Question

An IBM subcontractor was hired to make ceramic substrates, which are used to distribute power and signals to and from computer silicon chips. Specifications require resistance between 1,500 ohms and 2,500 ohms, but the population has normally distributed resistances with a mean of 1,978 ohms and a standard deviation of 0.172 ohms. What percentage of the ceramic substrates will not meet the manufacture’s specifications

Answer 1

Answer: 99.61%

Step-by-step explanation:

As per given , we have

$\mu=1.978\\\\ \sigma=0.172$

Let x be the random variable that represents the value of resistance .

Since, the resistance is normally distributed.

Then, the z-score corresponding to x on normal curve will be :-

$z=\dfrac{x-\mu}{\sigma}$

For x = 1.500

$z=\dfrac{1.500-1.978}{0.172}=-2.78$

For x = 2.500

$z=\dfrac{2.500-1.978}{0.172}=3.03$

The probability of the ceramic substrates will not meet the manufacture’s specifications :

P(1.500<x<2500)=P(-2.78<z<3.03)=P(z<3.03)-P(z<-2.78)

=P(z<3.03)-(1-P(z<2.78))

= 0.9987772-(1-0.9972821)   [Using z-table]

=0.9960593 =99.60593 % ≈ 99.61%

Hence, the percentage of the ceramic substrates will not meet the manufacture’s specifications = 99.61%