Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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ISBN: 9780079039897
Author: Carter
Publisher: McGraw Hill
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![**Statistical Analysis: Confidence Intervals**
**Problem Statement:**
8. A sample of 16 gas stations shows that the mean price for regular gasoline is \( \bar{x} = \$3.25 \). It is known that the standard deviation is \( \sigma = \$0.08 \).
a. Find a 95% confidence interval.
b. If we had used a 99% confidence interval, what effect, if any, would this have on the length of the confidence interval?
**Solution:**
**Part a: Finding a 95% Confidence Interval**
To find the 95% confidence interval for the mean price of regular gasoline, we use the formula for the confidence interval of the mean for a sample:
\[ CI = \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \]
Where:
- \( \bar{x} \) is the sample mean
- \( Z \) is the Z-value that corresponds to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Given:
- \( \bar{x} = \$3.25 \)
- \( \sigma = \$0.08 \)
- \( n = 16 \)
For a 95% confidence level, the Z-value is 1.96.
So, we calculate:
\[ CI = 3.25 \pm 1.96 \left( \frac{0.08}{\sqrt{16}} \right) \]
\[ CI = 3.25 \pm 1.96 \left( \frac{0.08}{4} \right) \]
\[ CI = 3.25 \pm 1.96 \left( 0.02 \right) \]
\[ CI = 3.25 \pm 0.0392 \]
Thus, the 95% confidence interval is:
\[ (\$3.2108, \$3.2892) \]
**Part b: Effect of Using a 99% Confidence Interval**
If we use a 99% confidence interval, the Z-value would change. For a 99% confidence interval, the Z-value is approximately 2.576.
The formula to recalculate the confidence interval with a 99% confidence level would be:
\[ CI = 3.25 \pm 2.576 \](https://content.bartleby.com/qna-images/question/0622a695-96c6-4977-a250-08b177cfa278/5c6d09e5-a702-4e04-8815-4e563b7ccd84/125yohq_processed.jpeg)
Transcribed Image Text:**Statistical Analysis: Confidence Intervals**
**Problem Statement:**
8. A sample of 16 gas stations shows that the mean price for regular gasoline is \( \bar{x} = \$3.25 \). It is known that the standard deviation is \( \sigma = \$0.08 \).
a. Find a 95% confidence interval.
b. If we had used a 99% confidence interval, what effect, if any, would this have on the length of the confidence interval?
**Solution:**
**Part a: Finding a 95% Confidence Interval**
To find the 95% confidence interval for the mean price of regular gasoline, we use the formula for the confidence interval of the mean for a sample:
\[ CI = \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \]
Where:
- \( \bar{x} \) is the sample mean
- \( Z \) is the Z-value that corresponds to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Given:
- \( \bar{x} = \$3.25 \)
- \( \sigma = \$0.08 \)
- \( n = 16 \)
For a 95% confidence level, the Z-value is 1.96.
So, we calculate:
\[ CI = 3.25 \pm 1.96 \left( \frac{0.08}{\sqrt{16}} \right) \]
\[ CI = 3.25 \pm 1.96 \left( \frac{0.08}{4} \right) \]
\[ CI = 3.25 \pm 1.96 \left( 0.02 \right) \]
\[ CI = 3.25 \pm 0.0392 \]
Thus, the 95% confidence interval is:
\[ (\$3.2108, \$3.2892) \]
**Part b: Effect of Using a 99% Confidence Interval**
If we use a 99% confidence interval, the Z-value would change. For a 99% confidence interval, the Z-value is approximately 2.576.
The formula to recalculate the confidence interval with a 99% confidence level would be:
\[ CI = 3.25 \pm 2.576 \
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