The Wall Street Journal reported that of taxpayers with adjusted gross incomes between and itemized deductions on their federal income tax return. The mean amount of deductions for this population of taxpayers was $.$ Assume that the standard deviation is $.$ Use z$-$table.

a$.$ What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within of the population mean for each of the following sample sizes: $,,,$ and $?$ Round your answers to four decimals.

b$.$ What is the advantage of a larger sample size when attempting to estimate the population mean?