In a random sample of seven people, the mean driving distance to work was 24.9 miles and the standard deviation was 7.1 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean μ. Interpret the results. ...
In a random sample of seven people, the mean driving distance to work was 24.9 miles and the standard deviation was 7.1 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean μ. Interpret the results. ...
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 95% Confidence Interval for the Population Mean
**Scenario:**
In a random sample of seven people, the mean driving distance to work was 24.9 miles and the standard deviation was 7.1 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean \( \mu \). Interpret the results.
**Steps and Results:**
1. **Identify the Margin of Error:**
The margin of error has been identified as **6.6 miles** (rounded to one decimal place as needed).
2. **Construct a 95% Confidence Interval for the Population Mean:**
The 95% confidence interval for the population mean has been calculated as **\( (18.3, 31.5) \)** miles (rounded to one decimal place as needed).
3. **Interpret the Results:**
You are given two options to interpret the results. The correct choice should be filled in with the appropriate values:
- **Option A:** It can be said that **95**% of the population has a driving distance to work (in miles) that is between the interval’s endpoints.
- **Option B:** With **95**% confidence, it can be said that the population mean driving distance to work (in miles) is between the interval’s endpoints.
**Correct Interpretation (Option B):** With 95% confidence, it can be said that the population mean driving distance to work (in miles) is between 18.3 and 31.5 miles.
### Explanation of Find Margin of Error and Confidence Interval:
1. **Calculation of Margin of Error (E):**
\[
E = t_{(\alpha/2, df)} \times \left(\frac{s}{\sqrt{n}}\right)
\]
- Where \( t_{(\alpha/2, df)} \) is the t-critical value from the t-distribution table with \( \alpha/2 \) and \( df = n - 1 \).
- \( s \) is the sample standard deviation (7.1 miles).
- \( n \) is the sample size (7).
2. **Constructing Confidence Interval:**
\[
\text{Confidence Interval} = \left( \bar{x} - E, \bar{x} + E \right)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F952bcd3e-4a7c-474a-8dd6-ddd08471460a%2F1a46b8da-0e8f-4b2e-86fb-da884bb495f3%2Fupghpr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Constructing a 95% Confidence Interval for the Population Mean
**Scenario:**
In a random sample of seven people, the mean driving distance to work was 24.9 miles and the standard deviation was 7.1 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean \( \mu \). Interpret the results.
**Steps and Results:**
1. **Identify the Margin of Error:**
The margin of error has been identified as **6.6 miles** (rounded to one decimal place as needed).
2. **Construct a 95% Confidence Interval for the Population Mean:**
The 95% confidence interval for the population mean has been calculated as **\( (18.3, 31.5) \)** miles (rounded to one decimal place as needed).
3. **Interpret the Results:**
You are given two options to interpret the results. The correct choice should be filled in with the appropriate values:
- **Option A:** It can be said that **95**% of the population has a driving distance to work (in miles) that is between the interval’s endpoints.
- **Option B:** With **95**% confidence, it can be said that the population mean driving distance to work (in miles) is between the interval’s endpoints.
**Correct Interpretation (Option B):** With 95% confidence, it can be said that the population mean driving distance to work (in miles) is between 18.3 and 31.5 miles.
### Explanation of Find Margin of Error and Confidence Interval:
1. **Calculation of Margin of Error (E):**
\[
E = t_{(\alpha/2, df)} \times \left(\frac{s}{\sqrt{n}}\right)
\]
- Where \( t_{(\alpha/2, df)} \) is the t-critical value from the t-distribution table with \( \alpha/2 \) and \( df = n - 1 \).
- \( s \) is the sample standard deviation (7.1 miles).
- \( n \) is the sample size (7).
2. **Constructing Confidence Interval:**
\[
\text{Confidence Interval} = \left( \bar{x} - E, \bar{x} + E \right)
![### Confidence Interval for Population Mean Driving Distance
In a random sample of seven people, the mean driving distance to work was 24.9 miles and the standard deviation was 7.1 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean μ. Interpret the results.
#### Identify the margin of error:
- **Margin of Error:** 6.6 (rounded to one decimal place as needed)
#### Construct the 95% confidence interval for the population mean:
- **Confidence Interval:** 18.3, 31.5 (rounded to one decimal place as needed)
#### Interpretation of the results:
Choose the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.)
- **Option A**: With 95% confidence, it can be said that the population mean driving distance to work (in miles) is between the interval’s endpoints.
- **Option B**: With 95% confidence, it can be said that the population mean driving distance to work (in miles) is \[Answer Box\].
**Note:** To provide a complete and accurate response, users must select the appropriate option and fill in any required calculations or values in the provided answer boxes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F952bcd3e-4a7c-474a-8dd6-ddd08471460a%2F1a46b8da-0e8f-4b2e-86fb-da884bb495f3%2Ffwrpae_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Confidence Interval for Population Mean Driving Distance
In a random sample of seven people, the mean driving distance to work was 24.9 miles and the standard deviation was 7.1 miles. Assume the population is normally distributed and use the t-distribution to find the margin of error and construct a 95% confidence interval for the population mean μ. Interpret the results.
#### Identify the margin of error:
- **Margin of Error:** 6.6 (rounded to one decimal place as needed)
#### Construct the 95% confidence interval for the population mean:
- **Confidence Interval:** 18.3, 31.5 (rounded to one decimal place as needed)
#### Interpretation of the results:
Choose the correct choice below and fill in the answer box to complete your choice. (Type an integer or a decimal. Do not round.)
- **Option A**: With 95% confidence, it can be said that the population mean driving distance to work (in miles) is between the interval’s endpoints.
- **Option B**: With 95% confidence, it can be said that the population mean driving distance to work (in miles) is \[Answer Box\].
**Note:** To provide a complete and accurate response, users must select the appropriate option and fill in any required calculations or values in the provided answer boxes.
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