$5.$ In the quality control setup, the process mean is monitored by taking a sample of

size n $=8$ every half an hour and stopping and adjusting the machine whenever X $<$ c $$ or

X $>$ c $+,$ where c$\backslash$pm $=0\backslash$pm b $\backslash$sigma

n

$.$ The cost of stopping and adjusting the machine is equal to

$1500,$ and every hour of production after a shift of the process mean incurs a cost of $1000.$

Assuming that mean shifts up or down from the standard in$-$control value by $1\mathrm{.}5\backslash$sigma at random

moments once in $20$ hours on average, find the value of the coefficient b minimizing the total

variable cost per unit time.