Question

Suppose X ~ N(3 , 4). If sample size = 100, P(∑ Xi > 250) = ?. Draw

a figure.

Answer 1

Suppose $X\sim N(\mu, \sigma^2).$

By the Central Limit Theorem for the random samples we take from the population, we can compute the mean of the sample means $\mu_{\bar{x}}=\mu$ and the standard deviation of the sample means $\sigma_{\bar{x}}=\sigma/\sqrt{n}.$

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

Given $\mu=3, \sigma^2=4, n=100.$

$\bar{x}\sim N(3, 2^2/100).$

$$P(\sum X_i>250)=P(\bar{x}>\dfrac{250}{100})=P(\bar{x}>2.5)$$

$$=1-P(\bar{x}\leq2.5)=1-P(Z\leq\dfrac{2.5-3}{2/\sqrt{100}})$$

$$=1-P(Z\leq\dfrac{2.5-3}{2/\sqrt{100}})=1-P(Z\leq-2.5)$$

$$\approx0.9938$$