# The amount of warpage in a type of wafer used in the manufacture of integrated circuits has mean 1.3 mm?

The amount of warpage in a type of wafer used in the manufacture of integrated circuits has mean 1.3 mm and standard deviation 0.1 mm. A random sample of 200 wafers is drawn.

a. What is the probability that the sample mean warpage exceeds 1.305 mm?

b. Find the 25th percentile of the sample mean.

c. How many wafers must be sampled so that the probability is 0.05 that the sample mean

exceeds 1.305?

• The standard error of the sample mean is

sigma/sqrt(N), so it’s

0.1 mm / sqrt(200) = 7.07 microns

(a) z = 5/7.07 = 0.707;

normal distribution table shows there is a 24.0% chance of exceeding this.

(b) 25th percentile is at z = -0.775 (see normal distr table), or a width of 1.3 mm – (0.775)(0.00707 mm) = 1.2945 mm

(c) You want to increase the z-score for 1.305 mm to 1.645, so the standard error of the mean must be decreased to

0.005 mm/1.645 = 0.00304 mm; then

0.00304 = 0.1 / sqrt(N) => N = 32.89^2 = about 1080