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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BlackTom AI · Accepted Answer

We start by identifying the hypotheses and the test context. The claim to test is that the true proportion p is less than 0.7 (alternative hypothesis: p < 0.7), against the null hypothesis that p = 0.7 (or equivalently p ≥ 0.7 for a left-tailed test).
Option 1: The proportion of doctors who recommend aspirin is less than 0.70. This would be the conclusion if the test statistic fell into the rejection region for the alternative p < 0.7. To check this, we compute the test statistic using p0 = 0.70 and n = 120. The sample proportion is phat = 77/120 ≈ 0.6417. The standard error under H0 is sqrt(p0(1 − p0)/n) = sqrt(0.7×0.3/120) ≈ sqrt(0.21/120) ≈ sqrt(0.00175) ≈ 0.04183. The z-value is (phat − p0)/SE ≈ (0.6417 − 0.7)/0.04183 ≈ −0.0583/0.04183 ≈ −1.39. A left-tailed test at α = 0.05 has a critical z of about −1.645. Since −1.39 is not as extreme as −1.645, the statistic does not fall in the rejection region. Therefore, we do not have enough evidence to conclude p < 0.7. This makes Option 1 incorrect.
Option 2: The proportion of doctors who recommend aspirin is not less than 0.70. Given the failure to reject H0 in a left-tailed test, this statement aligns with the conclusion that we cannot support p < 0.7; while it does not prove p ≥ 0.7, it reflects the lack of evidence for the alternative. Thus this option correctly mirrors the inference from the test.
Option 3: The proportion of doctors who recommend aspirin is greater than 0.70. This would require evidence in favor of p > 0.70, which is not what the data or the test were designed to test (the alternative was p < 0.7). Also, the observed phat is below 0.7, and the test is left-tailed, so this option is not supported by the results.
Option 4: The proportion of doctors who recommend aspirin is equal to 0.70. The data do not justify a precise equality claim; hypothesis tests do not prove equality but rather assess evidence against a stated value. With a non-significant result, we cannot claim exact equality.
Option 5: The proportion of doctors who recommend aspirin is not greater than 0.70. This would imply p ≤ 0.70. While our test did not provide evidence that p < 0.70, the standard conclusion from a non-rejection of H0 in a left-tailed test is not a definitive statement of p ≤ 0.70; it’s more accurate to say there is insufficient evidence to conclude p < 0.70. This makes Option 5 overly strong and not fully justified by the test outcome.
In summary, because the test statistic (approximately −1.39) does not reach the critical value for α = 0.05 in a left-tailed test, we fail to reject the null hypothesis that p ≥ 0.70, and the only option that faithfully reflects this conclusion is Option 2: 'The proportion of doctors who recommend aspirin is not less than 0.70.'

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