 # A Gallup Poll showed that 30% of Americans are satisfied with the way things are going in the United States (Gallup website, September 12, 2012). Suppose a sample of 20 Americans is selected as part of a study of the state of the nation. a. Compute the probability that exactly 4 of the 20 Americans surveyed are satisfied with the way things are going in the United States. b. Compute the probability that at least 2 of the Americans surveyed are satisfied with the way things are going in the United States. c. For the sample of 20 Americans, compute the expected number of Americans who are satisfied with the way things are going in the United States. d. For the sample of 20 Americans, compute the variance and standard deviation of the number of Americans who are satisfied with the way things are going in the United States. 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884 13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

#### Solutions

Chapter
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Chapter 5.5, Problem 42E
Textbook Problem
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## A Gallup Poll showed that 30% of Americans are satisfied with the way things are going in the United States (Gallup website, September 12, 2012). Suppose a sample of 20 Americans is selected as part of a study of the state of the nation. a. Compute the probability that exactly 4 of the 20 Americans surveyed are satisfied with the way things are going in the United States. b. Compute the probability that at least 2 of the Americans surveyed are satisfied with the way things are going in the United States. c. For the sample of 20 Americans, compute the expected number of Americans who are satisfied with the way things are going in the United States. d. For the sample of 20 Americans, compute the variance and standard deviation of the number of Americans who are satisfied with the way things are going in the United States.

Expert Solution

a.

To determine

Find the probability that exactly 4 of the 20 Americans surveyed are satisfied with the way things are going in the United States.

### Explanation of Solution

Calculation:

It was found that 30% of the Americans are satisfied with the way things are going in United States. A random sample of 20 Americans is selected.

Define the random variable x as the number of Americans who are satisfied with the way things are going in the United States. Here, a random sample (n) of 20 Americans is taken. Each American is independent of the other. Also, there are two possible outcomes, the American is satisfied with the way things are going in the United States or not (success or failure). It was found that 30% of the Americans are satisfied with the way things are going in the United States. Thus, the probability of success (p) is 0.30. Thus, x follows binomial distribution.

The probability of obtaining x successes in n independent trails of a binomial experiment is,

f(x)=(nx)px(1p)nx,x=0,1,2,...,n

Where, p is the probability of success.

Substitute values n=20 and p=0.30 in the above equation. Then,

f(x)=(20x)(0

Expert Solution

b.

To determine

Find the probability that at least 2 of the 20 Americans surveyed are satisfied with the way things are going in the United States.

Expert Solution

c.

To determine

Find the expected number of Americans surveyed who are satisfied with the way things are going in the United States.

Expert Solution

d.

To determine

Find the variance and standard deviation of the number of Americans surveyed who are satisfied with the way things are going in the United States.

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