AP Statistics Unit 6 Inference for Proportions

Topic 6.6: Concluding a Test for a Population Proportion

Making Conclusions Using Significance Level & p-Values
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:

Key Vocabulary

Significance Level (α) A predetermined boundary value used to determine if a p-value is small enough to reject H0. Common values: 0.05, 0.01, 0.10.
Reject H0 When the p-value ≤ α, we reject the null hypothesis and conclude there is convincing statistical evidence for Ha.
Fail to Reject H0 When the p-value > α, we fail to reject the null hypothesis and conclude there is not convincing statistical evidence for Ha.
Accepting the Null A common error: concluding that H0 is true when you fail to reject it. A lack of convincing evidence for Ha does not prove H0.

Video 1: Making a Conclusion

[0:00 – 8:43]
1. [0:30] Small p-values indicate that the observed test statistic would be if the null hypothesis were true, providing convincing statistical evidence for the .
2. [1:07] The significance level, denoted , is a boundary value that we use to determine if a p-value is small or not.
3. [1:21] Common significance levels include α = , α = , and α = .
Two Possible Conclusions: [1:29]
Context — "Does Green = More Natural?" (continued) [2:18] Same study from 6.4 and 6.5: 30 randomly selected students, 18 of 30 chose the green cup. From previous videos: H0: p = 0.50, Ha: p > 0.50. Conditions met. z = 1.10, p-value = 0.1357.
4. [3:32] Write the conclusion for the lemonade study at α = 0.05:
(a) Compare: The p-value of 0.1357 is α = 0.05.
(b) Decision: We the null hypothesis.
(c) Conclusion about Ha: There is convincing statistical evidence that more than half of all students at this school would choose the green cup.
Cautions About Conclusions: [3:52]
Practice — Football Favorites: [6:04] 40% of U.S. adults say football is their favorite sport. The mayor wonders if the proportion in his town differs from 0.40. Random sample of 100 adults: 29 named football. From previous videos: H0: p = 0.40, Ha: p ≠ 0.40. Conditions met. p-value = 0.0244. Use α = 0.10.
5. [7:01] Write the conclusion for the football study at α = 0.10:
(a) Compare: The p-value of 0.0244 is α = 0.10.
(b) Decision: We the null hypothesis.
(c) Conclusion about Ha: There convincing statistical evidence that the proportion of all adults in this town who would say football is their favorite sport differs from 0.40.

Video 2: Performing a Complete Significance Test

[0:00 – 6:33]
Context — 2005 AP Exam: Breakfast Cereal Vouchers [0:28] A cereal company claims 20% of boxes contain a voucher. A group of students suspects the true proportion is less than 0.20. They purchased 65 boxes and found 11 vouchers. Assume the 65 boxes are a random sample.
6. [1:37] State the hypotheses:
(a) H0: p =
(b) Ha: p 0.20
(c) where p = the proportion of
7. [2:22] Since no significance level is stated, we will use α = . The procedure is a .
8. [2:46] Check the conditions:
(a) Random: Stated as a .
(b) 10%: Assume there are more than boxes of this cereal.
(c) Large Counts: np0 = 65(0.2) = ≥ 10 and n(1 − p0) = 65(0.8) = ≥ 10.
9. [3:30] A common mistake is to use instead of p0 when checking the Large Counts condition.
10. [3:48] Calculations:
(a) p = 11/65 =
(b) z = (0.169 − 0.2) / √(0.2 × 0.8 / 65) =
(c) p-value = P(z ≤ −0.62) =
11. [5:14] Conclusion: Because the p-value of is α = 0.05, we the null hypothesis. There is convincing statistical evidence that the proportion of all boxes of this cereal with vouchers is less than 0.2.
Complete Significance Test Checklist: [5:44]
  1. State H0 and Ha,
  2. Identify significance level (α)
  3. Identify procedure (one-sample z-test for p)
  4. Verify and show evidence
  5. Calculate test statistic and
  6. Make a based on the p-value

Key Takeaways

[Summary]
  1. If p-value ≤ α, we H0. Conclude: there is convincing statistical evidence that [state Ha in context].
  2. If p-value > α, we H0. Conclude: there is not convincing statistical evidence that [state Ha in context].
  3. Always explicitly the p-value to α in your conclusion.
  4. Never "accept" H0 and never say you have "proven" Ha.

Reflection Questions

[AI-Graded]
R1. A student conducts a significance test and gets a p-value of 0.23 with α = 0.05. The student concludes: "Because the p-value is large, we accept H0 and conclude that the null hypothesis is true." Identify TWO errors in this conclusion and write a corrected version.
R2. Explain why "failing to reject H0" is different from "proving H0 is true." Use the cereal voucher example (where p = 0.169, p-value = 0.2676) to illustrate your explanation.

Exit Ticket

A local news station claims that 60% of residents support a new park. A city council member suspects the true proportion is less than 0.60. She surveys a random sample of 80 residents and finds that 42 support the new park.

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