A magazine conducts an annual survey in which readers rate their favorite cruise ship. All ships are rated on a 100-point scale, with higher values indicating better service. A sample of 36 ships that carry fewer than 500 passengers resulted in an average rating of 85.11, and a sample of 44 ships that carry 500 or more passengers provided an average rating of 81.40. Assume that the population standard deviation is 4.55 for ships that carry fewer than 500 passengers and 3.97 for ships that carry 500 or more passengers. (a) What is the point estimate of the difference between the population mean rating for ships that carry fewer than 500 passengers and the population mean rating for ships that carry 500 or more passengers? (Use smaller cruise ships - larger cruise ships.) (b) At 95% confidence, what is the margin of error? (Round your answer to two decimal places.) (c) What is a 95% confidence interval estimate of the difference between the population mean ratings for the two sizes of ships? (Use smaller cruise ships- larger cruise ships. Round your answers to two decimal places.) to
A magazine conducts an annual survey in which readers rate their favorite cruise ship. All ships are rated on a 100-point scale, with higher values indicating better service. A sample of 36 ships that carry fewer than 500 passengers resulted in an average rating of 85.11, and a sample of 44 ships that carry 500 or more passengers provided an average rating of 81.40. Assume that the population standard deviation is 4.55 for ships that carry fewer than 500 passengers and 3.97 for ships that carry 500 or more passengers. (a) What is the point estimate of the difference between the population mean rating for ships that carry fewer than 500 passengers and the population mean rating for ships that carry 500 or more passengers? (Use smaller cruise ships - larger cruise ships.) (b) At 95% confidence, what is the margin of error? (Round your answer to two decimal places.) (c) What is a 95% confidence interval estimate of the difference between the population mean ratings for the two sizes of ships? (Use smaller cruise ships- larger cruise ships. Round your answers to two decimal places.) to
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Annual Cruise Ship Rating Survey Analysis
A magazine conducts an annual survey in which readers rate their favorite cruise ships. All ships are rated on a 100-point scale, with higher values indicating better service. Below is the information from the survey results:
- **Smaller cruise ships (fewer than 500 passengers)**
- Sample size: 36 ships
- Average rating (mean): 85.11
- Population standard deviation: 4.55
- **Larger cruise ships (500 or more passengers)**
- Sample size: 44 ships
- Average rating (mean): 81.40
- Population standard deviation: 3.97
Using this data, we can address the following questions:
**(a) What is the point estimate of the difference between the population mean rating for ships that carry fewer than 500 passengers and the population mean rating for ships that carry 500 or more passengers?**
(Use smaller cruise ships – larger cruise ships.)
\[ \text{Point estimate} = 85.11 - 81.40 = 3.71 \]
**(b) At 95% confidence, what is the margin of error?**
(Round your answer to two decimal places.)
Assuming normal distribution and using the formula for the margin of error:
\[ ME = Z * \sqrt{\left(\frac{\sigma_1^2}{n_1}\right) + \left(\frac{\sigma_2^2}{n_2}\right)} \]
Where:
- \(Z\) is the Z-value corresponding to 95% confidence (1.96)
- \(\sigma_1\) and \(\sigma_2\) are the standard deviations
- \( n_1 \) and \( n_2 \) are the sample sizes
\[ ME = 1.96 * \sqrt{\left(\frac{4.55^2}{36}\right) + \left(\frac{3.97^2}{44}\right)} \]
**(c) What is a 95% confidence interval estimate of the difference between the population mean ratings for the two sizes of ships?**
(Use smaller cruise ships – larger cruise ships. Round your answers to two decimal places.)
The confidence interval is given by:
\[ \text{Point Estimate} \pm \text{Margin of Error} \]
\[CI = 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98fd5880-bd7f-4b90-9f43-35abbd9a1544%2F0d0a5c56-ac0b-44c2-b464-0e9581a8e238%2Fhtlbh0f_processed.png&w=3840&q=75)
Transcribed Image Text:### Annual Cruise Ship Rating Survey Analysis
A magazine conducts an annual survey in which readers rate their favorite cruise ships. All ships are rated on a 100-point scale, with higher values indicating better service. Below is the information from the survey results:
- **Smaller cruise ships (fewer than 500 passengers)**
- Sample size: 36 ships
- Average rating (mean): 85.11
- Population standard deviation: 4.55
- **Larger cruise ships (500 or more passengers)**
- Sample size: 44 ships
- Average rating (mean): 81.40
- Population standard deviation: 3.97
Using this data, we can address the following questions:
**(a) What is the point estimate of the difference between the population mean rating for ships that carry fewer than 500 passengers and the population mean rating for ships that carry 500 or more passengers?**
(Use smaller cruise ships – larger cruise ships.)
\[ \text{Point estimate} = 85.11 - 81.40 = 3.71 \]
**(b) At 95% confidence, what is the margin of error?**
(Round your answer to two decimal places.)
Assuming normal distribution and using the formula for the margin of error:
\[ ME = Z * \sqrt{\left(\frac{\sigma_1^2}{n_1}\right) + \left(\frac{\sigma_2^2}{n_2}\right)} \]
Where:
- \(Z\) is the Z-value corresponding to 95% confidence (1.96)
- \(\sigma_1\) and \(\sigma_2\) are the standard deviations
- \( n_1 \) and \( n_2 \) are the sample sizes
\[ ME = 1.96 * \sqrt{\left(\frac{4.55^2}{36}\right) + \left(\frac{3.97^2}{44}\right)} \]
**(c) What is a 95% confidence interval estimate of the difference between the population mean ratings for the two sizes of ships?**
(Use smaller cruise ships – larger cruise ships. Round your answers to two decimal places.)
The confidence interval is given by:
\[ \text{Point Estimate} \pm \text{Margin of Error} \]
\[CI = 3.
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