Question

A Gallup Poll showed that 30% of Americans are satisfied with the way things are going in the United States. Suppose a sample of 20 Americans is selected as part of a study of the state of the nation. 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. For the sample of 20 Americans, compute the standard deviation of the number of Americans who are satisfied with the way things are going in the United States.

Answer 1

The expected number of a distribution is its mean.

• The expected number of Americans who are satisfied with the way things are going in the United States is 6
• The standard deviation of Americans who are satisfied with the way things are going in the United States is 2.05

The given parameters are:

$\mathbf{p = 30\%}$ --- the proportion that are satisfied

$\mathbf{n = 20}$ --- the sample size

(a) The expected number that are satisfied

This is calculated using:

$\mathbf{E(x) = np}$

So, we have:

$\mathbf{E(x) = 20 \times 30\%}$

$\mathbf{E(x) = 6}$

The expected number of Americans who are satisfied with the way things are going in the United States is 6

(b) The standard deviation of those that are satisfied

This is calculated using:

$\mathbf{\sigma = \sqrt{n \times p \times (1 - p)}}$

So, we have:

$\mathbf{\sigma = \sqrt{20 \times 30\% \times (1 - 30\%)}}$

$\mathbf{\sigma = \sqrt{20 \times 30\% \times 70\%}}$

$\mathbf{\sigma = \sqrt{4.2}}$

$\mathbf{\sigma = 2.05}$

The standard deviation of Americans who are satisfied with the way things are going in the United States is 2.05

Read more about expected value and standard deviations at:

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