To find the 80% confidence interval for the mean commute time of all SVC students, we will use the t-distribution because the sample size is small (n = 9) and the population standard deviation is unknown.

Step 1: Identify the sample mean ($\bar{x}$), sample standard deviation (s), and sample size (n).
- $\bar{x} = 13.6$ minutes
- $s = 4.01$ minutes
- $n = 9$

Step 2: Determine the degrees of freedom (df).
- $df = n - 1 = 9 - 1 = 8$

Step 3: Find the t-value for an 80% confidence interval with 8 degrees of freedom.
- For an 80% confidence interval, the confidence level is 0.80, and the significance level ($\alpha$) is 0.20. Since the confidence interval is two-tailed, we divide $\alpha$ by 2, giving us 0.10 in each tail.
- Using a t-table or calculator, the t-value for 8 degrees of freedom and 0.10 in each tail is approximately 1.397.

Step 4: Calculate the standard error of the mean (SE).
- $SE = \frac{s}{\sqrt{n}} = \frac{4.01}{\sqrt{9}} = \frac{4.01}{3} \approx 1.337$

Step 5: Calculate the margin of error (ME).
- $ME = t \times SE = 1.397 \times 1.337 \approx 1.868$

Step 6: Determine the confidence interval.
- Lower limit = $\bar{x} - ME = 13.6 - 1.868 \approx 11.732$
- Upper limit = $\bar{x} + ME = 13.6 + 1.868 \approx 15.468$

Therefore, the 80% confidence interval for the mean commute time of all SVC students is approximately (11.732, 15.468) minutes.

Question

To find the 80% confidence interval for the mean commute time of all SVC students, we will use the t-distribution because the sample size is small (n = 9) and the population standard deviation is unknown.

Step 1: Identify the sample mean ($\bar{x}$), sample standard deviation (s), and sample size (n).
- $\bar{x} = 13.6$ minutes
- $s = 4.01$ minutes
- $n = 9$

Step 2: Determine the degrees of freedom (df).
- $df = n - 1 = 9 - 1 = 8$

Step 3: Find the t-value for an 80% confidence interval with 8 degrees of freedom.
- For an 80% confidence interval, the confidence level is 0.80, and the significance level ($\alpha$) is 0.20. Since the confidence interval is two-tailed, we divide $\alpha$ by 2, giving us 0.10 in each tail.
- Using a t-table or calculator, the t-value for 8 degrees of freedom and 0.10 in each tail is approximately 1.397.

Step 4: Calculate the standard error of the mean (SE).
- $SE = \frac{s}{\sqrt{n}} = \frac{4.01}{\sqrt{9}} = \frac{4.01}{3} \approx 1.337$

Step 5: Calculate the margin of error (ME).
- $ME = t \times SE = 1.397 \times 1.337 \approx 1.868$

Step 6: Determine the confidence interval.
- Lower limit = $\bar{x} - ME = 13.6 - 1.868 \approx 11.732$
- Upper limit = $\bar{x} + ME = 13.6 + 1.868 \approx 15.468$

Therefore, the 80% confidence interval for the mean commute time of all SVC students is approximately (11.732, 15.468) minutes.

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