Construct the indicated confidence interval for the population mean u using the t-distribution. Assume the population is normally distributed. c= 0.99, x 13.8, s=4.0, n=7 The 99% confidence interval using a t-distribution is (OD. (Round to one decimal place as needed.)
Construct the indicated confidence interval for the population mean u using the t-distribution. Assume the population is normally distributed. c= 0.99, x 13.8, s=4.0, n=7 The 99% confidence interval using a t-distribution is (OD. (Round to one decimal place as needed.)
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 Confidence Interval for a Population Mean**
When given the task to construct a confidence interval for the population mean \(\mu\) using the t-distribution, it's essential to assume the population is normally distributed. Here's how to construct a 99% confidence interval with the given data:
### Given Values:
- Confidence level (\(c\)): 0.99
- Sample mean (\(\bar{x}\)): 13.8
- Sample standard deviation (\(s\)): 4.0
- Sample size (\(n\)): 7
### Objective:
To find the 99% confidence interval using a t-distribution.
### Steps to Construct the Interval:
1. **Identify the t-score:**
- Use a t-distribution table or calculator to find the t-score that corresponds to a 99% confidence level with \(n-1\) degrees of freedom (in this case, 6 degrees of freedom).
2. **Calculate the Standard Error (SE):**
\[ SE = \frac{s}{\sqrt{n}} = \frac{4.0}{\sqrt{7}} \]
3. **Construct the Confidence Interval:**
\[
\text{Confidence Interval} = \bar{x} \pm (t \times SE)
\]
Where \(t\) is the t-score obtained from the t-distribution.
4. **Round to One Decimal Place:**
- Ensure the final confidence interval is rounded to one decimal place as specified.
The final confidence interval will look like \(( \_\_.\_ , \_\_.\_ )\), providing a range within which the population mean is likely to fall with 99% confidence.
### Note:
- It is critical to use the appropriate t-score and SE to ensure accuracy.
- Computers or statistical software can help automate these calculations and minimize errors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6158d43-334d-4161-8a0f-ff9657025746%2Ffdbb4f64-d685-423b-b9bf-76b98dede699%2Fpa25ba_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Constructing a Confidence Interval for a Population Mean**
When given the task to construct a confidence interval for the population mean \(\mu\) using the t-distribution, it's essential to assume the population is normally distributed. Here's how to construct a 99% confidence interval with the given data:
### Given Values:
- Confidence level (\(c\)): 0.99
- Sample mean (\(\bar{x}\)): 13.8
- Sample standard deviation (\(s\)): 4.0
- Sample size (\(n\)): 7
### Objective:
To find the 99% confidence interval using a t-distribution.
### Steps to Construct the Interval:
1. **Identify the t-score:**
- Use a t-distribution table or calculator to find the t-score that corresponds to a 99% confidence level with \(n-1\) degrees of freedom (in this case, 6 degrees of freedom).
2. **Calculate the Standard Error (SE):**
\[ SE = \frac{s}{\sqrt{n}} = \frac{4.0}{\sqrt{7}} \]
3. **Construct the Confidence Interval:**
\[
\text{Confidence Interval} = \bar{x} \pm (t \times SE)
\]
Where \(t\) is the t-score obtained from the t-distribution.
4. **Round to One Decimal Place:**
- Ensure the final confidence interval is rounded to one decimal place as specified.
The final confidence interval will look like \(( \_\_.\_ , \_\_.\_ )\), providing a range within which the population mean is likely to fall with 99% confidence.
### Note:
- It is critical to use the appropriate t-score and SE to ensure accuracy.
- Computers or statistical software can help automate these calculations and minimize errors.
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