According to the IRS, individuals filing federal income tax returns prior to March $31$ received an average refund of $$1,086$ in $2018.$ Consider the population of "last$-$minute" filers who mail their tax return during the last five days of the income tax period $($typically April $10$ to April $15).$

a$.$ A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention.

$H_0$:$is$ greater than $or$ equal $to1,086K$

$H_8$:$is$

b$.$ For a sample of $400$ individuals who filed a tax return between April $10$ and $15,$ the sample mean refund was $$910.$ Based on prior experience a population standard deviation of $=$$$1,800$ may be assumed. What is the $p-$value $($to $4$ decimals$)?$