5. Is the response rate for questionnaires affected by including some sort of incentive to respond along with the questionnaire? In one experiment, 110 questionnaires with no incentive re- sulted in 75 being returned, whereas 98 questionnaires that included a chance to win a lottery yielded 66 responses. Test the claim at a = 0.05 that including an incentive will increase the likelihood of a response using classical approach. Please show all 4 steps clearly and interpret your conclusion.
5. Is the response rate for questionnaires affected by including some sort of incentive to respond along with the questionnaire? In one experiment, 110 questionnaires with no incentive re- sulted in 75 being returned, whereas 98 questionnaires that included a chance to win a lottery yielded 66 responses. Test the claim at a = 0.05 that including an incentive will increase the likelihood of a response using classical approach. Please show all 4 steps clearly and interpret your conclusion.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Can someone please help me to solve the following question showing all work.
![### Question 5
**Research Question:** Is the response rate for questionnaires affected by including some sort of incentive to respond along with the questionnaire?
#### Experimental Details:
- **Without Incentive:**
- Total questionnaires: 110
- Returned: 75
- **With Incentive (chance to win a lottery):**
- Total questionnaires: 98
- Returned: 66
#### Task:
Test the claim at a significance level of \(\alpha = 0.05\) that including an incentive will increase the likelihood of a response using the classical approach.
**Instructions:**
Please show all 4 steps clearly and interpret your conclusion.
### Solution Steps (Classical Approach):
1. **State the Hypotheses:**
- Null Hypothesis (\(H_0\)): There is no difference in the response rates with or without incentive.
- Alternative Hypothesis (\(H_A\)): The response rate with an incentive is higher than the response rate without incentive.
2. **Choose the Significance Level:**
- \(\alpha = 0.05\)
3. **Select the Appropriate Test Statistic:**
- Since we are testing for proportions, we use the Z-test for the difference between two proportions.
4. **Calculate the Test Statistic and P-Value:**
\[ \hat{p}_1 = \frac{75}{110} \approx 0.682 \]
\[ \hat{p}_2 = \frac{66}{98} \approx 0.673 \]
\[ \hat{p} = \frac{75 + 66}{110 + 98} \approx 0.678 \]
\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left( \frac{1}{110} + \frac{1}{98} \right)}} \]
Insert the respective values to find the Z-score.
5. **Decision Rule:**
- Compare the Z-score with the critical value from the Z-table at \(\alpha = 0.05\) for a one-tailed test.
6. **Conclusion:**
- If the calculated Z value exceeds the critical value, reject \(H_0\) and conclude that offering an incentive indeed increases the response rate.
**Note:** Steps 4 and 5 would require mathematical computation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F322ab9b5-8b73-42ca-a79a-5c7ee3611144%2Faafe8b0b-3ac4-45b5-9e9a-3368d8349b54%2F47kspzl9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 5
**Research Question:** Is the response rate for questionnaires affected by including some sort of incentive to respond along with the questionnaire?
#### Experimental Details:
- **Without Incentive:**
- Total questionnaires: 110
- Returned: 75
- **With Incentive (chance to win a lottery):**
- Total questionnaires: 98
- Returned: 66
#### Task:
Test the claim at a significance level of \(\alpha = 0.05\) that including an incentive will increase the likelihood of a response using the classical approach.
**Instructions:**
Please show all 4 steps clearly and interpret your conclusion.
### Solution Steps (Classical Approach):
1. **State the Hypotheses:**
- Null Hypothesis (\(H_0\)): There is no difference in the response rates with or without incentive.
- Alternative Hypothesis (\(H_A\)): The response rate with an incentive is higher than the response rate without incentive.
2. **Choose the Significance Level:**
- \(\alpha = 0.05\)
3. **Select the Appropriate Test Statistic:**
- Since we are testing for proportions, we use the Z-test for the difference between two proportions.
4. **Calculate the Test Statistic and P-Value:**
\[ \hat{p}_1 = \frac{75}{110} \approx 0.682 \]
\[ \hat{p}_2 = \frac{66}{98} \approx 0.673 \]
\[ \hat{p} = \frac{75 + 66}{110 + 98} \approx 0.678 \]
\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left( \frac{1}{110} + \frac{1}{98} \right)}} \]
Insert the respective values to find the Z-score.
5. **Decision Rule:**
- Compare the Z-score with the critical value from the Z-table at \(\alpha = 0.05\) for a one-tailed test.
6. **Conclusion:**
- If the calculated Z value exceeds the critical value, reject \(H_0\) and conclude that offering an incentive indeed increases the response rate.
**Note:** Steps 4 and 5 would require mathematical computation
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 18 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman