# Sampling under normal distribution

The concentration of impurities in a semiconducter in the production of microprocessors for computer is a normally distributed random variable with mean 127 parts per million and standard deviation 22 parts per million. A semiconducter is acceptable only if its concentration of impurities is below 150 parts per million.

What is the proportion of semiconducters that are acceptable for use?(the area under normal curve for the value of $Z=1.5$ is $0.668$).

• Please see the guidelines for self-study questions. What is your attempt? Where do your difficulties lie? You should check all the details in your question, at least one of the pieces of information here is wrong. Commented Apr 26, 2014 at 16:33

Note that the acceptance value (150 ppm) is higher than the mean - which it had better be, if the supplier expects most of his materials to be accepted.

Note also that in the case were the impurities are < 127 ppm - or the lower half of the distribution - all the materials will pass. That lower half of the distribution - 50% of the materials - will always pass.

Now what about those materials on the upper half of the distribution - with impurities > 127 ppm? Only the fraction between 127 and 150 will pass. How many standard deviations is that from the mean?

$z = \frac{150-127}{22} = 1.0455$

What fraction of the cumulative normal distribution is that?

$P(0 < Z \leq 1.0455) = \Phi (1.0455) - \Phi (0) = 0.8508-0.5 = 0.3508$

So what is the total fraction that will pass?

This is the sum of the area where $P(Z \leq 0) = 0.5$ and the area where $P(0 < Z \leq 1.0455)=0.3508$. So then:

$P(Z \leq 0) + P(0 < Z \leq 1.0455) = 0.5 + 0.3588 = 0.8588$

This can be done in one step: $P(Z \leq 1.0455) = \Phi (1.0455) = 0.8588$

So why did I go to all the trouble of first solving it in two parts?

1. To give you better intuition into the answer by showing which parts of the area under the standard Normal curve that used.
2. Many times there are both a lower acceptance limit and and upper acceptance limit. Showing the answer in parts should help you solve the problem when there are an upper and lower limit.
• Please read the guidelines for answering self-study questions. This goes far beyond the recommended 'give helpful hints' and looks like it just does someone's homework for them. Once it's clear the OP has solved it, or in any case, after several days, this level of detail would be a good thing to put in an answer. Commented Apr 26, 2014 at 18:27
• (N.B. I am not suggesting you alter your answer at this point, but the guidelines would be useful to keep in mind for future self-study questions) Commented Apr 27, 2014 at 4:14