AP Statistics Unit 7 Inference for Means

Topic 7.5: Carrying Out a Test for a Population Mean

Test Statistic, P-Value, Interpretation, and Conclusions
Video Follow-Along Worksheet
↻ Already did this one? It stays open even after it’s checked off — just change any answer and click ✓ Check Answers again (no need to Reset). ✨ Grade with AI only ever raises your grade. Make sure you’re signed in so your changes save.
Learning Objectives: Essential Knowledge:

Key Vocabulary

Standardized Test Statistic A measure of how far the sample statistic is from the null value in standard-error units.
Degrees of Freedom For a one-sample t-test, the value n − 1 that determines which t-distribution to use.
p-Value The probability, assuming the null hypothesis is true, of getting evidence as extreme or more extreme than the evidence observed.
Two-Sided p-Value A p-value that adds probability from both tails because the alternative uses ≠.
One-Sided p-Value A p-value from just one tail because the alternative uses < or >.
Significance Level The cutoff value α used to decide whether a result is statistically significant.
Reject H0 The decision made when the p-value is small enough that the sample result would be unlikely if the null hypothesis were true.
Fail to Reject H0 The decision made when the p-value is too large to provide convincing evidence against the null hypothesis.
Matched Pairs A design in which two related measurements are taken together and then compared by their differences.
μD The true mean of the paired differences in a matched-pairs study.

Video 1: Calculating the Test Statistic and p-Value

[0:00 – 9:01]
1. [0:16] In this video, we learn how to calculate an appropriate test and a in a significance test for a population mean.
Context — Got Hops? [0:30 – 0:58] An article on the internet claims that the average vertical jump for teens is 15 inches. A random sample of 20 students at one large high school had a mean jump of 15.8 inches with a standard deviation of 2.33 inches. The question is whether the mean vertical jump for all students at this school differs from 15 inches.
2. [1:14] In the vertical jump study, the sample mean of 15.8 gives some evidence for the because 15.8 is 15.
3. [1:45] After verifying conditions, calculate the test statistic using statistic minus parameter divided by the of the statistic.

Test Statistic Formulas

Standardized test statistic = (statistic − parameter) / standard error
t = (x̄ − μ0) / (s / √n)
Approximate distribution: t with n − 1 degrees of freedom
4. [2:00] For a one-sample t-test for a mean, the test statistic follows an approximate with degrees of freedom.
5. [2:18] Using x̄ = 15.8, s = 2.33, and n = 20 gives t = with degrees of freedom.
6. [3:11] The p-value is the probability of obtaining a test statistic as or more extreme than the observed statistic when the null hypothesis and probability model are assumed to be .
7. [3:40 – 4:15] Because the Got Hops alternative is , the p-value uses of the t-distribution, giving p = .
8. [5:03 – 5:43] From Table B, the Got Hops tail probability is between and for one tail, so the two-sided p-value is between 0.10 and 0.20.

p-Value Tail Directions

Ha: μ > μ0  →  right-tail probability
Ha: μ < μ0  →  left-tail probability
Ha: μ ≠ μ0  →  probability in both tails
Practice — I Can Go for Miles and Miles [6:01 – 6:41] A tire manufacturer tests whether Tread40 tires last more than 40,000 miles on average. A random sample of 35 tires had a mean mileage of 42,348 miles with a standard deviation of 2,140 miles.
9. [6:58] For the tire example, plugging into the formula gives t = with degrees of freedom.
10. [7:28 – 8:15] Because the alternative is μ greater than 40,000, the p-value is a probability. Technology gives about , and Table B shows it is less than .

Video 2: Interpreting the p-Value and Stating a Conclusion

[0:00 – 5:48]
11. [0:14] In this video, we learn how to the p-value and state a in a test for a population mean.
Return to Got Hops? [1:06] For the vertical jump example, the hypotheses are H0: μ = 15 and Ha: μ ≠ 15, with x̄ = 15.8, s = 2.33, t = 1.535, 19 degrees of freedom, and p = 0.1412.
12. [1:06] The p-value measures how likely it is to get evidence for the alternative as strong as or stronger than the observed evidence by alone when the null is .
13. [1:52] To interpret a p-value, begin by that the null hypothesis is true and then describe getting a sample mean as or more extreme by chance alone.
14. [2:55] A p-value means the test statistic is unlikely to occur by random chance alone, while a p-value means it is likely to occur by random chance alone.
Conclusion Templates: [3:17]
  1. If p ≤ α, reject H0. There is convincing statistical evidence that the alternative is true in context.
  2. If p > α, fail to reject H0. There is not convincing statistical evidence that the alternative is true in context.
15. [3:17] If the p-value is less than or equal to α, we the null hypothesis and say there is statistical evidence for the alternative.
16. [3:17] If the p-value is greater than α, we the null hypothesis and say there is convincing statistical evidence for the alternative.
17. [4:21] For Got Hops, because p = 0.1412 is greater than α = , we fail to reject H0 and conclude there is not convincing evidence that the mean vertical jump is from 15 inches.
Tread40 Conclusion [4:51 – 5:48] For Tread40 tires, the one-sample t-test gave a p-value of 9.99 × 10−8, and the significance level for the test is α = 0.01.
18. [5:48] For Tread40 at α = , the tiny p-value means we H0 and conclude there is convincing evidence the tires last 40,000 miles on average.

Video 3: Performing a Complete Significance Test

[0:00 – 6:30]
Context — Bakin' Bacon! [0:25 – 1:12] Doug tested whether mango habanero seasoning helps bacon retain more weight after cooking. For each of 10 packages, one half-package was cooked with seasoning and the other half without seasoning, creating a matched-pairs experiment.
Package With Seasoning Without Seasoning Difference
127826018
226823830
3271281-10
4301306-5
525624016
62482444
724022020
82602528
9286287-1
1030929415
19. [1:27 – 1:45] Because the bacon data came from a matched-pairs experiment, they are an example of data, and we begin by finding the in each pair.
Matched-Pairs Strategy: [1:45 – 2:42]
  1. Compute one difference for each pair.
  2. Find the sample mean and standard deviation of the differences.
  3. Carry out a one-sample t-test for the population mean difference μD.
20. [2:14 – 2:41] From technology, the sample mean difference is x̄D = grams, the standard deviation of the differences is , and there are pairs.
21. [2:42 – 4:05] Inference is performed about μD, the true mean in weight () for packages of bacon like the ones in the study.
22. [3:17 – 4:16] The hypotheses are H0: μD = and Ha: μD 0 because the claim is that seasoning helps the bacon retain more weight.
23. [4:25 – 5:13] For conditions, the half-packages were randomly , there is no because this was not a random sample, and the dotplot showed no strong or .
24. [5:13 – 6:04] The bacon study gives t = with degrees of freedom and p = .
25. [6:04] Because 0.0199 is less than 0.05, we the null hypothesis and conclude there is statistical evidence that the seasoning causes bacon to retain more weight on average.

Key Takeaways

[Summary]
  1. For a one-sample test about a mean, compute t = (x̄ − μ0) / (s / √n) and use n − 1 degrees of freedom.
  2. The direction of the alternative tells you whether the p-value comes from the left tail, right tail, or both tails of the t-distribution.
  3. Interpret the p-value assuming the null is true, then compare it to α to reject or fail to reject H0.
  4. Matched-pairs data are handled by taking differences and then carrying out a one-sample t-test for μD.

Reflection Questions

[AI-Graded]
R1. Explain how to calculate the test statistic and p-value in a significance test for a population mean. Use the Got Hops and Tread40 examples to describe the formula, the degrees of freedom, how the alternative hypothesis determines the tail area, and how technology or Table B can be used.
R2. Explain how to interpret a p-value and state a conclusion in context. Use Got Hops and Tread40 to show the difference between failing to reject the null hypothesis and rejecting the null hypothesis, and include how the significance level affects the decision.

Exit Ticket

In the Bakin' Bacon matched-pairs experiment, the differences were defined as (with seasoning − without seasoning). The sample of 10 pairs had x̄D = 9.5 grams and sD = 12.51 grams. A dotplot of the differences showed no strong skewness or outliers, and the test results were t = 2.401 with p = 0.0199.

Class Responses (Esc to close)

Click a "Class" button next to a question to see responses.