Question

A factory produces pistons whose diameters follow a normal distribution with a mean of 50 mm and a standard deviation of 0.01 mm. For a piston to serve, its diameter must be between 49.98 and 50.02 mm. If the diameter is less than 49.98 mm it is rejected; if it is greater than 50.02 mm, it is reprocessed once and the new diameter follows a normal distribution of an average of 49.99 mm, a standard deviation of 0.01 mm.

I. The probability that the piston must be reprocessed is 0.0228.

II. The probability that a reprocessed piston will be rejected is 0.61

a. Only I is correct

b. Only II is correct.

c. None is correct.

d. Both are correct.

Answer 1

i) The mean is given as 50 and standard deviation as 0.01. To reprocess the diameter must be greater than 50.02. Hence we need to find the probability P(x>50.02) where x is the diameter of the piston which follows normal distribution.

We know that P(x>50.02) = 1-P(x<50.02)

$P(X<50.02) = P(z<\frac{x-\mu}{\sigma})$

where $\mu$ is the mean and $\sigma$ is the standard deviation. Hence we have the below

$P(X<50.02) = P(z<\frac{50.02-50}{0.01}) = P(z<2)$

From stnadard normal tables we know that P(z<2) = 0.9772

Hence P(X>50.02) = 1-0.9772 = 0.0228

Hence the probability that the piston must be reprocessed is 0.0228. Thus statement I is correct.

When the piston is reprocessed, if the diameter is less than 49.98 or more than 50.02 it will be rejected. For the reprocessed piston the mean is 49.99 and standard deviation is 0.01. Hence we need to find the probability 1-P(49.98<x<50.02)

$P(49.98<X<50.02) = P(\frac{49.98-49.99}{0.01}<z<\frac{50.02-49.99}{0.01})$

From standard normal tables we know that P(z<3) = 0.9987

We also know that P(-1<z) = 1-P(z<1)

From standard normal tables we know that P(z<1) = 0.8413

Hence P(-1<z) = 1-0.8413 = 0.1587

Hence P(49.98<x<50.02) = 0.9987 - 0.1587 = 0.84

Hence the probability that the reprocessed piston will be rejected is 1-0.84 = 0.16.

Hence statement II is not correct.

Thus the correct option is option a. Only I is correct.