Question

Critical Thinking: What does the survey tell us? Surveys have become an integral part of our lives. Because it is so important that every citizen has the ability to interpret survey results, surveys are the focus of this project. The Pew Research Center recently conducted a survey of 1007 U.S. adults and found that 85% of those surveyed know what Twitter is.

Analyzing the Data

Use the survey results to construct a 95% confidence interval estimate of the percentage of all adults who know what Twitter is.

Short answer

The 95% confidence interval estimate of the percentage of all adults, who know what Twitter is, is equal to (82.8%, 87.2%).

Step-by-step solution

Given information

A survey consisted of 1007 U.S. adults. 85% of those who were surveyed know what Twitter is.

Confidence interval of population proportion

The formula of the confidence interval for a population proportion is given as follows:

\(\begin{array}{c}CI = \left( {\hat p - E,\hat p + E} \right)\\ = \left( {\hat p - {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} ,\hat p + {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} } \right)\end{array}\)

Where,\(\hat p\)be the sample proportion, E is the margin of error, n is the sample size,\({z_{\frac{\alpha }{2}}}\)is the two-tailed critical value obtained from standard normal table.

Also,

\(\hat q = 1 - \hat p\)

Compute the confidence interval

The proportion of adults who know what Twitter is is shown below:

\(\begin{array}{c}\hat p = 85\% \\ = \frac{{85}}{{100}}\\ = 0.85\end{array}\)

The sample size (n) is equal to 1007.

The confidence level is given to be equal to 95%. This implies that the level of significance is equal to 0.05.

The value of\({z_{\frac{\alpha }{2}}}\)becomes equal to 1.96.

The following computation is made to construct the confidence interval estimate of the proportion of all adults who know what Twitter is:

\(\begin{array}{c}CI = \left( {\hat p - E,\hat p + E} \right)\\ = \left( {\hat p - {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} ,\hat p + {z_{\frac{\alpha }{2}}}\sqrt {\frac{{\hat p\hat q}}{n}} } \right)\\ = \left( {0.85 - \left( {1.96} \right)\sqrt {\frac{{\left( {0.85} \right)\left( {1 - 0.85} \right)}}{{1007}}} ,0.85 + \left( {1.96} \right)\sqrt {\frac{{\left( {0.85} \right)\left( {1 - 0.85} \right)}}{{1007}}} } \right)\\ = \left( {0.828,0.872} \right)\end{array}\)

In terms of percentage, the confidence interval becomes as follows:

\(\begin{array}{c}CI = \left( {0.828,0.872} \right)\\ = \left( {82.8\% ,87.2\% } \right)\end{array}\)

Thus, the 95% confidence interval estimate of the percentage of all adults, who know what Twitter is, is equal to (82.8%, 87.2%).