Questions: A college claims that the proportion, p, of students who commute more than fifteen miles to school is less than 20%. A researcher wants to test this. A random sample of 260 students at this college is selected, and it is found that 43 commute more than fifteen miles to school. Is there enough evidence to support the college's claim at the 0.05 level of significance? Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.) (a) State the null hypothesis H0 and the alternative hypothesis H1. H0: H1: (b) Determine the type of test statistic to use. (Choose one) (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the p-value. (Round to three or more decimal places.) (e) Is there enough evidence to support the claim that the proportion of students who commute more than fifteen miles to school is less than 20% ? Yes No Explanation

Solution Steps

We begin by stating the null and alternative hypotheses for the test: $$H_{0}: p \geq 0.20$$ $$H_{1}: p < 0.20$$

The appropriate test statistic for this hypothesis test is a Z-test, as we are dealing with proportions.

The test statistic $Z$ is calculated using the formula: $$Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$$ where:

• $\hat{p} = \frac{43}{260}$ (the sample proportion),
• $p_0 = 0.20$ (the hypothesized population proportion),
• $n = 260$ (the sample size).

Substituting the values, we find: $$Z = -1.395$$

The p-value associated with the test statistic $Z = -1.395$ is calculated to be: $$\text{P-value} = 0.081$$

To determine if there is enough evidence to support the college's claim, we compare the p-value to the significance level $\alpha = 0.05$:

• If $\text{P-value} < \alpha$, we reject the null hypothesis.
• If $\text{P-value} \geq \alpha$, we fail to reject the null hypothesis.

In this case: $$0.081 \geq 0.05$$ Thus, we fail to reject the null hypothesis.

Based on the results of the hypothesis test, there is not enough evidence to support the claim that the proportion of students who commute more than fifteen miles to school is less than $20\%$.

Final Answer