A survey claims that 7 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim against the alternative that the actual proportion of doctors who recommend aspirin is less than 0.7, a random sample of 120 doctors was selected. 77 out of the 120 doctors sampled recommended aspirin. Suppose the test statistic does not fall in the rejection region at α = 0.05. Which of the following conclusions is correct?

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Question restatement: A survey claims that 7 out of 10 doctors recommend aspirin for headaches. To test this against the alternative that the true proportion is less than 0.7, a random sample of 120 doctors was taken, with 77 (out of 120) recommending aspirin. At α = 0.05, the test statistic does not fall in the rejection region. Which conclusion is correct?

Option 1: The true proportion of doctors who recommend aspirin is equal to 0.70.
- Why this is not correct: Failing to reject the null does not prove equality to 0.70. A non-rejection only indicates insufficient evidence to conclude p < 0.7; it does not confirm p = 0.7.

Option 2: The true proportion of doctors who recommend aspirin is not greater than 0.70.
- Why this is not correct: This statement asserts p ≤ 0.70. The test was designed to detect p < 0.70 (one-sided). Failing to reject H0 does not allow us to claim p ≤ 0.70; it merely fails to provide enough evidence that p < 0.70.

Option 3: The true proportion of doctors who recommend aspirin is greater than 0.70.
- Why this is not correct: The data did not provide evidence supporting p > 0.70. The observed sample proportion is below 0.70 (77/120 ≈ 0.642), and the test did not reject the null in the direction of p < 0.70. This option contradicts the observed direction and the test outcome.

Option 4: The true proportion of doctors who recommend aspirin is not less than 0.70.
- Why this is correct: The alternative hypothesis is p < 0.7. Since the test statistic did not fall into the rejection region at α = 0.05, we fail to reject H0. Therefore, we cannot conclude that p < 0.7; the safest conclusion is that the true proportion is not less than 0.70 (i.e., p ≥ 0.70) based on this test’s result.

Option 5: The true proportion of doctors who recommend aspirin is less than 0.70.
- Why this is not correct: This would be the alternative hypothesis direction. Since the test statistic did not land in the rejection region to support Ha, we do not have evidence to claim p < 0.70.

Summary: The decision to fail to reject H0 at α = 0.05 implies we cannot conclude p < 0.70; the correct interpretation among the given choices is that the true proportion is not less than 0.70.

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