Answer ### 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 DistanceIn 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.

Answer Sample mean =24.9sample size =7standard deviation =7.1Confidence interval =95%

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)

### 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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