Question $7-7$: Based on long$-$standing observation and consensus, the speeds on a curve are

observed to average $57m\frac{i}{h}$ with a $6m\frac{i}{h}$ standard deviation. This is taken as "given." Some

active controls are put in place $($signing$,$ flashing lights, etc.$),$ and the engineer is sure that

the average will fall to at least $50m\frac{i}{h},$ with the standard deviation probably about the same.

$($a$)$ Formulate a null and alternative hypothesis based on taking "after" data only, and

determine the required sample size $n$ so that the Type I and Type II errors are each $0\mathrm{.}05.$

$($b$)$ Assume the data have been taken for the required N observations and that the average is

$52\mathrm{.}2m\frac{i}{h}$ and the standard deviation is $6\mathrm{.}0m\frac{i}{h}.$ What is your decision? What error may

have occurred, and what is its probability?

$($c$)$ Resolve part $($b$)$ with a mean of $52\mathrm{.}2m\frac{i}{h}$ and a standard deviation of $5\mathrm{.}4m\frac{i}{h}.$