Question

The study "Digital Footprints"(Pew Internet \& American Life Project, www.pewinternet.org, 2007 ) reported that \(47 \%\) of Internet users have searched for information about themselves online. The \(47 \%\) figure was based on a representative sample of Internet users. Suppose that the sample size was \(n=300\) (the actual sample size was much larger). a. Use the given information to estimate the proportion of Internet users who have searched for information about themselves online. What statistic did you use? b. Use the sample data to estimate the standard error of \(\hat{p}\). c. Calculate and interpret the margin of error associated with the estimate in Part (a).

Short answer

The proportion of internet users estimated to search for information about themselves online is 0.47 (or 47%). The standard error estimated from the sample data is calculated using the provided figures alongside the mentioned formula. The margin of error, calculated using the standard error and the z-score for a 95% confidence level, should be calculated following the mentioned formula.

Step-by-step solution

Calculate proportion

In order to estimate the proportion of Internet users who have searched for information about themselves online, the value given in the exercise should be used. As a percentage, this is given as 47%, or 0.47 when represented as a decimal. Thus, the estimate for the proportion, denoted as \(\hat{p}\), is 0.47. The statistic used here is the sample proportion.

Estimate standard error

The standard error of the sample proportion \(\hat{p}\) can be estimated using the formula \(SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\). Substituting the given values for \(\hat{p} = 0.47\) and \(n = 300\), the standard error \(SE_{\hat{p}} = \sqrt{\frac{0.47(1-0.47)}{300}}\).

Calculate the margin of error

The margin of error can be calculated using the formula \(E = Z * SE_{\hat{p}}\). Here Z value represents the z-score. We generally consider a 95% confidence level, it corresponds to a Z value of approximately 1.96. So, substitute Z = 1.96 and calculate \(E = 1.96 * SE_{\hat{p}}\)