Ciis questIOIT Deluw If n=24, (x-bar)=30, and s=7, construct a confidence interval at a 98% confidence level. Assume the data came from a normally distributed population. Give your answers to one decimal place.
Ciis questIOIT Deluw If n=24, (x-bar)=30, and s=7, construct a confidence interval at a 98% confidence level. Assume the data came from a normally distributed population. Give your answers to one decimal place.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
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![### Constructing a Confidence Interval
Given the problem:
- Sample size (\( n \)) = 24
- Sample mean (\( \bar{x} \)) = 30
- Sample standard deviation (\( s \)) = 7
We are asked to construct a confidence interval at a 98% confidence level, assuming the data comes from a normally distributed population.
### Solution Steps:
1. **Identify the Parameters:**
- We need to calculate the confidence interval for the mean (\( \mu \)).
2. **Determine the Appropriate Formula:**
Since the population standard deviation is unknown and \( n < 30 \), we use the t-distribution:
\[
\bar{x} \pm t \left(\frac{s}{\sqrt{n}}\right)
\]
Where:
- \( t \) is the t-score that corresponds to the desired confidence level and degrees of freedom (\( df = n-1 \)).
3. **Calculate the t-score:**
Look up the t-score for a 98% confidence level and 23 degrees of freedom.
4. **Calculate the Margin of Error:**
\[
t \left(\frac{7}{\sqrt{24}}\right)
\]
5. **Construct the Confidence Interval:**
Place the result into:
\[
30 \pm \text{Margin of Error}
\]
### Diagrams or Graphs:
The image contains several handwritten notes and calculations, primarily numbers and operations, possibly relating to the determination of the degrees of freedom or critical t-value. However, there are no explicit graphs or diagrams detailing the calculations further.
---
This transcription can be used for educational purposes to understand how to calculate a confidence interval using sample means, sample sizes, and the t-distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1d46466-3ecc-4254-a208-a4cd9fff6ffc%2F8049eda6-0c0e-4beb-ad27-ea81470db7af%2Fsallekq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a Confidence Interval
Given the problem:
- Sample size (\( n \)) = 24
- Sample mean (\( \bar{x} \)) = 30
- Sample standard deviation (\( s \)) = 7
We are asked to construct a confidence interval at a 98% confidence level, assuming the data comes from a normally distributed population.
### Solution Steps:
1. **Identify the Parameters:**
- We need to calculate the confidence interval for the mean (\( \mu \)).
2. **Determine the Appropriate Formula:**
Since the population standard deviation is unknown and \( n < 30 \), we use the t-distribution:
\[
\bar{x} \pm t \left(\frac{s}{\sqrt{n}}\right)
\]
Where:
- \( t \) is the t-score that corresponds to the desired confidence level and degrees of freedom (\( df = n-1 \)).
3. **Calculate the t-score:**
Look up the t-score for a 98% confidence level and 23 degrees of freedom.
4. **Calculate the Margin of Error:**
\[
t \left(\frac{7}{\sqrt{24}}\right)
\]
5. **Construct the Confidence Interval:**
Place the result into:
\[
30 \pm \text{Margin of Error}
\]
### Diagrams or Graphs:
The image contains several handwritten notes and calculations, primarily numbers and operations, possibly relating to the determination of the degrees of freedom or critical t-value. However, there are no explicit graphs or diagrams detailing the calculations further.
---
This transcription can be used for educational purposes to understand how to calculate a confidence interval using sample means, sample sizes, and the t-distribution.
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