# The Consumer Reports National Research Center conducted a telephone survey of 2000 adults to...

## Question:

The Consumer Reports National Research Center conducted a telephone survey of 2000 adults to learn about the major economic concerns for the future. The survey results showed that 1760 of the respondents think that the future health of Social Security is a major concern. Develop a 90% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern.

## Confidence interval

The confidence interval includes a lower and upper limit which is calculated with help the margin of error. The confidence interval is commonly calculated for the sample proportion, population mean and difference of two samples.

## Answer and Explanation:

Given information

Sample size: 2000

Number of success: 1760

The value of sample proportion is calculated as follow.

{eq}\begin{align*} \hat P &= \dfrac{X}{n}\\ & = \dfrac{{1760}}{{2000}}\\ &= 0.88 \end{align*}{/eq}

The 90% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern 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) &= 0.90\\ P\left( {0.88 - 1.645\sqrt {\dfrac{{0.88\left( {1 - 0.88} \right)}}{{2000}}} < p < 0.88 + 1.645\sqrt {\dfrac{{0.88\left( {1 - 0.88} \right)}}{{2000}}} } \right) &= 0.90\\ P\left( {0.868 < \mu < 0.891} \right) &= 0.90 \end{align*}{/eq}

Therefore, the required confidence interval is (0.868 to 0.891).