A credit card company has found that the monthly balances $x$ of its customers is

approximately normally distributed:

$xN(300,2500).$

Assume that the monthly balances of customers are independent draws from $x.$

$($a$)$ Using $R,$ calculate the probability that the monthly balance of a randomly chosen

customer is less than $$240.$

$($b$)$ Using $R,$ calculate the probability that the monthly balance of a randomly chosen

customer is between $$325$ and $$375.$

$($c$)$ Using $R,$ calculate the $90\%$ population quantile of $x.$

$($d$)$ Provide an $85\%$ probability interval for the monthly balance of a randomly chosen

customer. $($Use $R$ as needed.$)$

$1$

$($e$)$ Fill in the blank: The probability that the monthly balance of a randomly chosen

customer is greater than

is equal to $75\%.($Use $R$ as needed.$)$

$($f$)$ Write $x$ as a linear transformation of a standard normal random variable $Z$

$N(0,1).$

$($g$)$ Suppose that the credit card company's per$-$customer profit is $Y=5+0\mathrm{.}02x$

since each customer pays a $$5$ monthly fee for the card and the company receives

$2\%$ in transaction fees from its merchants. What is the expected value of $Y?$

$($h$)$ Same as part $($g$),$ but what is the standard deviation of $Y?$