14.2 Binomial tests for a proportion — Practise

Work through these on your own. Write the five steps for every full test, and name the significance level in each conclusion.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A pizza chain claims that 1 delivery in 5 takes more than 30 minutes. In 30 deliveries, 2 take more than 30 minutes. Test the claim at the 5% level against the alternative that the proportion is lower.
A manufacturer claims that 20% of components are faulty. A sample of 25 components contains 2 faulty ones. Test the claim at the 5% level against the alternative that the proportion is lower.
A survey claims that 40% of shoppers prefer brand A. In a sample of 30 shoppers, 16 prefer brand A. Test the claim at the 5% level against the alternative that the proportion is higher.
A coin is tossed 16 times and shows 12 heads. Test at the 5% level whether the coin is biased towards heads.
A spinner is claimed to land on red 1 time in 4. In 30 spins it lands on red 13 times. Test at the 5% level whether the spinner is unfair.
The same spinner in question 5 lands on red 12 times in 30 spins. Test at the 5% level whether the spinner is unfair.
A test gives a p-value of $0.0442$ . State, with a symbol, how this p-value compares with the 5% significance level, and name the decision.
The same test gives a p-value of $0.0442$ . State, with a symbol, how this p-value compares with the 1% significance level, and name the decision.
A chain claims that 1 delivery in 5 takes more than 30 minutes. Find $P(X \le 2)$ for a sample of 30 deliveries, to 4 decimal places.
A survey claims that 40% of shoppers prefer brand A. Find $P(X \ge 16)$ for a sample of 30 shoppers, to 4 decimal places.
A coin is tossed 20 times and shows 16 heads. Test at the 5% level whether the coin is fair.
A call centre claims that 70% of calls are answered within 30 seconds. In a sample of 40 calls, 32 are answered in time. Test at the 5% level whether the proportion is higher than the claim.

Answers and concise solutions

1. The p-value and the decision

Answer: $H_0: p = 0.2$ and $H_1: p < 0.2$ . Under $H_0$ , $X \sim \mathrm{B}(30, 0.2)$ . The p-value is $P(X \le 2) = 0.0442 < 0.05$ , so reject $H_0$ . There is sufficient evidence at the 5% level that fewer than 1 delivery in 5 takes more than 30 minutes.

2. The p-value and the decision

Answer: $H_0: p = 0.2$ and $H_1: p < 0.2$ . The p-value is $P(X \le 2) = 0.0982 > 0.05$ , so retain $H_0$ . There is not sufficient evidence at the 5% level that fewer than 20% of components are faulty.

3. The p-value and the decision

Answer: $H_0: p = 0.4$ and $H_1: p > 0.4$ . The p-value is $P(X \ge 16) = 0.0971 > 0.05$ , so retain $H_0$ . There is not sufficient evidence at the 5% level that more than 40% of shoppers prefer brand A.

4. The p-value and the decision

Answer: $H_0: p = 0.5$ and $H_1: p > 0.5$ . The p-value is $P(X \ge 12) = 0.0384 < 0.05$ , so reject $H_0$ . There is sufficient evidence at the 5% level that the coin is biased towards heads.

5. The two-tailed p-value and the decision

Answer: $H_0: p = 0.25$ and $H_1: p \ne 0.25$ . The smaller tail is $P(X \ge 13) = 0.0216$ , so the two-tailed p-value is $2 \times 0.0216 = 0.0432 < 0.05$ and the test rejects $H_0$ . There is sufficient evidence at the 5% level that the spinner is unfair.

6. The two-tailed p-value and the decision

Answer: The smaller tail is $P(X \ge 12) = 0.0507$ , so the two-tailed p-value is $2 \times 0.0507 = 0.1013 > 0.05$ and the test retains $H_0$ . There is not sufficient evidence at the 5% level that the spinner is unfair.

7. The comparison and the decision

Answer: $0.0442 < 0.05$ , so the test rejects $H_0$ at the 5% level.

8. The comparison and the decision

Answer: $0.0442 > 0.01$ , so the test retains $H_0$ at the 1% level. The same evidence gives two decisions at two levels.

9. The lower tail probability

Answer: $P(X \le 2) = 0.0442$ to 4 decimal places.

10. The upper tail probability

Answer: $P(X \ge 16) = 0.0971$ to 4 decimal places.

11. The two-tailed p-value and the decision

Answer: $H_0: p = 0.5$ and $H_1: p \ne 0.5$ . The smaller tail is $P(X \ge 16) = 0.0059$ , so the two-tailed p-value is $0.0118 < 0.05$ and the test rejects $H_0$ . There is sufficient evidence at the 5% level that the coin is not fair.

12. The p-value and the decision

Answer: $H_0: p = 0.7$ and $H_1: p > 0.7$ . The p-value is $P(X \ge 32) = 0.1110 > 0.05$ , so retain $H_0$ . There is not sufficient evidence at the 5% level that more than 70% of calls are answered within 30 seconds.