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 data provided. The claim to test is that the true proportion p is less than 0.70 (alternative hypothesis). The null hypothesis is p = 0.70. The data: sample size n = 120, number of doctors recommending aspirin x = 77, so the sample proportion phat = 77/120 ≈ 0.6417. Option 1: 'The proportion of doctors who recommend aspirin is less than 0.70' This would be the direct conclusion if the evidence against H0 was strong enough in favor of Ha. But the computed test statistic will determine whether we reject H0. Since the test statistic does not fall in the rejection region, this option cannot be the concluded result. Option 2: 'The proportion of doctors who recommend aspirin is not less than 0.70' This aligns with failing to reject the null hypothesis p = 0.70 at α = 0.05 for a left-tailed test. Given phat ≈ 0.642 and the z-statistic will not reach the critical threshold for rejecting H0, we do not have evidence that p < 0.70. Thus, the appropriate conclusion is that the proportion is not less than 0.70. Option 3: 'The proportion of doctors who recommend aspirin is greater than 0.70' This would require evidence that p > 0.70, which is not supported by the data. The sample proportion is below 0.70, and the alternative hypothesis is p < 0.70, not p > 0.70. Therefore this statement is inconsistent with the test setup and results. Option 4: 'The proportion of doctors who recommend aspirin is equal to 0.70' A point hypothesis p = 0.70 would require exact equality testing with potentially a different framework (e.g., a two-sided test or an exact method). Our test is a one-sided z-test for p < 0.70 (as implied by the Ha). The conclusion does not claim exact equality; instead, it uses the standard hypothesis testing framework. This option oversimplifies the inferential result and is not the appropriate interpretation of the test outcome. Option 5: 'The proportion of doctors who recommend aspirin is not greater than 0.70' This would correspond to a conclusion stating p ≤ 0.70. The typical test outcome for this data is either fail to reject H0 (p = 0.70) or reject in favor of p < 0.70 for a left-tail test. Since the result is not in the rejection region for Ha: p < 0.70, concluding p ≤ 0.70 is not the standard or supported inference from this test setup.

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