To construct an 80% confidence interval for the population mean, we will use the t-distribution since the sample size is small ($n = 25$), and we are given the sample's mean and standard deviation.

Given data:

• Sample mean ($\bar{x}$) = 30.2 minutes
• Sample standard deviation ($s$) = 7.1 minutes
• Sample size ($n$) = 25
• Confidence level = 80%

Steps:

1. Determine the degrees of freedom (df): $\text{df} = n - 1 = 25 - 1 = 24$

2. Find the critical value $t_{\alpha/2}$: For an 80% confidence interval, the area in each tail is $\alpha = 1 - 0.80 = 0.20$. Since it’s a two-tailed test, the area in each tail is $\alpha/2 = 0.10$.

   Using a t-table or a calculator to find $t_{0.10}$ with 24 degrees of freedom, the critical value $t_{0.10}$ is approximately 1.711.

3. Calculate the standard error (SE) of the mean: The standard error of the mean is given by: $SE = \frac{s}{\sqrt{n}} = \frac{7.1}{\sqrt{25}} = \frac{7.1}{5} = 1.42$

4. Construct the confidence interval: The confidence interval is given by: $\text{CI} = \bar{x} \pm t_{\alpha/2} \times SE$ Substituting the values: $\text{CI} = 30.2 \pm 1.711 \times 1.42$ $\text{CI} = 30.2 \pm 2.42$

5. Final confidence interval: $\text{CI} = (30.2 - 2.42, 30.2 + 2.42)$ $\text{CI} = (27.78, 32.62)$

Conclusion:

The 80% confidence interval for the population mean commute time is approximately (27.78 minutes, 32.62 minutes).

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Would you like a further explanation of any steps, or do you have any other questions?

Here are some related questions you might explore:

1. How does changing the confidence level affect the width of the confidence interval?
2. What would the confidence interval look like if we used a 95% confidence level instead?
3. How can we calculate the margin of error for the given confidence interval?
4. What is the difference between using the t-distribution and the z-distribution for confidence intervals?
5. How would the confidence interval change if the sample size was increased to 50?

Tip: As sample size increases, the standard error decreases, which leads to a narrower confidence interval.