# Of 118 randomly selected adults, 34 were found to have high blood pressure. Construct a 95%...

## Question:

Of {eq}118 {/eq} randomly selected adults, {eq}34 {/eq} were found to have high blood pressure. Construct a {eq}95\% {/eq} confidence interval for the true percentage of all adults that have high blood pressure.

## Confidence interval

The confidence interval provides the estimated range of values which contains an unknown population parameter. There are two types of a confidence interval, one-sided confidence interval and two-sided confidence interval.

Given information

• Sample size: 118
• Number of success: 34

The value of sample proportion is calculated as follow.

{eq}\begin{align*} \hat P &= \dfrac{X}{n}\\ &= \dfrac{{34}}{{118}}\\ &= 0.288 \end{align*} {/eq}

The critical value is obtained from the standard normal distribution at level of significance (0.05)

The Critical value is 1.96

The 95% confidence interval for the true percentage of all adults that have high blood pressure is calculated as follow.

{eq}\begin{align*} P\left( {\hat P - {Z_{\alpha /2}}\sqrt {\dfrac{{\hat P\left( {1 - \hat P} \right)}}{n}} < p < \hat P + {Z_{\alpha /2}}\sqrt {\dfrac{{\hat P\left( {1 - \hat P} \right)}}{n}} } \right) &= 0.95\\ P\left( {0.288 - 1.96\sqrt {\dfrac{{0.288\left( {1 - 0.288} \right)}}{{118}}} < p < 0.288 + 1.96\sqrt {\dfrac{{0.288\left( {1 - 0.288} \right)}}{{118}}} } \right) &= 0.95\\ P\left( {0.206 < p < 0.369} \right) &= 0.95 \end{align*}{/eq}

Interpretation: There is 95% confident that the true percentage of all adults that have high blood pressure is lies between confidence limits (0.206 to 0.369)