Question

What is the probability that the outdoor store will sell between 191 and 221 survival kits per month?

Answer 1

An outdoor store sells 200 survival kits per month, with a population standard deviation (σ) of 15 kits. Assume the selling of survival kits is normally distributed. What is the probability that the outdoor store will sell between 191 and 221 survival kits per month? 

Let $X=$ the number of survival kits sold per month: $X\sim N(\mu,\sigma^2)$.

Then

$$Z={X-\mu \over \sigma}\sim N(0,1)$$

Given that $\mu=200, \sigma=15.$

$$P(191<X<221)=P(X<221)-P(X\leq191)=$$

$$=P(Z<{221-200 \over 15})-P(Z\leq{191-200 \over 15})=$$

$$=P(Z<1.4)-P(Z\leq-0.6)\approx$$

$$\approx0.91924334-0.27425312\approx0.6450$$

The probability that the outdoor store will sell between 191 and 221 survival kits per month is $0.6450.$