College students with checking accounts typically write relatively few checks in any given month, whereas nonstudent residents typically write many more checks during a month. Suppose that $50 \%$ of a bank's accounts are held by students and that $50 \%$ are held by nonstudent residents. Let $x$ denote the number of checks written in a given month by a randomly selected bank customer.
a. Give a sketch of what the probability distribution of $x$ might look like.
b. Suppose that the mean value of $x$ is $22.0$ and that the standard deviation is $16.5 .$ If a random sample of $n=100$ customers is to be selected and $\bar{x}$ denotes the sample average number of checks written during a particular month, where is the sampling distribution of $\bar{x}$ centered, and what is the standard deviation of the $\bar{x}$ distribution? Sketch a rough picture of the sampling distribution.
c. Referring to Part (b), what is the approximate probability that $\bar{x}$ is at most 20 ? at least 25 ?