Practice
- A university is interested in estimating the average daily commute time for its students. A random sample of 50 students is selected, and their average commute time is found to be 32 minutes. The population standard deviation is known to be 8 minutes. Construct a 95% confidence interval for the true mean commute time of all students at the university.
Solution
From the problem we get the following information:
- Sample is random
- Sample size is \(n=50\)
- Sample mean is \(\bar{x}=32\)
- Population standard deviation is \(\sigma=8\)
1. Central Limit Theorem
- Is the sample random? Yes (stated in the problem)
- Is the sample large enough? Yes (\(n = 50 \ge 30\))
2. Critical Value
With a 95% confidence interval, the remaining 5% is divided between the two tails (2.5% each)
- The z-score dividing the center from the two tails is \(\pm\)1.96
3. Margin of Error
\(E = z_c\frac{\sigma}{\sqrt{n}} = 1.96\frac{8}{\sqrt{50}} = 2.22\)
4. Confidence Interval
\(\bar{x} + E = 32 + 2.22 = 34.22\)
\[\bar{x} - E = 32 - 2.22 = 29.78\]
The confidence interval is (29.78, 34.22).
5. Interpretation of the Confidence Interval
We are 95% confident that the true average daily commute is between 29.78 minutes and 34.22 minutes.
Return back to Lesson 18.3