×
Log in to StudySoup

Forgot password? Reset password here

Solved: Calculate the finite population corrections(a) For

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger ISBN: 9781118539712 55

Solution for problem 150E Chapter 3.8

Applied Statistics and Probability for Engineers | 6th Edition

  • Textbook Solutions
  • 2901 Step-by-step solutions solved by professors and subject experts
  • Get 24/7 help from StudySoup virtual teaching assistants
Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

Applied Statistics and Probability for Engineers | 6th Edition

4 5 0 250 Reviews
27
4
Problem 150E

Calculate the finite population corrections

(a) For Exercises 3-141 and 3-142, for which exercise should the binomial approximation to the distribution of X be better?

(b) For Exercise 3-141, calculate P(X = 1) and P(X = 4), assuming that X has a binomial distribution, and compare these results to results derived from the hypergeometric distribution.

(c) For Exercise 3-142, calculate P(X = 1) and P(X = 4), assuming that X has a binomial distribution, and compare these results to the results derived from the hypergeometric distribution.

(d) Use the binomial approximation to the hypergeometric distribution to approximate the probabilities in Exercise 3-146. What is the finite population correction in this exercise?

Step-by-Step Solution:

Step 1 of 5:

Here, the values for N, n and K are given.We need to find the finite population correction and also we need to find out the required values.

Step 2 of 5:

(a)

Here for the given two values of N and n, we need to find the finite population correction and also we have to check which of the given distribution has the Binomial distribution.

The finite population correction is given by

For N=100 and n=4,the finite population correction is given by

=

         =0.9697

For N=20 and n=4, the finite population correction is given by

=

        =0.8421

Here we can observe that the finite population correction for the population with N=100 and n=4 is greater than the population with N=20 and n=4.

So the population with larger finite population correction has the better approximation to the Binomial distribution.

Thus, the finite population correction is 0.9697 for N=100,n=4,0.8421 for N=20,n=4 and the first sample has the better approximation to the Binomial distribution.

Step 3 of 5:

(b)

It is given that X has the Binomial distribution with N=100,n=4,K=20.

We need to find the values of P(X=1) and P(X=4).

Also, we have to assume that X has the Hypergeometric distribution and we have to find the values of P(X=1),P(X=4) to compare it with the values obtained from the Binomial assumption.

We have X~B(n=4,p)

The probability of success p is given by

p=

  =

  =0.2

So P(X=x) becomes

P(X=x)=

           

So,P(X=1) becomes

P(X=1)=

            =(4)(0.2)(0.512)

            =0.4096

P(X=4)=

           =(1)(0.0016)(1)

           =0.0016

Now using the Hypergeometric distribution P(X=x) is given by

P(X=x)=

Now,P(X=1) becomes

P(X=1)=

            =

           =

           =0.4190

P(X=4)=

            =

            =0.00124

Thus, using the Binomial approximation P(X=1)=4096 and P(X=4)=0.0016

Using the Hypergeometric approximation,P(X=1)=0.4190 and P(X=4)=0.0012.

Here we can observe that the probabilities obtained using the Binomial approximation agrees well with the probabilities obtained using the Hypergeometric approximation.

Step 4 of 5

Chapter 3.8, Problem 150E is Solved
Step 5 of 5

Textbook: Applied Statistics and Probability for Engineers
Edition: 6
Author: Douglas C. Montgomery, George C. Runger
ISBN: 9781118539712

Unlock Textbook Solution

Enter your email below to unlock your verified solution to:

Solved: Calculate the finite population corrections(a) For

×
Log in to StudySoup
Get Full Access to Statistics - Textbook Survival Guide

Forgot password? Reset password here

Join StudySoup for FREE
Get Full Access to Statistics - Textbook Survival Guide
Join with Email
Already have an account? Login here
Reset your password

I don't want to reset my password

Need an Account? Is not associated with an account
Sign up
We're here to help

Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or support@studysoup.com

Got it, thanks!
Password Reset Request Sent An email has been sent to the email address associated to your account. Follow the link in the email to reset your password. If you're having trouble finding our email please check your spam folder
Got it, thanks!
Already have an Account? Is already in use
Log in
Incorrect Password The password used to log in with this account is incorrect
Try Again

Forgot password? Reset it here