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

First, restating the problem helps ensure understanding: we test H0: p = 0.70 versus H1: p < 0.70 using a sample of n = 120, where x = 77 doctors recommended aspirin, so the sample proportion is p̂ = 77/120 ≈ 0.6417. The test uses α = 0.05 and a one-sided (left-tail) z-test for a proportion. The standard error under H0 is sqrt(p0(1 - p0)/n) = sqrt(0.70×0.30/120) = sqrt(0.21/120) ≈ sqrt(0.00175) ≈ 0.04183. The test statistic is z = (p̂ − p0) / SE = (0.6417 − 0.70) / 0.04183 ≈ (−0.0583) / 0.04183 ≈ −1.395.

Now evaluate each option against this result and the decision rule.

Option 1: 'The proportion of doctors who recommend aspirin is less than 0.70.' This would require rejecting H0 in favor of H1, i.e., z ≤ zα for a left-tailed test, where zα ≈ −1.645 at α = 0.05. Our computed z ≈ −1.395 is not as far left as −1.645, so we do not reject H0. Therefore, this statement is not supported by the test outcome.

Option 2: 'The proportion of doctors who recommend aspirin is not less than 0.70.' This aligns with failing to reject H0, meaning we do not have sufficient evidence to claim p < 0.70. In the context of a one-sided test, the complement statement to H1 is consistent with the decision to not reject H0. Since the observed z-statistic does not cross the critical boundary, this conclusion matches the result of the test.

Option 3: 'The proportion of doctors who recommend aspirin is greater than 0.70.' This would imply p̂ > 0.70 and would require evidence against H0 in the positive direction, which is not the case here. The sample proportion is actually less than 0.70, and the test is designed to detect values smaller than 0.70, so this option contradicts both the data and the test direction.

Option 4: 'The proportion of doctors who recommend aspirin is equal to 0.70.' The data provide a sample proportion different from 0.70, and hypothesis tests do not establish exact equality to a parameter; they test whether there is statistically significant evidence to conclude p differs from or is below 0.70. With a non-rejection of H0, we cannot claim equivalence; nonetheless, the standard interpretation here is not to claim exact equality.

Option 5: 'The proportion of doctors who recommend aspirin is not greater than 0.70.' This statement asserts p ≤ 0.70. While it is not the same as p < 0.70, the test specifically targets whether p < 0.70; failing to reject H0 does not provide evidence for p being ≤ 0.70 beyond 0.70 or less. The wording is subtly different and less aligned with the test's direction; the conventional takeaway from the result is about not having evidence that p < 0.70, rather than asserting p ≤ 0.70.

In summary, given that the test statistic does not enter the rejection region at α = 0.05, we do not have statistical evidence to conclude that the true proportion is less than 0.70, which makes Option 2 the statement that corresponds to the test outcome. The other options either overstate the direction of the effect, contradict the data, or misinterpret what a failure to reject implies.

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