AP Statistics Confidence Intervals for a Proportion Worksheet Solutions (PDF)

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

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This confidence interval proportion 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 lists the conditions, the standard error, the margin of error, and the interval, followed by an interpretation in context. Values are rounded to three decimals unless a sample size is requested.

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 answers: exercises 1 to 10

Exercise 1 – Critical values

The critical value z* = 1.96 captures the middle 95% of the standard normal curve.

The middle C% of a standard normal curve lies between -z* and z*.

z* = 1.645, 1.960, 2.576. More confidence requires capturing more area, so the interval must extend farther.

Exercise 2 – Check conditions

Random: given. 10%: 80 < 400. Large counts: \(n\hat{p}=12\) and \(n(1-\hat{p})=68\), both at least 10.

All conditions are met.

Exercise 3 – Compute an interval

95% interval: (0.283, 0.437).

\(\hat{p}=\frac{54}{150}=0.36\). \(SE=\sqrt{\frac{0.36(0.64)}{150}}=0.0392\). \(ME=1.96(0.0392)=0.0768\).

Interval: \((0.283,\ 0.437)\).

(0.283, 0.437).

Exercise 4 – Interpret the interval

Name the parameter and the population.

We are 95% confident that the interval from 0.283 to 0.437 captures the true proportion of all teens in the region who use the app daily.

Exercise 5 – Interpret the confidence level

It describes the method.

If we repeatedly took random samples of 150 teens and made a 95% interval from each, about 95% of the intervals would capture the true proportion.

Exercise 6 – Identify errors

The parameter is fixed, not random; the interval is what varies from sample to sample.

Correct wording: we are 95% confident the interval captures the true proportion, because 95% of such intervals do.

Exercise 7 – 90 percent interval

\(\hat{p}=0.60\). \(SE=0.0262\). \(ME=1.645(0.0262)=0.0431\).

(0.557, 0.643).

Exercise 8 – 99 percent interval

\(ME=2.576(0.0262)=0.0675\). Interval: (0.533, 0.667).

Widths: 99%: 0.135; 90%: 0.086.

The 99% interval is wider because higher confidence requires a larger margin of error.

Exercise 9 – Margin of error

\(48\%\pm 3.1\%=(44.9\%,\ 51.1\%)\).

(44.9%, 51.1%). The sample proportion is typically within 3.1 percentage points of the true proportion at the stated confidence level.

Exercise 10 – Find p-hat and ME from an interval

\(\hat{p}=\frac{0.43+0.57}{2}=0.50\). \(ME=\frac{0.57-0.43}{2}=0.07\).

p-hat = 0.50; margin of error = 0.07.

Exercises 11 and up

Exercise 11 – Sample size

Use \(p^{*}=0.5\). \(n\ge\left(\frac{1.96}{0.04}\right)^{2}(0.5)(0.5)=600.25\).

Round up: n = 601.

Exercise 12 – Sample size with an estimate

\(n\ge\left(\frac{1.96}{0.03}\right)^{2}(0.3)(0.7)=896.37\).

n = 897.

Exercise 13 – Effect of n

Quadrupling n halves the margin of error.

Quadrupling n divides the margin of error by 2.

About 0.025.

Exercise 14 – Claim justification

The value 0.50 is above the upper bound 0.437.

No. 0.50 is not a plausible value, so the interval provides convincing evidence that the proportion is less than 0.50.

Exercise 15 – Claim not rejected

\(SE=\sqrt{\frac{0.46(0.54)}{100}}=0.0498\). \(ME=1.96(0.0498)=0.0977\). Interval: (0.362, 0.558).

0.50 is inside the interval, so the claim is plausible; the data do not provide convincing evidence against it.

Exercise 16 – Large counts fails

\(n\hat{p}=3\), which is less than 10.

The large counts condition fails, so the normal model for p-hat is not appropriate.

Exercise 17 – Wider or narrower?

ME = z* sqrt(p-hat(1 – p-hat)/n).

(a) smaller; (b) larger; (c) larger (since p-hat(1 – p-hat) is largest at 0.5).

Exercise 18 – Convenience sample

The Random condition fails for a call-in poll.

No. The interval is based on a voluntary response sample, so its confidence level is meaningless; the margin of error accounts for sampling variability, not bias.

Exercise 19 – Multi-step

(a) \(\hat{p}=0.37,\ SE=0.0216,\ ME=0.0423\), interval (0.328, 0.412).

(b) We are 95% confident that this interval captures the true proportion of homes with a smart thermostat. (c) 0.40 lies inside the interval, so it is plausible.

(0.328, 0.412); 40% is plausible.

Exercise 20 – Two sizes

n = 100: ME = 0.0960. n = 400: ME = 0.0480.

The n = 400 interval is half as wide (ME 0.0480 compared with 0.0960).

Key facts behind this confidence interval proportion 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 answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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