Question

10% of all college students volunteer their time. Is the percentage of college students who are volunteers different for students receiving financial aid? Of the 387 randomly selected students who receive financial aid, 23 of them volunteered their time. What can be concluded at the a=0.10 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) H: ? ? (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 ? va f. Based on this, we should Select an answer the null hypothesis. g. Thus, the final conclusion is that ... O The data suggest the population proportion is not significantly different from 10% at a = 0.10, so there is insufficient evidence to conclude that the percentage of financial aid recipients who volunteer is different from 10%. The data suggest the populaton proportion is significantly different from 10% at a = 0.10, so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is different from 10%. The data suggest the population proportion is not significantly different from 10% at a = 0.10, so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is equal to 10%.

Answer 1

Given ; n=387 , X=23

The point estimate of the sample proportion is ,

$\hat{p}=\frac{X}{n}=\frac{23}{387}=0.0594$

a. For this study , we should use normal distribution.

i.e. Z-test

b. The null and alternative hypothesis is ,

$H_0:p=0.10;H_1:p\neq 0.10$

The test is two-tailed test.

c. The test statistic is ,

$Z_{stat}=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}=\frac{0.0594-0.10}{\sqrt{\frac{0.10(1-0.10)}{387}}}=-2.660$

d. The p-value is ,

p-value=$2*P(Z>|Z_{stat}|)=2*P(Z>2.660)=2*(1-P(Z\leq 2.660))$

$=2*(1-\Phi(2.660))=2*(1-0.9961)=0.0078$ ; From standard normal distribution table

e. The p-value is less than

a = 0.10

f. Based on this , we should reject the null hypothesis.

g. Conclusion : The data suggest that the population proportion is significantly different from 10%. At the , so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is different from 10%.

a = 0.10