A machine is set to manufacture circular metal discs of diameter $\displaystyle 10 cm$; in fact, the diameter (in cm) of the discs it produces is a random variable $\displaystyle Y$ with normal distribution having a mean of $\displaystyle 10.05$ and a standard deviation of $\displaystyle 0.10$.

The demand of quality control place "acceptable limits between $\displaystyle 9.90 cm$ and $\displaystyle 10.20 cm$; all disks out of this range are rejected.

a) Given that a particular disk is accepted, find the probability that its diameter is more than $\displaystyle 10 cm$.

b) It is decided that too many discs are rejected by the standard. If a new rejection rate is to be 10%, find the new acceptable limits, equally spaced on either side of the mean.

c) Use the fact that $\displaystyle Var(Y) =E(Y^2) - [E(Y)]^2$ to estimate the mean area of the discs.