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A city commissioner claims that 80% of the people living

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer ISBN: 9780495110811 47

Solution for problem 184SE Chapter 3

Mathematical Statistics with Applications | 7th Edition

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Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Mathematical Statistics with Applications | 7th Edition

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Problem 184SE

Problem 184SE

A city commissioner claims that 80% of the people living in the city favor garbage collection by contract to a private company over collection by city employees. To test the commissioner’s claim, 25 city residents are randomly selected, yielding 22 who prefer contracting to a private company.

a If the commissioner’s claim is correct, what is the probability that the sample would contain at least 22 who prefer contracting to a private company?

b If the commissioner’s claim is correct, what is the probability that exactly 22 would prefer contracting to a private company?

c Based on observing 22 in a sample of size 25 who prefer contracting to a private company, what do you conclude about the commissioner’s claim that 80% of city residents prefer contracting to a private company?

Step-by-Step Solution:

Solution :

Step 1 of 3:

Let p denotes the 80% of the people living in the city favour garbage collection by contract to a private company.

So p=80%=0.80.

The 25 residents are selected randomly.

So n=25.

Then the number of 22 people who prefer contracting to a private company.

Our goal is:

a). If the commissioner’s claim is correct, we need to find the probability that the sample

     would contain at least 22 who prefer to contracting a private company.

b). We need to find the probability that exactly 22 would prefer contracting to a private company.

c). Based on observing 22 in a sample of size 25 who prefer contracting to a private

     company, we need to conclude about the commissioner's claim that 80% of city      

     residents prefer contracting to a private company.

a).

If the commissioner’s claim is correct.

We have to find the probability that the sample would contain at least 22 who prefer to contracting a private company.

The formula for the binomial probability is

P(X=a)=

Then the probability of at least 22 is

P(X22)=P(X=22)+P(X=23)+P(X=24)+P(X=25)

We know that n=25, p=0.80.

Where, P(X=22),P(X=23),P(X=24),P(X=25) is obtained from Excel by using the function “=Binomdist(x,n,p,false)”

Then the table is given below.

X

p(x  22)

22

0.135768036

23

0.070835497

24

0.023611832

25

0.003777893

Total

0.233993259

Now,

P(X22)= P(X=22)+P(X=23)+P(X=24)+P(X=25)

P(X22)= 0.23399

Therefore the probability of at least 22 is 0.23399.


Step 2 of 3

Chapter 3, Problem 184SE is Solved
Step 3 of 3

Textbook: Mathematical Statistics with Applications
Edition: 7
Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
ISBN: 9780495110811

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A city commissioner claims that 80% of the people living