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Gears produced by a grinding process are categorized

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 14E Chapter 4.2

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 14E

Gears produced by a grinding process are categorized either as conforming (suitable for their intended purpose), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). Suppose that 80% of the gears produced arc conforming, 15% are degraded, and 5% are scrap. Ten gears are selected at random.

a. What is the probability that one or more is scrap?

b. What is the probability that eight or more are not scrap?

c. What is the probability that more than two are either degraded or scrap?

d. What is the probability that exactly nine are either conforming or degraded?

Step-by-Step Solution:
Step 1 of 3

Solution 14E

Step1 of 5:

We have random variable X which presents the number of gears that are scrap among the ten selected. Here X follows binomial distribution with parameters “n and p” that is X B(n, p),

The probability mass function of binomial distribution is given by

, x = 0,1,2,...,n.

Where,

n = sample size

   = 10

x = random variable

p = probability of success

   = 5%

   = 0.05.

q = 1 - p (probability of failure)

   = 1 - 0.05

   = 0.95

Here our goal is:

a).We need to find the probability that one or more is scrap.

b).We need to find the probability that eight or more are not scrap.

c).We need to find the probability that more than two are either degraded or scrap.

d).We need to find the probability that exactly nine are either conforming or degraded.

 


Step2 of 5:

a).

P(One or more is scrap) = P(X1)

                                        = 1 - P(X < 1)

                                        = 1 - P(X = 0)

Consider,

       P(X) =

P(X = 0) =  

             = (1)(1)(0.5987)

             = 0.5987

Now,

P(one or more is scrap) = 1 - P(X = 0)

                                      = 1 - 0.5987

                                      = 0.4012

Hence, P(one or more is scrap) = 0.4012.


Step3 of 5:

b).

We have total gears n = 10 in that we need to find the probability that eight or more are not scrap

Hence,

P(eight or more are not scrap) = P(two or more are scrap)

                                           = P(X2)

1).Mean of the binomial distribution is

                                                            =

                                                            = 0.5

     Hence, = 0.5.  

 

2).Standard deviation of binomial distribution is

                                                                        =

                                                                        =  

                                                                        = 0.6892

    Hence, =0.6892.

 Now,

P(X2) =

              =

              =

value is obtained from standard normal table(area under normal curve) then

= 0.9850 (In area under normal curve we have to see row 2.1 under column 0.07)

 Therefore,  0.9850.


Step4 of 5:

c).

From the given information we have 15% degraded and 5% are Scrap and total selected gears

n = 10.

That is p = 0.15 + 0.05

               = 0.20

P(more than two are either degraded or scrap) = P(X)

                                                                = 1 - P(X2)

                                                                = 1 - {P(X=0)+P(X=1)+P(X=2)}

Consider,

{P(X=0)+P(X=1)+P(X=2)} =

                                                             

                                                             

                                             = {1(1)(0.1073)+10(0.2)(0.1342)+45(0.04)(0.1677)}

                                       

                                             = {0.1073+0.2684+0.3018}

                       

                                             = 0.6775

Now,

P(X) = 1 - {P(X=0)+P(X=1)+P(X=2)}

               = 1 - 0.6775

               = 0.3224

Hence, P(X) = 0.3224.


Step5 of 5:

d).

We have n = 10 and p = 0.95

Consider,

P(exactly nine are either conforming or degraded) = P(X = 9)

                                                                      =

                                                                      = 10(0.6302)(0.05)

                                                                      = 0.3151

Hence, P(X = 9) = 0.3151.


Step 2 of 3

Chapter 4.2, Problem 14E is Solved
Step 3 of 3

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4. This full solution covers the following key subjects: scrap, Probability, degraded, conforming, purpose. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. The full step-by-step solution to problem: 14E from chapter: 4.2 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. Since the solution to 14E from 4.2 chapter was answered, more than 1370 students have viewed the full step-by-step answer. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. The answer to “Gears produced by a grinding process are categorized either as conforming (suitable for their intended purpose), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). Suppose that 80% of the gears produced arc conforming, 15% are degraded, and 5% are scrap. Ten gears are selected at random.a. What is the probability that one or more is scrap?________________b. What is the probability that eight or more are not scrap?________________c. What is the probability that more than two are either degraded or scrap?________________d. What is the probability that exactly nine are either conforming or degraded?” is broken down into a number of easy to follow steps, and 99 words.

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