Question

Let X1, X2,... ,X16 be a random sample from a N(μ, σ²) with σ² = 25 and sample mean X= 60 . Find a 95% confidence interval for μ.

Answer 1

$X_1, X_2, ..., X_n$ is a random sample drawn from a normal distribution with $X_i\thicksim N(\mu,\sigma^2)$ for all $i$. Then

$${\bar{X}-\mu \over \sigma/\sqrt{n}}\thicksim N(0,1)$$

Using this probability distribution, we have 

$$P(-z_{0.025}<Z<z_{0.025})=0.95$$

$$P(-z_{0.025}<{\bar{X}-\mu \over \sigma/\sqrt{n}}<z_{0.025})=0.95$$

95% CI for µ when $\sigma^2$ known and drawing from a normally distributed population:

$$\bar{X}-z_{0.025}{\sigma \over \sqrt{n}}\le \mu \le \bar{X}+z_{0.025}{\sigma \over \sqrt{n}}$$

$$\bar{X}=60, \sigma=\sqrt {25}=5, n=16, z_{0.025}=1.96$$

$$60-1.96{5 \over \sqrt{16}}\le \mu \le 60+1.96{5 \over \sqrt{16}}$$

$$57.55\le \mu \le 62.45$$