AP Statistics Significance Tests for a Proportion Test Solutions and Scoring (PDF)

AP Statistics math test solutions: Significance Tests for a Proportion – Maths PDF

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This test proportion test answers for AP Statistics covers significance tests for a proportion (Common Core AP STAT 6.4, AP STAT 6.5, AP STAT 6.6, AP STAT 6.7, HSS-IC.A.2, HSS-IC.B.5), with a free PDF to print.

Each solution includes a scoring guide. Give credit for the correct hypotheses, the conditions, the z statistic and p-value, and a conclusion that links the p-value to alpha and to the context.

AP Statistics · Common Core State Standards: AP STAT 6.4, AP STAT 6.5, AP STAT 6.6, AP STAT 6.7, HSS-IC.A.2, HSS-IC.B.5

Test proportion test answers: worked solutions

Question 1 – One-proportion z test [5 pts]

P-value region.

State: \(H_0: p=0.30\), \(H_a: p > 0.30\), p is the true proportion of customers who return products.

Plan: one-proportion z test; random sample, 200 is less than 10% of the customers, \(np_0=60\) and \(n(1-p_0)=140\) at least 10.

Do: \(\hat{p}=0.35,\ SE=0.0324,\ z=1.54,\ p\text{-value}=0.0618\).

Conclude: p-value > 0.05, so fail to reject H0; there is not convincing evidence that the return rate exceeds 30%.

Scoring: State 1; Plan 1 (test and conditions); Do 2 (z and p-value); Conclude 1 (link to alpha and context).

Question 2 – Interpret the p-value [2 pts]

If the true return rate were 30%, the probability of a sample return rate of 35% or more in a random sample of 200 customers is 0.0618.

It is the probability of such a result if H0 is true.

Scoring: 1 point for conditioning on H0 true, 1 point for context and ‘as extreme or more’.

Question 3 – Errors [3 pts]

(a) Concluding the fertilizer increases yield when it does not. (b) Not concluding it increases yield when it does. (c) 0.05.

As described; 0.05.

Scoring: 1 point per part.

Question 4 – Two-sided p-value [2 pts]

Two-sided p-value.

\(2P(Z > 2.10)=2(0.0179)=0.0357\).

0.0357 < 0.05: reject H0.

Scoring: 1 point for doubling the tail area, 1 point for the conclusion.

Question 5 – Conclusion language [2 pts]

0.27 is greater than 0.05.

We fail to reject H0. There is not convincing evidence that the new app increases the proportion of users who finish a lesson.

Scoring: 1 point for the comparison with alpha, 1 point for the conclusion in context.

Question 6 – Alternative hypothesis [1 pt]

A tail.

Only the right tail is shaded.

p > p0.

Scoring: 1 point for p > p0.

Question 7 – Conditions [2 pts]

\(np_0=150(0.02)=3\), less than 10.

No: the large counts condition fails.

Scoring: 1 point for np0, 1 point for the conclusion.

Question 8 – Decision [3 pts]

Decision rule.

(a) \(P(Z > 1.40)=0.0808\). (b) 0.0808 < 0.10: reject H0. (c) 0.0808 > 0.05: fail to reject H0.

0.0808; reject; fail to reject.

Scoring: 1 point per part.

Total score

Total

20 points

Key facts behind this test proportion test answers

Key facts

H0 states no effect (p = p0); Ha is p > p0, p < p0, or p not equal to p0.
z = (p-hat – p0) / sqrt(p0(1 – p0)/n); check random, 10%, and large counts with p0.
p-value: probability of a result at least as extreme as observed, assuming H0 is true.
p-value < alpha: reject H0 (convincing evidence); otherwise fail to reject; never 'accept H0'.
Type I error: reject true H0 (probability alpha). Type II: fail to reject false H0; power = 1 – P(Type II).

How to use this test 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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