Question

question 10 1 pts 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? the alternative hypothesis is hₐ: p < 0.37. the researcher did not compute the test statistic appropriately. there is not enough evidence to reject the null hypothesis at α = 0.10. the results of the hypothesis test are statistically significant at α = 0.10 but not at α = 0.05. none of the above answer options are correct.

Explanation:

• For the alternative hypothesis: The researcher believes the value is "inaccurate", which is a two - tailed test. The alternative hypothesis should be \(H_{a}:p

eq0.37\), not \(H_{a}:p < 0.37\). So the first option is wrong.

• For the test statistic calculation:
• The formula for the test statistic \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\), where \(\hat{p}=0.34\), \(p = 0.37\), and \(n = 600\).
• \(z=\frac{0.34 - 0.37}{\sqrt{\frac{0.37\times(1 - 0.37)}{600}}}=\frac{- 0.03}{\sqrt{\frac{0.37\times0.63}{600}}}\approx\frac{-0.03}{\sqrt{\frac{0.2331}{600}}}=\frac{-0.03}{\sqrt{0.0003885}}\approx\frac{-0.03}{0.0197}\approx - 1.52\approx - 1.5\). So the researcher computed the test statistic appropriately, and the second option is wrong.
• For the significance at \(\alpha=0.10\) and \(\alpha = 0.05\):
• For a two - tailed test:
• At \(\alpha=0.10\), the critical values are \(z=\pm1.645\). Since \(|z| = 1.5<1.645\), we fail to reject the null hypothesis at \(\alpha = 0.10\).
• At \(\alpha=0.05\), the critical values are \(z=\pm1.96\). Since \(|z|=1.5 < 1.96\), we also fail to reject the null hypothesis at \(\alpha=0.05\). So the fourth option is wrong.
• Since we fail to reject the null hypothesis at \(\alpha = 0.10\), the third option "There is not enough evidence to reject the null hypothesis at \(\alpha=0.10\)" is correct.

Answer:

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