Question

1. a national survey reports that 18% of undergraduate students work out in the morning before attending classes. a researcher at a large university believes that a larger percentage of students work out in the mornings. the researcher collects data from a random sample of 500 students and finds that 105 of these students work out in the morning. based on this information, which one of the following statements is false? a. the difference between ( p ) and ( hat{p} ) would not be considered large enough if ( alpha ) is set at 0.01. b. as long as ( alpha ) is set at 0.10 or 0.05, the researcher can conclude that chance alone is an unlikely explanation for the hypothesis testing results. c. there is evidence against the null hypothesis when ( alpha ) is set at 0.05, but not when it is set at 0.01. d. because the sample proportion is less than two standard deviations away from ( p ), there is insufficient evidence against the null hypothesis. e. the standard error is approximately 0.0172.

Explanation:

Step1: Calculate sample proportion

The sample proportion \(\hat{p}=\frac{105}{500} = 0.21\)
The population proportion \(p = 0.18\)

Step2: Calculate standard error

The formula for standard error \(SE=\sqrt{\frac{p(1 - p)}{n}}\)
Substitute \(p = 0.18\), \(n = 500\)
\(SE=\sqrt{\frac{0.18\times(1 - 0.18)}{500}}=\sqrt{\frac{0.18\times0.82}{500}}\approx0.0172\) (matches option E)

Step3: Calculate z - score

The formula for z - score \(z=\frac{\hat{p}-p}{SE}\)
Substitute \(\hat{p}=0.21\), \(p = 0.18\), \(SE\approx0.0172\)
\(z=\frac{0.21 - 0.18}{0.0172}\approx1.74\)

Step4: Analyze significance levels

For \(\alpha=0.01\), the critical z - value for a one - tailed test is \(z_{0.01}=2.33\). Since \(z = 1.74<2.33\), we fail to reject the null hypothesis at \(\alpha = 0.01\)
For \(\alpha=0.05\), the critical z - value for a one - tailed test is \(z_{0.05}=1.645\). Since \(z = 1.74>1.645\), we reject the null hypothesis at \(\alpha = 0.05\)
For \(\alpha=0.10\), the critical z - value for a one - tailed test is \(z_{0.10}=1.28\). Since \(z = 1.74>1.28\), we reject the null hypothesis at \(\alpha = 0.10\)

Answer:

D. Because the sample proportion is less than two standard deviations away from \(p\), there is insufficient evidence against the null hypothesis.