Chapter 11.5, Problem 27E

To determine

To calculate:

Number of people would have to be surveyed.

Answer to Problem 27E

Number of people to be surveyed to be confident that sport $X$ is truly performed by more than half of the population is $2398$ .

Explanation of Solution

Given information:

People who prefer sport drink $X$ : $52\%$

People who prefer sport drink $Y$ : $48\%$

Calculation:

Proportion of people who prefer sport drink $X$ , $p=0.52$

To be confident that $X$ is truly preferred, the margin of error must be less than $0.52-0.50=0.02$ .

Margin of error is calculated as,

  $E=Z.\sqrt{\frac{p\left(1-p\right)}{n}}$

Where,

  $z=$

  $z-$ Score.

  $\widehat{p}=$ Sample proportion.

  $n=$ Sample size.

Lest assume, $95\%$ confidence level.

Thus, $z=1.96$

  $\begin{array}{l}0.02=\mathrm{1.96.}\sqrt{\frac{0.52\left(1-0.52\right)}{n}}\\ \sqrt{n}=48.96\\ n=2398\end{array}$

Therefore, number of people to be surveyed to be confident that sport $X$ is truly performed by more than half of the population is $2398$ .