Chapter 19, Problem 19.1AYK

(a)

To determine

To explain:

The population in words that the parameter $p$ of the given sample.

Answer to Problem 19.1AYK

  $p\approx 0.568$

Explanation of Solution

Given:

Sample of cultures from individuals in Florida $=1714$

Concept used:

Formula

  $p=\frac{x}{n}$

Significance level

  $\begin{array}{l}\alpha =1-\text{confidence}\\ \alpha =1-0.95\end{array}$

Calculation:

Parameter $p=$ true proportion of individuals in population

The best point of estimate of $p$ is sample proportion

  $\begin{array}{l}p=\frac{x}{n}\\ p=\frac{973}{1714}\\ p\approx 0.567678\\ p\approx 0.568\end{array}$

(b)

To determine

To find:The population in words that the numerical value of the statistic $p$ of the given sample.

Answer to Problem 19.1AYK

  $\left(0.544,0.591\right)$

Explanation of Solution

Given:

Sample of cultures from individuals in Florida $=1714$

Concept used:

Formula

  $p=\frac{x}{n}$

Significance level

  $\begin{array}{l}\alpha =1-\text{confidence}\\ \alpha =1-0.95\end{array}$

Calculation:

Parameter $p=$ true proportion of individuals in population

The best point of estimate of $p$ is sample proportion

  $\begin{array}{l}p=\frac{x}{n}\\ p=\frac{973}{1714}\\ p\approx 0.567678\\ p\approx 0.568\end{array}$

Significance level

  $\begin{array}{l}\alpha =1-\text{confidence}\\ \alpha =1-0.95\\ \alpha =0.05\end{array}$

Critical $z$ value

  $\begin{array}{l}=z_{\alpha /2}\\ =z_{0.025}\\ =1.96\end{array}$

Margin of error

  $\begin{array}{l}E=z_{\alpha /2}\sqrt{\frac{\hat{p}\times \left(1-\hat{p}\right)}{n}}\\ E=1.96\sqrt{\frac{973/1714\times \left(1-973/1714\right)}{1714}}\\ E=1.96\times 0.011966\\ E\approx 0.023453\end{array}$

At $95\%$ confidence interval for the proportion $p$

  $\begin{array}{l}\hat{p}-E<P<\hat{p}+E\\ 973/1714-0.0234<P<973/1714+0.0234\\ 0.544<P<0.591\\ \left(0.544,0.591\right)\end{array}$