Chapter 8.2, Problem 58E

To determine

To explain confidence level.

Answer to Problem 58E

The confidence level is 82%

Explanation of Solution

Given:

Sample size = n = 100

Confidence interval for proportion:

0.565 < p < 0.695

Formula:

Sample proportion:

  $\widehat{p}=\frac{Lower\lim it+Upper\lim it}{2}$

Confidence interval for proportion:

  $\widehat{p}\pm Z_{\alpha /2}\times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$

The sample proportion is,

  $\widehat{p}=\frac{0.565+0.695}{2}=0.63$

Using the formula of confidence interval,

  $\begin{array}{l}0.63+Z_{\alpha /2}\times \sqrt{\frac{0.63(1-0.63)}{100}}=0.695\\ Z_{\alpha /2}=\frac{0.065}{\sqrt{\frac{0.63(1-0.63)}{100}}}=1.35\end{array}$

Therefore,

We need to determine the probability that,

  $\begin{array}{l}P(-1.35<Z<1.35)=P(Z<1.35)-P(Z<-1.35)\\ P(-1.35<Z<1.35)=0.9115-0.0885\\ P(-1.35<Z<1.35)=0.8230=82.30\%=82\%\end{array}$

Hence, confidence level is 82%