 2 are equal, based on the information provided by the samples. The test has two non-overlapping hypotheses, the null and the alternative hypothesis.

The null hypothesis is a statement about the population means, corresponding to the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample z-test for two population means are:

• Depending on our knowledge about the "no effect" situation, the z-test can be two-tailed, left-tailed or right-tailed
• The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
• The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
• In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

How to calculate the test statistic for the two samples? We have that the formula for a z-statistic for two population means is:

The above formula allows you to assess whether or not there is a statistically significant difference between two means. The null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).

In case that the population standard deviations are not known, you can use a t-test for two sample means calculator .

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