Question

The mean and standard deviation for the quality grade point averages of a random sample of 28 college seniors are calculated to be 2.6 and 0.3 respectively. Find the 95% confidence interval for the mean of the entire senior class. How large a sample is required if we want to be 95% confident that our estimate of µ is not off by more than 0.05?

Answer 1

Question #15348The mean and standard deviation for the quality grade point averages of a random sample of 28 college seniors are calculated to be 2.6 and 0.3 respectively. Find the 95% confidence interval for the mean of the entire senior class. How large a sample is required if we want to be 95% confident that our estimate of $\mu$ is not off by more than 0.05?

Solution.Let $\mu$ be the mean of the entire senior class. Given: $n = 28$, $s^2 = 0.3$, $\overline{x} = 2.6$, $(1 - \alpha) = 0.95$.

(a) A 95% confidence interval estimate for the is

$$\bar{x} - z_{0.025}s / \sqrt{n} \leq \mu \leq \bar{x} + z_{0.025}s / \sqrt{n},$$ $$2.6 - 1.96 \cdot 0.3 / 5.3 \leq \mu \leq 2.6 + 1.96 \cdot 0.3 / 5.3$$

or $2.49 \leq \mu \leq 2.71$.

(b) Let $n$ be the required sample size. To be 95% confident that if off by less than 0.05 would implies $z_{0.025}s / \sqrt{n} < 0.05$, or $n \geq \left[\frac{0.3 \cdot 1.96}{0.05}\right]^2 \approx 139$.

Answer.a) [2.49, 2.71], b) 139.