A chees type of cheese. The estimate must be within 0.71 milligram of the population mean. (a) Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.05 milligrams. (b) The sample mean is 25 milligrams. Using the minimum sample size with a 95% level of confidence, does it seem likely that the population mean could be within 3% of the sample mean? within 0.3% of the sample mean? Explain. Click here to view page 1 of the Standard Normal Table. Click here view page 2 of the Standard Normal Table. (a) The minimum sample size required to construct a 95% confidence interval is 71 servings. (Round up to the nearest whole number.) (b) The 95% confidence interval is (24.29-25.71). It does not seem likely that the population mean could be within 3% of the sample mean because the interval formed by the values 3% away from the sample-mean overlaps but does not entirely contain the confidence interval. It population mean could be within 0.3% of the sample mean because the interval formed by the values 0.3% seems seem likely that the entirely contains the confidence interval. away from the sample mean donimal places as needed.)
A chees type of cheese. The estimate must be within 0.71 milligram of the population mean. (a) Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.05 milligrams. (b) The sample mean is 25 milligrams. Using the minimum sample size with a 95% level of confidence, does it seem likely that the population mean could be within 3% of the sample mean? within 0.3% of the sample mean? Explain. Click here to view page 1 of the Standard Normal Table. Click here view page 2 of the Standard Normal Table. (a) The minimum sample size required to construct a 95% confidence interval is 71 servings. (Round up to the nearest whole number.) (b) The 95% confidence interval is (24.29-25.71). It does not seem likely that the population mean could be within 3% of the sample mean because the interval formed by the values 3% away from the sample-mean overlaps but does not entirely contain the confidence interval. It population mean could be within 0.3% of the sample mean because the interval formed by the values 0.3% seems seem likely that the entirely contains the confidence interval. away from the sample mean donimal places as needed.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Problem 1P
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![### Estimating the Mean Cholesterol Content for Cheese
A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.71 milligrams of the population mean.
**(a) Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.05 milligrams.**
**(b) The sample mean is 25 milligrams. Using the minimum sample size with a 95% level of confidence, does it seem likely that the population mean could be within 3% of the sample mean? Within 0.3% of the sample mean? Explain.**
*Click here to view page 1 of the Standard Normal Table.*
*Click here to view page 2 of the Standard Normal Table.*
---
**Solutions:**
**(a) The minimum sample size required to construct a 95% confidence interval is 71 servings.**
(Round up to the nearest whole number.)
**(b) The 95% confidence interval is [24.29, 25.71].**
- It **does not seem** likely that the population mean could be within 3% of the sample mean because the interval formed by the values 3% away from the sample mean overlaps but does not entirely contain the confidence interval.
- It **seems** likely that the population mean could be within 0.3% of the sample mean because the interval formed by the values 0.3% away from the sample mean entirely contains the confidence interval.
*(Round to two decimal places as needed.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F952bcd3e-4a7c-474a-8dd6-ddd08471460a%2Fd95d6db4-0e00-486b-86b6-ad9aa746484f%2F539rwhd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Estimating the Mean Cholesterol Content for Cheese
A cheese processing company wants to estimate the mean cholesterol content of all one-ounce servings of a type of cheese. The estimate must be within 0.71 milligrams of the population mean.
**(a) Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 3.05 milligrams.**
**(b) The sample mean is 25 milligrams. Using the minimum sample size with a 95% level of confidence, does it seem likely that the population mean could be within 3% of the sample mean? Within 0.3% of the sample mean? Explain.**
*Click here to view page 1 of the Standard Normal Table.*
*Click here to view page 2 of the Standard Normal Table.*
---
**Solutions:**
**(a) The minimum sample size required to construct a 95% confidence interval is 71 servings.**
(Round up to the nearest whole number.)
**(b) The 95% confidence interval is [24.29, 25.71].**
- It **does not seem** likely that the population mean could be within 3% of the sample mean because the interval formed by the values 3% away from the sample mean overlaps but does not entirely contain the confidence interval.
- It **seems** likely that the population mean could be within 0.3% of the sample mean because the interval formed by the values 0.3% away from the sample mean entirely contains the confidence interval.
*(Round to two decimal places as needed.)*
- [Click here to view page 2 of the Standard Normal Table](#)
---
### Solution:
#### (a) Minimum Sample Size Calculation
The minimum sample size \( n \) required to construct a 95% confidence interval is:
\[ n = \frac{(Z_{\alpha/2} \cdot \sigma)^2}{E^2} = 71 \]
Where:
- \( Z_{\alpha/2} \) is the critical value from the Z-distribution
- \( \sigma \) is the population standard deviation (3.05 milligrams)
- \( E \) is the margin of error (0.71 milligrams)
*Rounded up to the nearest whole number.*
#### (b) Confidence Interval and Population Mean Proximity
The 95% confidence interval for the population mean, using the sample mean of 25 milligrams, is:
\[ (24.29, 25.71) \]
It **does not seem** likely that the population mean could be within 3% of the sample mean because the interval overlaps but does not entirely contain the confidence bound.
For a more detailed analysis, consider the values within 0.3% of the sample mean and whether these values fall within the calculated confidence interval of (24.29, 25.71). This interval is used to imply that the population mean is not within 0.3% range of 25 milligrams.
*Round to two decimal places as needed.*
The above analysis helps in determining the feasibility and statistical significance of the sample size and confidence intervals](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F952bcd3e-4a7c-474a-8dd6-ddd08471460a%2Fd95d6db4-0e00-486b-86b6-ad9aa746484f%2Fqvpv8m9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Estimating the Mean Cholesterol Content in Cheese
A cheese processing company aims to estimate the mean cholesterol content of all one-ounce servings of a particular type of cheese. The estimate must be within 0.71 milligrams of the population mean.
#### Problem Statement:
1. **Determine the minimum sample size required to construct a 95% confidence interval for the population mean.** Assume the population standard deviation is 3.05 milligrams.
2. **Given a sample mean of 25 milligrams, use the minimum sample size with a 95% level of confidence to determine if it is likely that the population mean could be within 3% of the sample mean.** Is it within 0.3% of the sample mean?
To assist in calculations, refer to the Standard Normal Table:
- [Click here to view page 1 of the Standard Normal Table](#)
- [Click here to view page 2 of the Standard Normal Table](#)
---
### Solution:
#### (a) Minimum Sample Size Calculation
The minimum sample size \( n \) required to construct a 95% confidence interval is:
\[ n = \frac{(Z_{\alpha/2} \cdot \sigma)^2}{E^2} = 71 \]
Where:
- \( Z_{\alpha/2} \) is the critical value from the Z-distribution
- \( \sigma \) is the population standard deviation (3.05 milligrams)
- \( E \) is the margin of error (0.71 milligrams)
*Rounded up to the nearest whole number.*
#### (b) Confidence Interval and Population Mean Proximity
The 95% confidence interval for the population mean, using the sample mean of 25 milligrams, is:
\[ (24.29, 25.71) \]
It **does not seem** likely that the population mean could be within 3% of the sample mean because the interval overlaps but does not entirely contain the confidence bound.
For a more detailed analysis, consider the values within 0.3% of the sample mean and whether these values fall within the calculated confidence interval of (24.29, 25.71). This interval is used to imply that the population mean is not within 0.3% range of 25 milligrams.
*Round to two decimal places as needed.*
The above analysis helps in determining the feasibility and statistical significance of the sample size and confidence intervals
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