The problem asks for the margin of error and confidence interval for the population mean $\mu$ based on a sample, assuming normal distribution and using the t-distribution. Let's break down the given information and compute the values.

Given:

• Sample size $n = 28$
• Sample mean $\bar{x} = 31.1$ minutes
• Sample standard deviation $s = 7.3$ minutes
• Confidence level = 90%

We need to:

1. Calculate the margin of error (ME).
2. Construct the 90% confidence interval for the population mean $\mu$.

Step 1: Find the t-critical value

Since the sample size $n = 28$, degrees of freedom $df = n - 1 = 27$. For a 90% confidence level, we can use the t-distribution table or a calculator to find the t-critical value for $df = 27$.

Step 2: Margin of Error formula

$ME = t_{\alpha/2} \times \frac{s}{\sqrt{n}}$

Where:

• $t_{\alpha/2}$ is the t-critical value
• $s = 7.3$ is the sample standard deviation
• $n = 28$ is the sample size

Step 3: Confidence Interval

The confidence interval is given by: $CI = \left( \bar{x} - ME, \bar{x} + ME \right)$ Where $\bar{x} = 31.1$ is the sample mean.

Let me calculate the margin of error and the confidence interval for you.### Results:

• Margin of error (ME): $2.35$ minutes (rounded to two decimal places).
• 90% Confidence Interval: $(28.8, 33.4)$ minutes (rounded to one decimal place).

This means we are 90% confident that the true population mean commute time lies between 28.8 minutes and 33.4 minutes.

Would you like more details or explanations on any part of this process?

Here are 5 related questions to deepen your understanding:

1. How does the sample size affect the width of the confidence interval?
2. What happens to the margin of error if the confidence level is increased to 95%?
3. How do you interpret a confidence interval in practical terms?
4. Why is the t-distribution used instead of the normal distribution in this case?
5. How would the result change if the standard deviation was smaller?

Tip: Always use the t-distribution when dealing with small sample sizes (typically $n < 30$) unless the population standard deviation is known.