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.

Answer and Explanation:

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)


Learn more about this topic:

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Finding Confidence Intervals for Proportions: Formula & Example

from Statistics 101: Principles of Statistics

Chapter 9 / Lesson 8
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