Here are summary statistics for randomly selected weights of newborn girls: n= 191, x 26.4 hg, s = 7.2 hg. Construct a confidence interval estimate of the mean. Use a 99% confidence level. Are these results very different from the confidence interval 24.4 hg
Here are summary statistics for randomly selected weights of newborn girls: n= 191, x 26.4 hg, s = 7.2 hg. Construct a confidence interval estimate of the mean. Use a 99% confidence level. Are these results very different from the confidence interval 24.4 hg
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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![### Confidence Interval for Population Mean
**Problem Statement:**
Here are summary statistics for randomly selected weights of newborn girls:
- Sample size (n) = 191
- Sample mean (x̄) = 26.4 hg
- Sample standard deviation (s) = 7.2 hg
Construct a confidence interval estimate of the mean at a 99% confidence level. Determine if these results are significantly different from the confidence interval 24.4 hg < μ < 28.0 hg with only 18 sample values, x̄ = 26.2 hg, and s = 2.7 hg.
**Question:**
What is the confidence interval for the population mean (μ)?
\[ \text{__ hg} < \mu < \text{__ hg} \]
(Round to one decimal place as needed.)
### Solution:
We use the formula for the confidence interval for a population mean when the standard deviation is unknown:
\[ \text{Confidence Interval} = x̄ \pm t \frac{s}{\sqrt{n}} \]
Where:
- \( x̄ \) is the sample mean
- \( t \) is the t-score for the 99% confidence level and degrees of freedom (\( df = n-1 \))
- \( s \) is the sample standard deviation
- \( n \) is the sample size
Steps:
1. **Determine the t-score:**
Given the degrees of freedom \( df = 191 - 1 = 190 \), and the confidence level of 99%, the t-score can be found using a t-distribution table or statistical software.
2. **Calculations:**
- Sample mean \( x̄ = 26.4 \) hg
- Sample standard deviation \( s = 7.2 \) hg
- Sample size \( n = 191 \)
### Graphical Representation:
In this context, a detailed explanation of the confidence interval calculation is performed, often represented using error bars on a graph showing the sample mean and the confidence interval margins. Here is the breakdown:
**Formula Application:**
\[
\text{Margin of Error (E)} = t \times \frac{s}{\sqrt{n}}
\]
By finding the exact t-score (typically using software or t-distribution tables), input the values and solve for the confidence interval range. Round final answers to one decimal place for consistency](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11fb6ed7-9250-48c6-8339-b5d076b64c81%2F1a6bf5db-8196-4303-8b6e-96262ab23b2f%2Ffcusx7n.jpeg&w=3840&q=75)
Transcribed Image Text:### Confidence Interval for Population Mean
**Problem Statement:**
Here are summary statistics for randomly selected weights of newborn girls:
- Sample size (n) = 191
- Sample mean (x̄) = 26.4 hg
- Sample standard deviation (s) = 7.2 hg
Construct a confidence interval estimate of the mean at a 99% confidence level. Determine if these results are significantly different from the confidence interval 24.4 hg < μ < 28.0 hg with only 18 sample values, x̄ = 26.2 hg, and s = 2.7 hg.
**Question:**
What is the confidence interval for the population mean (μ)?
\[ \text{__ hg} < \mu < \text{__ hg} \]
(Round to one decimal place as needed.)
### Solution:
We use the formula for the confidence interval for a population mean when the standard deviation is unknown:
\[ \text{Confidence Interval} = x̄ \pm t \frac{s}{\sqrt{n}} \]
Where:
- \( x̄ \) is the sample mean
- \( t \) is the t-score for the 99% confidence level and degrees of freedom (\( df = n-1 \))
- \( s \) is the sample standard deviation
- \( n \) is the sample size
Steps:
1. **Determine the t-score:**
Given the degrees of freedom \( df = 191 - 1 = 190 \), and the confidence level of 99%, the t-score can be found using a t-distribution table or statistical software.
2. **Calculations:**
- Sample mean \( x̄ = 26.4 \) hg
- Sample standard deviation \( s = 7.2 \) hg
- Sample size \( n = 191 \)
### Graphical Representation:
In this context, a detailed explanation of the confidence interval calculation is performed, often represented using error bars on a graph showing the sample mean and the confidence interval margins. Here is the breakdown:
**Formula Application:**
\[
\text{Margin of Error (E)} = t \times \frac{s}{\sqrt{n}}
\]
By finding the exact t-score (typically using software or t-distribution tables), input the values and solve for the confidence interval range. Round final answers to one decimal place for consistency
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