The photoresist thickness in semiconductor manufacturing has

The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is normally distributed and that the thicknesses of different wafers are independent.

(a) Determine the probability that the average thickness of 10 wafers is either greater than 11 or less than 8 micrometers.

(b) Determine the number of wafers that needs to be measured such that the probability that the average thickness exceeds 11 micrometers is 0.01.

(c) If the mean thickness is 10 micrometers, what should the standard deviation of thickness equal so that the probability that the average of 10 wafers is either greater than 11 or less than 9 micrometers is 0.001?

Solution

Random variable X (thickness of wafer) has mean = 10, and standard deviation ?

Your link got cut off, but using an online normal distribution calculator, I get
z = 3.291

3.291 s.d. = 1
s.d. = 1/3.291

Now your solution would be correct if we were dealing with only 1 wafer, but we are dealing with the average of 10 wafers.

Let Y = (X? + X? + . . . + X??)/10

Standard deviation found above (1/3.291) is standard deviation for Y, not X

Expected value of Y is also 10. We can show this using following identities:
E[X?+X?] = E[X?] + E[X?]
E[aX] = a E[X]

E[X?] = E[X?] = . . . = E[X??] = 10
E[Y] = E[(X? + X? + . . . + X??)/10]
E[Y] = 1/10 E[X?] + 1/10 E[X?] + . . . + 1/10 E[X??] = 10

So mean doesn\'t change

Var[X?] = Var[X?] = . . . = Var[X??] = ?