91Ó°ÊÓ

The quantity injected per container is approximately normally distributed with mean 10 units and standard deviation .2 units. Each unit of fill costs $20 per unit. If a container contains less than 10 units (i.e., is underfilled), it must be reprocessed at a cost of $10. A properly filled container sells for $230.

Let x be the random variable follow normal distribution with mean 10 units and standard deviation 0.2 units.

$x~N(10,0.2)$

a.

The probability of underfill is given by,

role="math" localid="1660285516190" $$\begin{gathered} P(x<10)=P\left(\frac{x-10}{0.2}<\frac{10-10}{0.2}\right) \\ =P(z<0) \\ =0.50 \end{gathered}$$

The probability is 0.50

The probability of not underfill is given by,

$$\begin{gathered} P(xâ‰\yen 10)=P\left(\frac{x-10}{0.2}â‰\yen \frac{10-10}{0.2}\right) \\ =1-P(z<0) \\ =1-0.50 \\ =0.50 \end{gathered}$$

Thus, the required probability is 0.50

b.

The cost of each unit is $20

The reprocessed cost is $10

The selling cost is $230

The profit make the company upon refilling the container is,

$$\begin{gathered} profit=230-10-10.60×20 \\ =8 \end{gathered}$$

Thus, required the profit in refilling the container is 8$.

The mean of filling process upward is 10.60 units. So,

$$\begin{gathered} Expectedprofit=230-10.60×20 \\ =18 \end{gathered}$$

Thus, the required he expected profit in each container is $18.