Q$8.$H$.$W An electronics company wanted to estimate its monthly operating expenses in thousands riyals $(M).$ It is assumed that the monthly operating expenses $($in thousands riyals$)$ are distributed according to a normal distribution with variance $S^2=0\mathrm{.}584.$

$($I$)$ Suppose that a random sample of $49$ months produced a sample mean $\stackrel{}{x}=5\mathrm{.}47.$

$($a$)$ Find a point estimate for $m.$

$($b$)$ Find the standard error of $\stackrel{}{x}.$

$($c$)$ Find a $90\%$ confidence interval for $m.$

$($II$)$ Suppose that they want to estimate $M$ by $\stackrel{}{x}.$ Find the sample size $($n$)$ required if they want their estimate to be within $0\mathrm{.}15$ of the actual mean with probability equals to $0\mathrm{.}95.$

Q$9.$ The tensile strength of a certain type of thread is approximately normally distributed with standard deviation of $6\mathrm{.}8$ kilograms. A sample of $20$ pieces of the thread has an average tensile strength of $72\mathrm{.}8$ kilograms. Then,

$($a$)$ A point estimate of the population mean of the tensile strength $(m)$ is:

$($A$)72\mathrm{.}8$

$($B$)20$

$($C$)6\mathrm{.}8$

$($D$)46\mathrm{.}24$

$($E$)$

None of these

$($b$)$ Suppose that we want to estimate the population mean $(M)$ by the sample mean ${?}^{(\stackrel{}{x})}.$ To be $95\%$ confident that the error of our estimate of the mean of tensile strength will be less than $3\mathrm{.}4$ kilograms, the minimum sample size should be at least:

$($A$)4$

$($B$)16$

$($C$)20$

$($D$)$

$18$

$($E$)$ None of these