Question

in a certain school is known to be normally distributed with a mean of P70 and a standard deviation of P20. A random sample of 16 students will be taken.

What are the two values that are within P10 of the true mean?

Answer 1

$\bar{X}\sim N(\mu, \sigma^2/n)$

Given $\mu=70, \sigma=20, n=16.$ Then

$$\mu_{\bar{X}}=\mu=70$$

$$\sigma_{\bar{X}}=\sigma/\sqrt{n}=20/\sqrt{16}=5$$

$$\mu_{\bar{X}}\pm10=\mu_{\bar{X}}\pm2\sigma_{\bar{X}}=70\pm10$$

$$60<\bar{X}<80$$

For example $64, 72$