AP Statistics Unit 6 Inference for Proportions

Topic 6.3: Justifying a Claim Based on a Confidence Interval for p

Interpreting CIs, Confidence Level, Margin of Error Factors & the 5-Step Process
Video Follow-Along Worksheet
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Learning Objectives:

Key Vocabulary

Interpreting a CI “We are C% confident that the interval from ___ to ___ captures the [parameter in context].”
Confidence Level In repeated random sampling, approximately C% of C% CIs will capture the true population parameter
Justifying a Claim If all values in the CI are consistent with a claim → convincing evidence; if any are not → not convincing
Margin of Error Half the width of a CI; ME = z* × SE. Affected by sample size and confidence level.
Width of a CI Exactly twice the margin of error. Decreases with larger n; increases with higher confidence level.

Video 1: Interpreting a CI & Justifying Claims

[Video: ~7 min]
1. [0:00] In this video you’ll learn two skills: how to a confidence interval and how to a claim based on a confidence interval.
Proposition 100 Signature Example (from Topic 6.2): [0:35] Supporters submitted 9,388 signatures. Administrators randomly selected 500 and verified 364 were from registered voters. The 95% CI was (0.689, 0.767).
CI Interpretation Template: [1:23] “We are C% confident that the interval from ___ to ___ captures the [population parameter in context].”
2. [1:23] Applied to the example: “We are % confident that the interval from 0.689 to 0.767 captures the of all submitted signatures that are from registered voters in the city.”
3. [2:21] Propositions need at least 6,000 valid signatures. With 9,388 submitted, they need at least 6,000 / 9,388 ≈ % of signatures to be valid.
4. [2:21] Because all values in the CI (0.689 to 0.767) are than 0.639, there is convincing evidence that the supporters gathered enough valid signatures.
Practice — Will Proposition 100 Pass? [3:16] A news program reports a 95% CI of 0.518 ± 0.044 for the proportion of registered voters who plan to vote for Prop 100.
5. [4:27] Converting: lower = 0.518 − 0.044 = , upper = 0.518 + 0.044 = .
6. [4:27] Interpretation: “We are 95% confident that the interval from 0.474 to 0.562 captures the proportion of all in the city who plan to vote for Proposition 100.”
7. [5:06] To determine if a majority supports Prop 100, we check if the interval contains the value .
8. [5:06] Because the interval the value 0.5 (and values less than 0.5), there is convincing evidence that a majority plan to vote for Prop 100.
Key Rule for Justifying Claims: [5:55] If all values in the CI are consistent with the claim → convincing evidence. If one or more values are inconsistent → not convincing evidence.
9. [5:55] Summary: If all values are consistent with the claim, the evidence is . If one or more values are inconsistent, the evidence is .

Video 2: Interpreting Confidence Level & Factors Affecting ME

[Video: ~9 min]
Driver’s License Example: [0:45] In a high school with 2,000 students, 30% have a driver’s license (p = 0.30). We take random samples of size 50 and build 95% CIs.
10. [0:45] The true population proportion is p = . Samples are of size n = .
11. [1:19] Sample 1: p̂ = 12/50 = 0.24, CI = (0.12, 0.36). This interval capture p = 0.30.
12. [2:58] Sample 4: p̂ = 22/50 = . CI = (0.302, 0.578). This interval capture p = 0.30. (Just missed! 0.302 > 0.30)
13. [3:31] After many repeated samples, about % of the 95% CIs captured the true proportion.
Interpreting Confidence Level: [4:06] “In repeated random sampling with the same sample size, approximately C% of C% confidence intervals will capture the population proportion.”
14. [4:06] Confidence level describes what happens in sampling — approximately C% of intervals will the population proportion.
15. [4:56] Common misconception: The confidence level does NOT give the that a particular interval captures the population proportion.
16. [5:18] A specific CI that includes p captures it with probability . A specific CI that misses captures it with probability .
17. [6:15] The width of a confidence interval is exactly the margin of error.
18. [6:54] Factor 1: Assuming everything else stays the same, ME is smaller when the sample size is . Width is proportional to 1/√.
19. [6:54] To cut the margin of error in half, you must multiply the sample size by . (Quadruple it!)
20. [7:31] Factor 2: ME is smaller when the confidence level is . For 90% confidence, z* = , which is less than z* = 1.96 for 95%.
21. [8:32] Summary: Increasing sample size the margin of error. Increasing the confidence level the margin of error.

Video 3: Constructing & Interpreting a CI — Full Worked Example

[Video: ~7 min]
2010 AP Exam Problem: [0:30] Mike and Lori share a digital music player with a random-selection feature. A total of 2,384 songs are loaded. In a random sample of 50 songs played, 13 were loaded by Lori.
22. [0:30] The music player has total songs. Of the 50 randomly selected, were loaded by Lori.
5-Step Process for Constructing & Interpreting a CI:
  1. Define the parameter
  2. Identify the procedure
  3. Verify conditions (with evidence!)
  4. Calculate the confidence interval
  5. Interpret in context
23. [1:52] Step 1 — Define the parameter: p = the proportion of on the player that were loaded by Lori.
24. [2:26] Step 2 — Identify the procedure: .
25. [2:37] Step 3 — Check conditions:
Random: Songs were selected ✓
10%: 50 ≤ 10% of ✓
Large Counts: Successes = ≥ 10 and Failures = ≥ 10 ✓
26. [3:25] Step 4 — Calculate: p̂ = 13/50 = .
27. [3:36] For a 90% CI, z* = . Margin of error = 1.645 × √(0.26 × 0.74 / 50) ≈ .
28. [4:05] The 90% CI: (, ).
29. [4:26] Step 5 — Interpret: “We are % confident that the interval from 0.158 to 0.362 captures the proportion of on the player that were loaded by Lori.”
30. [5:00] Part (b): The distinction between with/without replacement doesn’t matter because the sample of 50 is less than % of 2,384, so the 10% condition is satisfied.
31. [6:14] The 5-step process: (1) Define the , (2) Identify the , (3) Verify (with evidence!), (4)  the CI, (5)  in context.
32. [6:46] Important: You do not need to interpret the confidence unless specifically asked. Doing so unnecessarily can lead to mistakes.

Reflection Questions

[AI-Graded]
R1. A survey of 200 randomly selected adults in a large city finds that 124 support expanding public transit. A 95% confidence interval for the proportion of all adults who support expansion is (0.558, 0.682). (a) Interpret this confidence interval in context. (b) A city council member claims that at least 50% of adults support expansion. Does this interval provide convincing evidence for this claim? Explain. (c) Another council member claims that more than 70% support expansion. Does the interval support this claim? Explain.
R2. Explain what it means to say we are “95% confident” in the context of confidence intervals. In your explanation, address: (a) What does the 95% refer to in terms of repeated sampling? (b) Can we say there is a 95% probability that a specific, already-calculated confidence interval contains the true proportion? Why or why not? (c) Name two ways to decrease the margin of error of a confidence interval and explain why each works.

Exit Ticket

A researcher randomly selects 80 songs from a digital playlist of 1,200 songs and finds that 24 are jazz.

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