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?[Fill in the blank]

Question

BlackTom AI · Accepted Answer

To analyze this hypothesis testing scenario, I’ll go through each option and connect them to what the sample data and the test decision imply.
Option 1: 'The proportion of doctors who recommend aspirin is less than 0.70'. This would be a conclusion if we rejected the null in favor of the alternative p < 0.70. However, the problem states the test statistic does not fall in the rejection region at α = 0.05, meaning we do not have enough evidence to conclude p < 0.70. Therefore, this option is not supported by the given test 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 H0: p ≥ 0.70 (or equivalently concluding p ≥ 0.70, depending on framing) in the context of a left-tailed test where the alternative is p < 0.70. Since we did not reject H0 at α = 0.05, saying that the proportion is not less than 0.70 is consistent with the result. This option correctly reflects the conclusion implied by the non-rejection.
Option 3: 'The proportion of doctors who recommend aspirin is greater than 0.70'. This would require evidence supporting p > 0.70, which is not the direction of our alternative (the alternative is p < 0.70) and is not what a non-rejection at α = 0.05 would imply. Hence this option is incorrect given the test setup and result.
Option 4: 'The proportion of doctors who recommend aspirin is equal to 0.70'. A hypothesis test rarely yields a conclusion of exact equality to a population parameter, especially with a finite sample. The test outcome of non-rejection does not prove equality; it merely fails to show p < 0.70. This makes this option incorrect as a direct conclusion from the test.
Option 5: 'The proportion of doctors who recommend aspirin is not greater than 0.70'. This statement is essentially equivalent to p ≤ 0.70. Under a standard test where the alternative is p < 0.70, failing to reject H0 (which typically sets a boundary at 0.70) does not support the claim that p is not greater than 0.70; in fact, the non-rejection is more naturally interpreted as p ≥ 0.70. This option swaps the logical direction and therefore is inconsistent with the test outcome and hypothesis structure.
In summary, examining how the null and alternative hypotheses are set up (p ≥ 0.70 vs p < 0.70) and noting that the test statistic did not fall into the rejection region at α = 0.05, the conclusion that best matches the result is that the proportion is not less than 0.70. The other statements either contradict the non-rejection, assert a direction not supported by the alternative, or misstate what a non-significant result implies about the parameter.

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