An endless production line is turning out solid-state diodes and every 100th dio

Question

An endless production line is turning out solid-state diodes and every 100th diode is tested for reverse current -i and forward current i at diode voltages of -1 and +1, respectively,

A. If the random variable -i has a true mean value of 10^-6 and a variance of 10^-12, how many diodes must be tested to obtain a sample mean whose standard deviation is 5% of the true mean?

B. If the random variable i has a true mean value of 0.1 and a variance of 0.025, how many diodes must be tested to obtain a sample mean whose standard deviation is 2%of the true mean?

C. If the larger of the two numbers found in A and B is used for both tests, what will the standard deviation of the sample mean be for each test?

Explanation / Answer

the random variable -i has a true mean value of 10-6 and a variance of 10-12.

let n diodes are tested.

then the variance of the sample mean is 10-12/n

hence the standard deviation of sample mean is sqrt[10-12/n]=10-6/sqrt(n)

now it is said that this standard deviation is 5% of the true mean

hence 10-6/sqrt(n)=10-6*5/100

or, sqrt(n)=100/5=20

or, n=202=400

hence 400 diodes must be tested [answer]

b) the random variable i has true mean 0.1 and variance 0.025

let m diodes are tested

then the variance of sample mean is 0.025/m

henc the standard deviation of sample mean is sqrt[0.025/m]=0.15811/sqrt(m)

now it is said that this standard deviation is 2% of the true mean

hence

0.15811/sqrt(m)=0.1*2/100=0.002

hence sqrt(m)=0.15811/0.002=79.055

or, =79.0552=6249.69=6250 [approx]

hence 6250 diodes must be tested. [answer]

c) the larger number is 6250

hence the standard deviation of sample mean for A if 6250 diodes are tested is

sqrt[10-12/6250]=1.2649*10-8      [answer]

hence the standard deviation of sample mean for B if 6250 diodes are tested is

sqrt[0.025/6250]=0.002 [answer]

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