Practice
- A national education researcher wants to estimate the average SAT Math score for high school seniors in a particular state. A random sample of 60 students yields a sample mean score of 540. The population standard deviation is known to be 100 points. Construct a 92% confidence interval for the true mean SAT Math score of all high school seniors in the state.
Solution
From the problem we get the following information:
- Sample is random
- Sample size is \(n=60\)
- Sample mean is \(\bar{x}=540\)
- Population standard deviation is \(\sigma=100\)
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 92% confidence interval, the remaining 8% is divided between the two tails (4% each)
- Using a Z-Table or a calculator, we find that the z-score dividing the center from the two tails is \(\pm\)1.75
3. Margin of Error
\(E = z_c\frac{\sigma}{\sqrt{n}} = 1.75\frac{100}{\sqrt{60}} = 22.59\)
4. Confidence Interval
\(\bar{x} + E = 540 + 22.59 = 562.59\)
\[\bar{x} - E = 540 - 22.59 = 517.41\]
The confidence interval is (517.41, 562.59).
5. Interpretation of the Confidence Interval
We are 92% confident that the average daily commute is between 517.41 minutes and 562.59 minutes.
Return back to Lesson 18.3