A pizza house claims that their online delivery service only takes at most an hour. Recent experiences of Bert on the times until he gets his orders made online, however, seem to always exceed an hour. So Bert decides to conduct a hypothesis test. He supposed that the number of hours for a successful delivery $($i$.$e$.,$ from ordering online until delivery of the item$/$s$)$ X$$ follows an Exp$(\backslash$lambda$),$ for $\backslash$lambda in $(0,\backslash$infty $)$ unknown. He wishes to test if the true mean number of hours for the delivery exceeds an hour or not. Suppose Bert wishes to test the following hypotheses Ho : E$($X$)<=1$ against $$H$1$: $$E$($X$)>1$ and

decided that the test shall have a critical region C $=\{$x: x $>1\mathrm{.}5\}.$

a$.$ Obtain the power function of the test.

b$.$ Obtain the size of the test.

c$.$ Is the test unbiased? Show$/$justify your answer.

d$.$ Is the test ideal? Show$/$justify your answer.

e$.$ What is the size of Type II error of the test at $\backslash$alpha $=0\mathrm{.}5?$

f$.$ Suppose, during his last online transaction, Richie observed that his order was delivered in an hour and $25$ minutes. Given this, compute for the p$-$value. At the worst, can Richie conclude that the pizza house$$s claim is erroneous? Test at $0\mathrm{.}10$ level.