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Statistics / Applied Statistics and Probability for Engineers 6 / Chapter 5.4 / Problem 67E

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

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Solved: In the manufacture of electroluminescent lamps,

Chapter 5, Problem 67E

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QUESTION:

Problem 67E

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 mm and a standard deviation of 0.00031 mm, and Y is also normally distributed with a mean of 0.23 mm and a standard deviation of 0.00017 mm. 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 0.2337 mm?

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

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QUESTION:

Problem 67E

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 mm and a standard deviation of 0.00031 mm, and Y is also normally distributed with a mean of 0.23 mm and a standard deviation of 0.00017 mm. 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 0.2337 mm?

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

ANSWER:

Answer

Step 1 of 2

(a)

We are asked to find the probability that the total ink thickness is less than $0.2337\ mm.$

Let $T$ denote the total thickness.

Let $X\ and\ Y$ denote the thickness of two different layers of ink.

Then, $T\ =X+Y$ ……….(1)

We need to find $P(T<0.2337).$

If ${X}_{1},\ {X}_{2},........,{X}_{p}$ are independent, normal random variables with $E({X}_{i})={\mu{}}_{i}\ and\ V({X}_{i})={\sigma{}}_{i}^{2},\ for\ i=1,\ 2,....p,$

$Y\ ={c}_{1}{X}_{1}+{c}_{2}{X}_{2}+.........+{c}_{p}{X}_{p}$

Is a normal random variable with

$E(Y)\ ={c}_{1}{\mu{}}_{1}+{c}_{2}{\mu{}}_{2}+.........+{c}_{p}{\mu{}}_{p}$ …….(2)

$V(Y)={{c}^{2}}_{1}{\sigma{}}_{1}^{2}+{{c}^{2}}_{2}{\sigma{}}_{2}^{2}+.........+{{c}^{2}}_{p}{\sigma{}}_{p}^{2}$ ………(3)

Hence the mean and standard deviation of the total thickness of the two halves using equation $(2)\ and\ (3).$

$E(T)\ ={c}_{X}{\mu{}}_{X}+{c}_{Y}{\mu{}}_{Y}$ ……….(4)

Compare equation (1) and (4), we get ${C}_{X}={C}_{Y}=1\ and\ {\mu{}}_{X}=0.1\ mm,{\ \mu{}}_{Y}=0.23\ mm\ given\$

$E(T)\ =0.1+0.23=0.33\ mm$

$V(T)=0.{00031}^{2}+0.{00017}^{2}=1.25\times{}1{0}^{-7}\ m{m}^{2}$

${\sigma{}}_{T}=\sqrt{V(T)}=0.000354\ mm.$

Since total thickness follows normal distribution, we can write,

$P(T<X)\ =P(Z<\frac{X-\mu{}}{\sigma{}})$

$P(T<0.2337)\ =P(Z<\frac{0.2337-0.33}{0.000354})$

$P(T\leq{}0.2337)\ =P(Z\leq{}-272.377)$

Using $z-score$ table, the area to the left of $z=-272.377$ is $0.$

Hence the probability that the total ink thickness is less than $0.2337\ mm$ is $0.$

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