Construct a 90% confidence interval of the mean birth weight of 195 boys with sample mean = 32.7 hg, and standard deviation = 6.6 hg. O 0.777
Construct a 90% confidence interval of the mean birth weight of 195 boys with sample mean = 32.7 hg, and standard deviation = 6.6 hg. O 0.777
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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![### Constructing a 90% Confidence Interval for Mean Birth Weight
The following exercise involves constructing a 90% confidence interval for the mean birth weight of 195 boys. The sample mean is 32.7 hg and the standard deviation is approximately 6.6 hg.
Given Data:
- **Sample Size (n)**: 195
- **Sample Mean (x̄)**: 32.7 hg
- **Standard Deviation (σ)**: 6.6 hg
### Confidence Interval Options
1. **Option A**: \( 0.777 < \text{mean} < 1 \)
2. **Option B**: \( 31.923 < \text{mean} < 33.477 \)
3. **Option C**: \( 32\% < \text{mean} < 33\% \)
4. **Option D**: \( 1.645 < \text{mean} < 1.777 \)
### Explanation of Options
- **Option A**: A very small range which is impractical given the sample mean of 32.7 hg.
- **Option B**: Provides a realistic range around the given sample mean.
- **Option C**: Represented as percentages which is not appropriate for this context.
- **Option D**: Another small range not feasible for the given sample statistics.
### Correct Configuration
To compute the correct confidence interval, use the formula for the confidence interval of the mean:
\[ \text{CI} = \bar{x} ± Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where Z is the Z-value for the desired confidence level (for 90%, Z ≈ 1.645).
Using the given data:
\[ \text{Margin of Error} = 1.645 \left(\frac{6.6}{\sqrt{195}}\right) \approx 0.777 \]
Hence, the 90% confidence interval would be:
\[ 32.7 ± 0.777 \]
Therefore, the interval is:
\[ 31.923 < \text{mean} < 33.477 \]
**Conclusion**: The correct answer is **Option B**: \( 31.923 < \text{mean} < 33.477 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ed366a-05c2-4728-8c6d-70ad10b89152%2F111cf501-1e87-4f3b-b91e-3f527016267b%2Fx90ixr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a 90% Confidence Interval for Mean Birth Weight
The following exercise involves constructing a 90% confidence interval for the mean birth weight of 195 boys. The sample mean is 32.7 hg and the standard deviation is approximately 6.6 hg.
Given Data:
- **Sample Size (n)**: 195
- **Sample Mean (x̄)**: 32.7 hg
- **Standard Deviation (σ)**: 6.6 hg
### Confidence Interval Options
1. **Option A**: \( 0.777 < \text{mean} < 1 \)
2. **Option B**: \( 31.923 < \text{mean} < 33.477 \)
3. **Option C**: \( 32\% < \text{mean} < 33\% \)
4. **Option D**: \( 1.645 < \text{mean} < 1.777 \)
### Explanation of Options
- **Option A**: A very small range which is impractical given the sample mean of 32.7 hg.
- **Option B**: Provides a realistic range around the given sample mean.
- **Option C**: Represented as percentages which is not appropriate for this context.
- **Option D**: Another small range not feasible for the given sample statistics.
### Correct Configuration
To compute the correct confidence interval, use the formula for the confidence interval of the mean:
\[ \text{CI} = \bar{x} ± Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where Z is the Z-value for the desired confidence level (for 90%, Z ≈ 1.645).
Using the given data:
\[ \text{Margin of Error} = 1.645 \left(\frac{6.6}{\sqrt{195}}\right) \approx 0.777 \]
Hence, the 90% confidence interval would be:
\[ 32.7 ± 0.777 \]
Therefore, the interval is:
\[ 31.923 < \text{mean} < 33.477 \]
**Conclusion**: The correct answer is **Option B**: \( 31.923 < \text{mean} < 33.477 \).
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