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.

(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.

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In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light are to be met. Let X and Y denote the thickness of two different layers of ink. It is known that X is normally distributed with a mean of 0.1 millimeter and a standard deviation of 0.00031 millimeter and Y is also normally distributed with a mean of 0.23 millimeter and a standard deviation of 0.00017 millimeter. Assume that these variables are independent.

(a) If a particular lamp is made up of these two inks only, what is the probability that the total ink thickness is less than 0.2451 millimeter?

(b) A lamp with a total ink thickness exceeding 0.2405 millimeters lacks the uniformity of color demanded by the customer. Find the probability that a randomly selected lamp fails to meet customer specifications.

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A mechanical assembly used in an automobile engine contains four major components. The weights of the components are independent and normally distributed with the following means and standard deviations (in ounces):

(a) What is the probability that the weight of an assembly exceeds 29.5 ounces?

(b) What is the probability that the mean weight of eight independent assemblies exceeds 29 ounces?

Solution

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)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    8      
x2 = upper bound =    11      
u = mean =    10      
n = sample size =    10      
s = standard deviation =    1      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -6.32455532      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    3.16227766      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    1.26981E-10      
P(z < z2) =    0.999217299      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.999217299      

Thus, those outside this interval is the complement =    0.000782701   [ANSWER]

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b)

Note that the z score corresponding to a right tailed area of 0.01 is, by table/technology, is

z = 2.326347874

Thus,

z = (x-u)*sqrt(n)/s

2.326347874 = (11-10)*Sqrt(n)/1

Thus,

n = 2.326347874^2 = 5.411894431

Rounding up,

n = 6 [ANSWER]  

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