AP Statistics Confidence Intervals for a Proportion Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Confidence Intervals for a Proportion – Maths PDF

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This confidence interval proportion test answers for AP Statistics covers confidence intervals for a proportion (Common Core AP STAT 6.1, AP STAT 6.2, AP STAT 6.3, HSS-IC.B.4), with a free PDF to print.

Each solution includes a scoring guide. Award points for checking conditions, for the correct interval, and for an interpretation that names the parameter and the population.

AP Statistics · Common Core State Standards: AP STAT 6.1, AP STAT 6.2, AP STAT 6.3, HSS-IC.B.4

Confidence interval proportion test answers: worked solutions

Question 1 – Build an interval [4 pts]

Interval for the proportion.

(a) Random sample; 300 is less than 10% of a city’s adults; \(n\hat{p}=135\) and \(n(1-\hat{p})=165\) are at least 10.

(b) \(\hat{p}=0.45,\ SE=0.0287,\ ME=0.0563\), interval (0.394, 0.506).

(c) We are 95% confident that the interval from 0.394 to 0.506 captures the true proportion of adults in the city who own or plan to buy a charger.

(0.394, 0.506).

Scoring: 1 point for random and 10%, 1 point for large counts, 1 point for the interval, 1 point for the interpretation in context.

Question 2 – Name the margin [1 pt]

Interval for the proportion.

\(\frac{0.51-0.39}{2}=0.06\).

0.06.

Scoring: 1 point for 0.06.

Question 3 – Interpret the level [2 pts]

The level refers to the method used.

If many random samples of size 300 were taken and a 95% interval computed from each, about 95% of those intervals would capture the true proportion.

Scoring: 1 point for referring to repeated sampling, 1 point for the 95% capture rate.

Question 4 – Claim [2 pts]

A confidence interval.

0.55 is above the upper endpoint 0.52.

No. 0.55 is not a plausible value, so there is convincing evidence the proportion is less than 0.55.

Scoring: 1 point for comparing 0.55 with the interval, 1 point for the conclusion.

Question 5 – Sample size [3 pts]

\(n\ge\left(\frac{2.576}{0.02}\right)^{2}(0.25)=4147.4\).

n = 4148.

Scoring: 1 point for z*, 1 point for p* = 0.5 setup, 1 point for rounding up.

Question 6 – Conditions [2 pts]

\(n\hat{p}=4\) is less than 10.

No: the large counts condition fails.

Scoring: 1 point for computing the count, 1 point for the conclusion.

Question 7 – Effect of changes [3 pts]

(a) wider; (b) narrower (half as wide); (c) narrower, since p-hat(1 – p-hat) decreases from 0.25 to 0.09.

Wider; half as wide; narrower.

Scoring: 1 point each.

Question 8 – Process [3 pts]

Four steps.

STATE, PLAN, DO, CONCLUDE. In PLAN, name the procedure and check the random, 10%, and large counts conditions.

State, plan, do, conclude; PLAN = name the interval and check conditions.

Scoring: 1 point for the four steps, 1 point for naming the procedure, 1 point for the three conditions.

Total score

Total

20 points

Key facts behind this confidence interval proportion test answers

Key facts

Interval: p-hat +/- z* sqrt(p-hat(1 – p-hat)/n); z* = 1.645, 1.960, 2.576 for 90%, 95%, 99%.
Conditions: Random, 10% (n < 10% of population), Large counts (n p-hat and n(1 - p-hat) at least 10).
Interpret the interval in context; interpret the level as the long-run capture rate.
Higher confidence or smaller n gives a wider interval; quadrupling n halves the margin of error.
A claimed value outside the interval is not plausible at that confidence level.

How to use this confidence interval proportion test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All AP Statistics chapters: AP Statistics math.

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