Chapter 6, Problem 17TP

Solution

To State:

The number of people surveyed, and whether it is known for certain that majority of adults in the United States follow football.

  $2500$ people were surveyed. It is not known for certain that majority of adults in the United States follow football.

Given:

According to a certain poll $51\%$ of adults in the United States follow professional football. The margin of error is $\pm 2\%$ .

Concepts Used:

The margin of error is $\pm \frac{1}{\sqrt{n}}$ where $n$ is the number of people surveyed.

The population proportion may lie anywhere between $p-\frac{1}{\sqrt{n}}$ and $p+\frac{1}{\sqrt{n}}$ . Where $p$ is the estimate of population proportion and $\pm \frac{1}{\sqrt{n}}$ is the margin for error.

Calculations:

Determine the number of people surveyed.

  $\begin{array}{c}\pm 2\%=\pm \frac{1}{\sqrt{n}}\\ 0.0004=\frac{1}{n}\\ n=\frac{10000}{4}\\ n=2500\end{array}$

  $2500$ people were surveyed.

Note that the actual proportion of football followers can be anywhere between $51\%-2\%=49\%$ and $51\%+2\%=53\%$ . Since the minimum value of the proportion is $49\%$ , it is not known for certain that majority of adults in the United States follow football.

Conclusion:

  $2500$ people were surveyed. It is not known for certain that majority of adults in the United States follow football.