Question

A study investigates whether the percentage of college students who volunteer their time is smaller for students receiving financial aid. Out of 393 randomly selected students who receive financial aid, 31 of them volunteered their time. The significance level is set at 0.10. What can be concluded?

a. What type of statistical test should be used for this study?
b. What are the null and alternative hypotheses in terms of the population proportion (please enter decimal values)?
c. What is the test statistic (please show your answer to 3 decimal places)?
d. What is the p-value (please show your answer to 4 decimal places)?
e. Should the null hypothesis be rejected or not?
f. What is the final conclusion based on the results of the study?

Answer 1

The study should use a one-proportion z-test to determine if the proportion of college students on financial aid who volunteer their time is less than some established proportion. The null hypothesis is that the proportion of these students volunteering is equal to a specified proportion, while the alternative is that it is less. The test statistic and p-value can be calculated from the sample data, determining whether to reject the null hypothesis.

Step-by-step explanation:

To answer the student's question, we will break it down into parts:

Statistical Test

The type of statistical test that should be used for this study is a one-proportion z-test, which is used to determine whether the observed proportion of students volunteering is significantly different from a comparison proportion (in this case, implied to be the national average or another established proportion).

Null and Alternative Hypotheses

The null hypothesis (H0) is that the proportion of college students receiving financial aid who volunteer their time (π) is equal to (≥) the proportion of all college students who volunteer. Hence, H0: π = π_0. The alternative hypothesis (H1) is that the proportion is less than the comparison proportion, H1: π < π_0. We don't have the specific comparison proportion (π_0) which should be based on external data or research.

Test Statistic and P-Value

The test statistic for a one-proportion z-test is calculated using the formula: z = (p - π_0) / √(π_0(1-π_0)/n) where p is the sample proportion, π_0 is the null hypothesis proportion, and n is the sample size. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed statistic under the null hypothesis. The p-value would be calculated using this test statistic and a z table or statistical software.

Hypothesis Decision and Conclusion

To determine if the null hypothesis should be rejected, compare the p-value with the significance level (α). If the p-value is less than α, we reject the null hypothesis. Without the actual calculation provided in the problem, we cannot directly answer questions (c) to (f). However, the general conclusion after a hypothesis test would tell us whether there is evidence to suggest that the proportion of volunteering among students with financial aid is different from the specified proportion (π_0).