Answer to Question #201321 in Statistics and Probability for Yougan

Question

The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of$110 and a standard deviation of $20. If the management decides to give free gifts to all those customers who spend more than  $150 at this supermarket during a visit, what percentage of the customers are expected to get free gifts?

Answer 1

The random variable $X$ can be defined as the expenditures for all customers at the supermarket: $X\sim N(\mu, \sigma^2).$

Given $\mu=110, \sigma=20.$

$$P(X>150)=1-P(X\leq 150)$$

$$=1-P(Z\leq \dfrac{150-110}{20})=1-P(Z\leq2)$$

$$\approx0.02275$$

$2.275\%$ of the customers are expected to get free gifts.