AP Statistics (AP Stats): Confidence Intervals for One Proportion (p? ± z* × SE)

AP Statistics – Confidence Intervals for One Proportion (p? ± z* × SE)

AP Statistics: Confidence Intervals for One Proportion (p? ± z* × SE) – Exam-Ready Study Guide

What This Is

A confidence interval for one proportion estimates the true proportion (p) of a population based on a sample proportion (p?). It answers questions like: "What percentage of voters support a policy?" or "What’s the defect rate in a factory?" The AP exam tests this concept in multiple-choice and FRQs, often requiring you to construct, interpret, and justify intervals. Example: A pollster surveys 500 voters and finds 60% support a candidate—what’s the plausible range for the true support level?

Key Terms & Formulas

• Population proportion (p): The true proportion of interest in the entire population.
• Sample proportion (p?): The observed proportion in the sample; p? = (number of successes) / n.
• Standard error (SE): SE = ?(p?(1 – p?) / n); measures variability of p?.
• Confidence interval (CI) for p: p? ± z × SE, where z is the critical value for the desired confidence level.
• Critical value (*z): The z-score corresponding to the confidence level (e.g., *z = 1.96 for 95% confidence).
• Calculator command: invNorm(area to left) (e.g., invNorm(0.975) for 95% CI).
• Confidence level (e.g., 95%): The long-run success rate of the method; not the probability the interval contains p.
• Conditions for inference (BINS):
• Binary: Data are yes/no (success/failure).
• Independent: Sampled observations are independent (or n < 10% of population if sampling without replacement).
• Normal: np?-10 and n(1 – p?)-10 (ensures sampling distribution of p? is approximately normal).
• Simple random sample (SRS): Data come from a random sample.
• Margin of error (ME): z × SE; the maximum expected difference between p? and p*.
• Interpretation template: "We are [C]% confident that the true proportion of [context] is between [lower bound] and [upper bound]."

Step-by-Step / Process Flow

For a typical FRQ (e.g., "Construct and interpret a 95% CI for the proportion of students who prefer online learning"):

1. State the parameter and statistic:
2. Define p = true proportion of [context].

3. Calculate p? = (number of successes) / n.

4. Check conditions (BINS):

5. Binary: Responses are yes/no (e.g., "prefer online" vs. "do not prefer").
6. Independent: Sample size n-10% of population (if sampling without replacement).
7. Normal: Verify np?-10 and n(1 – p?)-10.

8. SRS: Assume random sampling unless stated otherwise.

9. Calculate the interval:

10. Compute SE = ?(p?(1 – p?) / n).
11. Find z* using invNorm(area to left) (e.g., invNorm(0.975) for 95% CI).

12. CI = p? ± z* × SE.

13. Interpret in context:

14. "We are 95% confident that the true proportion of students who prefer online learning is between [lower bound] and [upper bound]."

15. Address follow-ups (if asked):

16. Margin of error: ME = z* × SE.
17. Effect of sample size: Larger n-narrower interval.
18. Effect of confidence level: Higher confidence-wider interval.

Common Mistakes

• Mistake: Forgetting to check the 10% condition for independence.

• Correction: Always verify n-10% of the population when sampling without replacement. The AP exam loves to test this!

• Mistake: Using p instead of p? in the SE formula.

• Correction: SE = ?(p?(1 – p?) / n). p is unknown (that’s why we’re estimating it!).

• Mistake: Misinterpreting the confidence level (e.g., "There’s a 95% chance p is in the interval").

• Correction: The correct interpretation is about the method, not the interval: "We are 95% confident that the interval captures p."

• Mistake: Rounding p? too early (e.g., using p? = 0.6 for n = 50, x = 30).

• Correction: Keep p? as 30/50 = 0.6 exactly until the final step to avoid rounding errors.

• Mistake: Using t instead of z for proportions.

• Correction: Proportions use z (normal distribution); t is for means.

AP Exam Insights

• FRQ setups: Expect a 2-part question:
• Construct a CI (show work: conditions, formula, interval).
• Interpret the interval or explain how to reduce the margin of error.
• Tricky distinctions:
• Confidence level vs. interval: The level (e.g., 95%) describes the method; the interval is the range (e.g., 0.55 to 0.65).
• z vs. t: Always use z for proportions (even if n* is small, as long as Normal condition is met).
• Calculator pitfalls:
• 1-PropZInt (TI-84): Use this for quick checks, but show work (conditions, SE, z*) on FRQs.
• invNorm: Remember to use the area to the left (e.g., 0.975 for 95% CI).
• Real-world context: The AP exam often embeds CIs in surveys, experiments, or quality control (e.g., "A factory tests 200 widgets and finds 12 defective").

Quick Check Questions

1. Multiple Choice: A survey of 400 teens finds 60% own a smartphone. Which condition is not met for a 95% CI for p?
2. (A) Binary
3. (B) Independent (n-10% of population)
4. (C) Normal (np?-10 and n(1 – p?)-10)

5. (D) SRS Answer: (B) The 10% condition may not be met if the population of teens is small (e.g., < 4,000).

6. FRQ Part: A poll of 1,200 voters finds 52% support Candidate A. Construct a 99% CI for the true proportion of supporters.

7. Conditions: Binary (support/oppose), Independent (n = 1,200-10% of voters), Normal (1,200 × 0.52 = 624-10, 1,200 × 0.48 = 576-10), SRS (assumed).
8. Calculation: p? = 0.52, SE = ?(0.52 × 0.48 / 1,200)-0.0144, z* = invNorm(0.995)-2.576. CI = 0.52 ± 2.576 × 0.0144? (0.483, 0.557).

9. Interpretation: We are 99% confident the true proportion of voters supporting Candidate A is between 48.3% and 55.7%.

10. Multiple Choice: Which change would decrease the margin of error for a 95% CI for p?

11. (A) Increasing the confidence level to 99%
12. (B) Decreasing the sample size
13. (C) Increasing the sample size
14. (D) Using p = 0.5 instead of p? in the SE formula Answer: (C) Larger n-smaller SE-smaller ME.

Last-Minute Cram Sheet

1. Formula: CI = p? ± z × ?(p?(1 – p?) / n*).
2. Conditions: BINS (Binary, Independent (n-10% of population), Normal (np?-10, n(1 – p?)-10), SRS).
3. Critical value: invNorm(area to left) (e.g., invNorm(0.975) for 95% CI).
4. Calculator shortcut: 1-PropZInt (but show work on FRQs!).
5. Interpretation: "We are [C]% confident that the true proportion of [context] is between [LB] and [UB]."
6. Margin of error: ME = z × SE; larger n*-smaller ME.
7. Always check the 10% condition (even if the problem doesn’t mention it!).
8. Never use *t for proportions—only *z!
9. Round p? only at the end (keep exact value for calculations).
10. Confidence level-probability (it’s about the method, not the interval).