You take a random sample of 49 Toyota Tacomas. Their mean highway fuel efficiency is 20.4. From prior studies you know the population standard deviation is 0.8 mpg. The population appears to be normally distributed. If you construct a 90% confidence interval of the population mean, what is the margin of error?

To calculate the margin of error for a confidence interval, we use the formula:

\[
\text{Margin of Error} = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

Where:
- \( z_{\alpha/2} \) is the critical value for a 90% confidence interval (from the standard normal distribution),
- \( \sigma \) is the population standard deviation (0.8 mpg),
- \( n \) is the sample size (49),
- \( \frac{\sigma}{\sqrt{n}} \) is the standard error of the mean.

### Step-by-step:

1. **Determine the critical value** \( z_{\alpha/2} \) for a 90% confidence interval.
   - For a 90% confidence interval, the critical value \( z_{\alpha/2} \) is 1.645.

2. **Calculate the standard error** \( \frac{\sigma}{\sqrt{n}} \):
   \[
   \frac{0.8}{\sqrt{49}} = \frac{0.8}{7} = 0.1143
   \]

3. **Calculate the margin of error**:
   \[
   \text{Margin of Error} = 1.645 \times 0.1143 = 0.188
   \]

Thus, the margin of error for the 90% confidence interval is **0.188 mpg**.

Would you like more details on this process or have any further questions?

---

Here are 5 related questions for practice:
1. How would the margin of error change with a 95% confidence interval?
2. What would the margin of error be if the sample size was increased to 100?
3. How does the population standard deviation affect the margin of error?
4. If the sample mean was 21.0 instead of 20.4, how would that affect the confidence interval?
5. Why is it important to know whether the population is normally distributed?

**Tip**: A larger sample size decreases the margin of error because it reduces the standard error of the mean.

Calculate the margin of error for a 90% confidence interval based on a sample of 49 Toyota Tacomas with a mean highway fuel efficiency of 20.4 mpg and a population standard deviation of 0.8 mpg. This problem applies key statistical concepts such as the standard normal distribution and confidence intervals, ideal for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation