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A city commissioner claims that 80% of the people living
Chapter 3, Problem 184SE(choose chapter or problem)
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?
Questions & Answers
QUESTION:
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?
ANSWER:
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 |
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.