$1.$ Process distribution and sampling distribution. An industrial process that makes $3-$foot sections of plastic pipe with an average inside diameter of $1$ inch and a standard deviation of $\mathrm{.}05$ inch.

a$.$ If you randomly select one piece of pipe, what is the probability that its inside diameter will exceed $1\mathrm{.}02$ inches, assuming the population is normal?

b$.$ If you select a random sample of $25$ pieces of pipe, what is the probability that the sample me will exceed $1\mathrm{.}02$ inches?

$\backslash$mu $=1\mathrm{.}00,\backslash$sigma $=\mathrm{.}05$

$2.$ Computing Cpk$.$ A process has a mean of $9\mathrm{.}20$ grams and a standard deviation of $\mathrm{.}30$ gram. The lower specification limit is $7\mathrm{.}50$ grams and the upper specification limit is $10\mathrm{.}50.$ Compute Cpk$.$

$3.$ Planners for a company makes several models of skateboards are about to prepare the aggregate plans that will cover six periods. They have assembled the following information:

Period $123456$ Total

Forecast $2002003004005002001,800$

Cost

Output

Regular time $=$ $$20$ per skateboard

Overtime $=$ $$30$ per skateboard

Subcontract $=$ $$60$ per skateboard

Inventory $=$ $$1$ per skateboard per period on average inventory

Back orders $=$ $$5$ per skateboard per period

They now want to evaluate a plan that calls for a steady rate of regular$-$time output, mainly using inventory to absorb the uneven demand but allowing some backlog. overtime and subcontracting are no used because they want steady output. The intend to start with zero inventory on hand in the first period. Prepare an aggregate plan and determine its cost per period with regular time $($i$.$e$.,1,800/6=300).$ Note that the planned ending inventory is zero. There are two workers, and each can produce $150$ skateboards per period.

$4.$ What are Linear Programming models?