Topic 7.9: Carrying Out a Test for the Difference of Two Population Means
Checking Conditions, Computing the Test Statistic, Finding the P-value, and Writing a Conclusion
Video Follow-Along Worksheet
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Learning Objectives:
Check whether the conditions for a test about the difference of two population means are satisfied.
Compute the test statistic for a two-sample t test using sample statistics.
Interpret the sign and magnitude of a test statistic in context.
Find and interpret a P-value for a test about the difference of two means.
Use the P-value to decide whether to reject or fail to reject the null hypothesis.
Write a conclusion in context and explain why failing to reject H0 does not prove the means are equal.
Essential Knowledge:
After hypotheses are stated, carrying out a test includes checking conditions, calculating the test statistic, finding the P-value, and making a decision.
For a difference in two population means, the test statistic compares the observed sample difference to the null value of 0 using the estimated standard error.
The P-value is the probability, assuming the null hypothesis is true, of getting a result at least as extreme as the one observed.
If the P-value is less than or equal to the significance level, reject H0; otherwise, fail to reject H0.
The sign of the test statistic depends on the order of subtraction used for the sample means.
A non-significant result does not prove the population means are equal; it only means the sample does not provide convincing evidence of a difference.
Key Vocabulary
Two-Sample t Test
A test used to compare two population means when the population standard deviations are unknown.
Conditions
The assumptions that must be checked before using the test, such as random sampling, independence, and a large sample or roughly normal distribution.
Standard Error
The estimated standard deviation of the sampling distribution of the difference in sample means.
Test Statistic
A standardized value that tells how many standard errors the observed difference is from the null value.
Degrees of Freedom
A value used with the t distribution when finding the P-value.
P-value
The probability, assuming the null is true, of getting a result at least as extreme as the observed one.
Significance Level
The cutoff, often 0.05, used to decide whether the evidence against the null is strong enough.
Reject H0
A decision made when the P-value is small enough to show convincing evidence against the null hypothesis.
Fail to Reject H0
A decision made when the sample does not provide convincing evidence against the null hypothesis.
Statistically Significant
A description for a result when the P-value is less than or equal to the chosen significance level.
Conclusion in Context
A final statement that refers to the population means and the original question, not just the sample values.
Video 1: Carrying Out the Test
[0:00 – 5:40]
1.[0:08]
After writing the hypotheses, we check the and then carry out the two-sample .
Context — Chemistry vs. Physics Textbooks[0:20 – 0:42]
A statistics major sampled 200 words from a chemistry textbook and 200 words from a physics textbook to investigate whether the mean word lengths differ. Use a significance level of α = 0.05.
Textbook
n
x̄
s
Chemistry
200
5.71
3.02
Physics
200
6.03
3.58
2.[0:44]
Because the question asks whether the mean word lengths , the alternative hypothesis is two-sided.
3.[0:56]
The null hypothesis sets the difference equal to , and the alternative uses because the question does not specify a direction.
4.[1:14]
To check the first condition, we confirm that the words in each sample were selected at .
5.[1:28]
For independence, the 200 sampled words should be less than of all the words in each textbook.
6.[1:42]
Because both sample sizes are 200, the large-sample condition is satisfied and the sampling distribution is approximately .
Conditions Checklist[1:14 – 1:55]
Check that each sample came from a random process.
Check independence within each sample, usually with the 10% condition when sampling without replacement.
Check that the samples are large enough or the population distributions are roughly normal.
7.[2:05]
The observed difference in sample means is x̄C − x̄P = 5.71 − 6.03 = .
8.[2:20]
The denominator of the test statistic is the estimated of the difference in sample means.
The null value is 0 because the null hypothesis says the population means are equal.
The standard error combines the estimated variability from both samples.
9.[2:52]
The denominator uses information from samples, not just one group.
10.[3:06]
Using the sample statistics, the test statistic is about .
11.[3:18]
The negative sign means the chemistry sample mean is than the physics sample mean because we subtracted chemistry minus physics.
12.[3:32]
The P-value is found by assuming the null hypothesis is and asking how surprising the observed result would be.
13.[3:46]
Because the alternative is two-sided, we consider results in directions away from 0.
14.[4:00]
For this test, the P-value is about .
15.[4:14]
Since the P-value is greater than 0.05, we the null hypothesis.
16.[4:28]
The result is statistically significant at the 5% level.
Decision Rule[4:14 – 4:35]
If the P-value is less than or equal to α, reject H0.
If the P-value is greater than α, fail to reject H0.
The decision should be followed by a conclusion written in context.
17.[4:42]
There is not evidence that the mean word lengths of chemistry and physics textbooks differ.
18.[4:58]
Failing to reject H0 does prove that the two population means are equal.
19.[5:12]
A good conclusion must be written in and should refer to the population means, not only the sample statistics.
20.[5:28]
If we had subtracted physics minus chemistry instead, the test statistic would have the opposite but the same two-sided P-value.
Key Takeaways
[Summary]
To carry out a test for the difference of two population means, check conditions, calculate the test statistic, find the P-value, and make a decision.
The test statistic compares the observed difference in sample means to the null value of 0 using the estimated standard error.
The P-value is calculated under the assumption that the null hypothesis is true.
If the P-value is greater than the significance level, fail to reject H0 and state that there is not convincing evidence of a difference.
Failing to reject the null does not prove the population means are equal.
Reflection Questions
[AI-Graded]
R1.Explain how to carry out the two-sample t test for the chemistry-vs.-physics textbook example. Include the hypotheses, the conditions, how the test statistic is computed and interpreted, the approximate P-value, and the final decision at α = 0.05.
R2.Explain how to interpret the P-value and the final conclusion in this example. Describe why the test is two-sided, what the negative test statistic means, and why failing to reject H0 does not prove that the population means are equal.
Exit Ticket
A statistics major sampled 200 words from a chemistry textbook and 200 words from a physics textbook. The chemistry sample had x̄C = 5.71 letters and sC = 3.02 letters. The physics sample had x̄P = 6.03 letters and sP = 3.58 letters. Use α = 0.05.
(a) State the null and alternative hypotheses for testing whether the textbooks differ in mean word length.
(b) Check the conditions needed for a two-sample t test.
(c) Compute or describe the test statistic and the approximate P-value.
(d) State the decision and write a conclusion in context.
(e) Explain why failing to reject H0 does not prove that the mean word lengths are equal.
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