19% of all college students volunteer their time. Is the percentage of college students who are volunteers smaller for students receiving financial aid? Of the 362 randomly selected students who receive financial aid, 47 of them volunteered their time. What can be concluded at the a = 0.01 level of significance? a. For this study, we should use z-test for a population proportion) b. The null and alternative hypotheses would be: Ho: P (please enter a decimal) H₁: po (Please enter a decimal) c. The test statistic (z✓ (please show your answer to 3 decimal places.) d. The p-value = answer to 4 decimal places.) (Please show your
19% of all college students volunteer their time. Is the percentage of college students who are volunteers smaller for students receiving financial aid? Of the 362 randomly selected students who receive financial aid, 47 of them volunteered their time. What can be concluded at the a = 0.01 level of significance? a. For this study, we should use z-test for a population proportion) b. The null and alternative hypotheses would be: Ho: P (please enter a decimal) H₁: po (Please enter a decimal) c. The test statistic (z✓ (please show your answer to 3 decimal places.) d. The p-value = answer to 4 decimal places.) (Please show your
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![### Hypothesis Testing for Proportion of College Student Volunteers
**Question:**
19% of all college students volunteer their time. Is the percentage of college students who are volunteers smaller for students receiving financial aid? Of the 362 randomly selected students who receive financial aid, 47 of them volunteered their time. What can be concluded at the \(\alpha = 0.01\) level of significance?
#### Solution:
**a. Type of Test:**
For this study, we should use a **z-test for a population proportion**.
**b. Hypotheses:**
The null and alternative hypotheses are:
\[ H_0: p = 0.19 \]
\[ H_A: p < 0.19 \]
**c. Test Statistic:**
The z-test statistic is calculated as follows:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]
Where:
- \(\hat{p}\) is the sample proportion.
- \(p_0\) is the null hypothesis population proportion.
- \(n\) is the sample size.
(Please show your answer to 3 decimal places.)
**d. P-value:**
The p-value corresponding to the test statistic.
(Please show your answer to 4 decimal places.)
**e. Decision Rule:**
If the p-value \(\leq \alpha\), we reject the null hypothesis.
**f. Conclusion:**
Based on this, we should **reject** the null hypothesis.
**g. Final Conclusion:**
Select the appropriate statement based on the p-value and conclusion from the decision rule:
- The data suggest the population proportion is not significantly lower than 19% at \(\alpha = 0.01\), so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is equal to 19%.
- The data suggest the population proportion **is** significantly lower than 19% at \(\alpha = 0.01\), so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is less than 19%.
**Note:**
Make sure to calculate the test statistic \(z\) and corresponding p-value accurately from the given data before making the final conclusion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2d30e49-de0d-418d-b702-a4b3e53b4c91%2F5cc7b7c6-d146-4b35-9bf0-4aa3eceba0bb%2Fkttb6rf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing for Proportion of College Student Volunteers
**Question:**
19% of all college students volunteer their time. Is the percentage of college students who are volunteers smaller for students receiving financial aid? Of the 362 randomly selected students who receive financial aid, 47 of them volunteered their time. What can be concluded at the \(\alpha = 0.01\) level of significance?
#### Solution:
**a. Type of Test:**
For this study, we should use a **z-test for a population proportion**.
**b. Hypotheses:**
The null and alternative hypotheses are:
\[ H_0: p = 0.19 \]
\[ H_A: p < 0.19 \]
**c. Test Statistic:**
The z-test statistic is calculated as follows:
\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]
Where:
- \(\hat{p}\) is the sample proportion.
- \(p_0\) is the null hypothesis population proportion.
- \(n\) is the sample size.
(Please show your answer to 3 decimal places.)
**d. P-value:**
The p-value corresponding to the test statistic.
(Please show your answer to 4 decimal places.)
**e. Decision Rule:**
If the p-value \(\leq \alpha\), we reject the null hypothesis.
**f. Conclusion:**
Based on this, we should **reject** the null hypothesis.
**g. Final Conclusion:**
Select the appropriate statement based on the p-value and conclusion from the decision rule:
- The data suggest the population proportion is not significantly lower than 19% at \(\alpha = 0.01\), so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is equal to 19%.
- The data suggest the population proportion **is** significantly lower than 19% at \(\alpha = 0.01\), so there is sufficient evidence to conclude that the percentage of financial aid recipients who volunteer is less than 19%.
**Note:**
Make sure to calculate the test statistic \(z\) and corresponding p-value accurately from the given data before making the final conclusion.
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