Question

1. a sociologist conducts a hypothesis test because she has a theory that the proportion of college students who study abroad falls below 0.45. after surveying a random sample of college students, and conducting a hypothesis test, the sociologist reports a p - value of 0.01. what can we conclude based on this information? a. the sample proportion is at least three standard deviations away from 0.45. b. there must be a large difference between p and \\( \hat { p } \\). c. if \\( \alpha \\) is set at 0.01, the researcher should conclude there is no evidence against the null hypothesis. d. the sample size must have been large. e. none of the above answer options are correct.

Explanation:

Step1: Analyze Option A

For a one - sided test, if \(P - value = 0.01\), the \(z\) - score is \(z=- 2.33\) (from standard normal distribution tables for a left - tailed test). The formula for the \(z\) - score is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\). A \(z\) - score of \(-2.33\) is not at least \(3\) standard deviations away from the hypothesized proportion \(p = 0.45\).

Step2: Analyze Option B

The \(P\) - value is related to the probability of getting a sample statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. It is not directly equivalent to a “large difference” between \(p\) (population proportion) and \(\hat{p}\) (sample proportion). The \(P\) - value also depends on the sample size \(n\) (since the standard deviation of the sampling distribution of \(\hat{p}\) is \(\sqrt{\frac{p(1 - p)}{n}}\)).

Step3: Analyze Option C

The decision rule for a hypothesis test is: if \(P - value\leq\alpha\), we reject the null hypothesis. If \(\alpha=0.01\) and \(P - value = 0.01\), we reject the null hypothesis (not conclude there is no evidence against it).

Step4: Analyze Option D

The \(P\) - value is a function of the sample proportion \(\hat{p}\), the hypothesized proportion \(p\), and the sample size \(n\) (\(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), \(P - value = P(Z\lt z)\) for a left - tailed test). A small \(P\) - value can be due to a large difference between \(\hat{p}\) and \(p\) or a large sample size. We cannot be sure that it is only due to a large sample size.

Answer:

E. None of the above answer options are correct.