In a sample of 100 patients that did not receive the flu shot, the average recovery time after being infected with the flu was 7 days with a standard deviation of 4 days. What is a 95% confidence interval for the population average recovery time?
In a sample of 100 patients that did not receive the flu shot, the average recovery time after being infected with the flu was 7 days with a standard deviation of 4 days. What is a 95% confidence interval for the population average recovery time?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Question: 2 In a sample of 100 patients that did not receive the flu shot, the average recovery time after being infected with the flu was 7 days with a standard deviation of 4 days. What is a 95% confidence interval for the population average recovery time?
![**Question:**
In a sample of 100 patients that did not receive the flu shot, the average recovery time after being infected with the flu was 7 days with a standard deviation of 4 days. What is a 95% confidence interval for the population average recovery time?
**Options:**
- (6.97, 7.03)
- (5.97, 8.03)
- (6.21, 7.79)
- (6.22, 7.78)
**Explanation:**
This question involves calculating the 95% confidence interval for the population mean recovery time. We know the sample mean, sample size, and standard deviation, which allows us to use the formula for the confidence interval:
\[ \text{CI} = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \(\bar{x}\) is the sample mean (7 days)
- \(z\) is the z-score corresponding to the desired confidence level (approximately 1.96 for 95%)
- \(\sigma\) is the standard deviation (4 days)
- \(n\) is the sample size (100)
Substituting the values:
\[ \text{Margin of error} = 1.96 \times \left(\frac{4}{\sqrt{100}}\right) = 1.96 \times 0.4 = 0.784 \]
95% CI: \( 7 \pm 0.784 = (6.216, 7.784) \)
Thus, the correct option is (6.22, 7.78).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefa768ab-5fb9-4472-87b8-1839dd8202b8%2F040729e0-0fa1-40eb-9920-33e8bb7e86d7%2F2i7b0ha_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
In a sample of 100 patients that did not receive the flu shot, the average recovery time after being infected with the flu was 7 days with a standard deviation of 4 days. What is a 95% confidence interval for the population average recovery time?
**Options:**
- (6.97, 7.03)
- (5.97, 8.03)
- (6.21, 7.79)
- (6.22, 7.78)
**Explanation:**
This question involves calculating the 95% confidence interval for the population mean recovery time. We know the sample mean, sample size, and standard deviation, which allows us to use the formula for the confidence interval:
\[ \text{CI} = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \(\bar{x}\) is the sample mean (7 days)
- \(z\) is the z-score corresponding to the desired confidence level (approximately 1.96 for 95%)
- \(\sigma\) is the standard deviation (4 days)
- \(n\) is the sample size (100)
Substituting the values:
\[ \text{Margin of error} = 1.96 \times \left(\frac{4}{\sqrt{100}}\right) = 1.96 \times 0.4 = 0.784 \]
95% CI: \( 7 \pm 0.784 = (6.216, 7.784) \)
Thus, the correct option is (6.22, 7.78).
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