Question

1. a published report claims that the proportion of college students who prefer online courses to in - person courses is 0.37. an educational researcher believes this value is inaccurate. the researcher gathers data from a random sample of 600 college students and finds the sample proportion to be 0.34. the researcher goes on to compute a test statistic and reports that statistic as ( z=-1.5 ). which one of the following conclusions is most appropriate?

a. the alternative hypothesis is ( h_{a}: p<0.37 ).
b. the researcher did not compute the test statistic appropriately.
c. there is not enough evidence to reject the null hypothesis at ( alpha = 0.10 ).
d. the results of the hypothesis test are statistically significant at ( alpha = 0.10 ) but not at ( alpha = 0.05 ).
e. none of the above answer options are correct.

Explanation:

• Option A: Since the researcher believes the value is inaccurate (not specifically less), the alternative hypothesis should be \(H_a: p

eq0.37\), not \(H_a: p < 0.37\). So, A is incorrect.

• Option B: The formula for the test statistic in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\). Here, \(\hat{p}=0.34\), \(p = 0.37\), \(n = 600\).

$$z=\frac{0.34 - 0.37}{\sqrt{\frac{0.37\times(1 - 0.37)}{600}}}=\frac{- 0.03}{\sqrt{\frac{0.37\times0.63}{600}}}\approx\frac{-0.03}{\sqrt{\frac{0.2331}{600}}}=\frac{-0.03}{\sqrt{0.0003885}}\approx\frac{-0.03}{0.0197}\approx - 1.52\approx - 1.5$$

The researcher computed the test statistic appropriately. So, B is incorrect.

• Option C: For a two - tailed test (\(H_a: p

eq0.37\)) at \(\alpha=0.10\), the critical values are \(z=\pm1.645\). Since \(z=-1.5\) and \(|z| = 1.5<1.645\), we fail to reject the null hypothesis \(H_0:p = 0.37\) at \(\alpha = 0.10\).

• Option D: At \(\alpha=0.05\), the critical values for a two - tailed test are \(z=\pm1.96\). Since \(|z|=1.5<1.96\), and at \(\alpha = 0.10\) (critical values \(z=\pm1.645\)) \(|z| = 1.5<1.645\), the results are not statistically significant at either \(\alpha=0.10\) (for a two - tailed test) or \(\alpha=0.05\).

Answer:

C. There is not enough evidence to reject the null hypothesis at \(\alpha = 0.10\).