Question

17% of all college students volunteer their time. Is the percentage of college students who are volunteers larger for students receiving financial aid? Of the 307 randomly selected students who receive financial aid, 68 of them volunteered their time. What can be concluded at the a = 0.01 level of significance? a. For this study, we should use Select an answer b. The null and alternative hypotheses would be: Ho: ? ? (please enter a decimal) Hj: ? ? Please enter a decimal) c. The test statistic ? = (please show your answer to 3 decimal places.) d. The p-value = (Please show your answer to 4 decimal places.) e. The p-value is ? a f. Based on this, we should Select an answer the null hypothesis. g. Thus, the final conclusion is that ... The data suggest the populaton proportion is significantly higher than 17% at a= 0.01, so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is higher than 17%. The data suggest the population proportion is not significantly higher than 17% at a = 0.01, so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is equal to 17%. The data suggest the population proportion is not significantly higher than 17% at a = 0.01, so there is insufficient evidence to conclude that the percentage of financial aid recipients who volunteer is higher than 17%.

Answer 1

(a) One sample proportion test

The following information is provided: The sample size is n=347, the number of favorable cases is x=68, and the sample proportion is p​= x/n​ = 68/347​ = 0.196, and the significance level is α=0.01

(b) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p=0.17

Ha:p>0.17

This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used.

(c) Test Statistics

The z-statistic is computed as follows:

z=(p​−p0​​) / sqrt(p0​(1−p0​)/n)

=(0.196-0.17)/sqrt(0.17(1-0.17)/347)

= 1.288

(d)  p-value is p = 0.0989

(e) p = 0.0989 ≥ α=0.01

(f)  Decision about the null hypothesis

Based on the information provided, the significance level is α=0.01, and the critical value for a right-tailed test is zc​=2.33.

The rejection region for this right-tailed test is R={z:z>2.33}

Since it is observed that z=1.288 ≤ zc​=2.33, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.0989, and since p=0.0989 ≥ 0.01, it is concluded that the null hypothesis is not rejected.

Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is greater than p0​, at the α=0.01 significance level.

(g) Option 2nd is right.

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