Question

The report "2005 Electronic Monitoring \& Surveillance Survey: Many Companies Monitoring, Recording, Videotaping-and Firing-Employees" (American Management Association, 2005 ) summarized the results of a survey of 526 U.S. businesses. The report stated that 137 of the 526 businesses had fired workers for misuse of the Internet and 131 had fired workers for email misuse. For purposes of this exercise, assume that it is reasonable to regard this sample as representative of businesses in the United States. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of U.S. businesses that have fired workers for misuse of the Internet. b. What are two reasons why a \(90 \%\) confidence interval for the proportion of U.S. businesses that have fired workers for misuse of email would be narrower than the \(95 \%\) confidence interval computed in Part (a).

Short answer

The 95% confidence interval for the proportion of U.S. businesses that have fired workers for misuse of the internet is calculated using the formula for confidence intervals and the calculated proportion. The 90% confidence interval would be narrower because a lower degree of confidence results in more accepted uncertainty and a lower z-score.

Step-by-step solution

Calculate the Internet Misuse Proportion

The total number of businesses surveyed is 526, and out of these 137 had fired workers for misuse of the internet. The proportion of businesses (p) which have fired workers for internet misuse is thus calculated as \(p = \frac{137}{526} \)

Calculate the Confidence Interval for Internet Misuse

The confidence interval is obtained through the formula \(CI = p ± Zα/2 * SE \), where p is the proportion previously calculated in step 1, Zα/2 is the Z score that corresponds to the desired level of confidence (In this case, it would be 1.96 for a 95% confidence level), and SE (Standard Error) is calculated as \(SE = \sqrt{\frac{p(1-p)}{n}}\), where n is the sample size and p the proportion.

Interpret the Confidence Interval

The result from step 2 gives us an interval with a lower and higher bound. This is interpreted as the range in which we are 95% confident that the true proportion of U.S. businesses that have fired workers for internet misuse falls.

Reasons for a Narrower 90% Confidence Interval

The first reason why a 90% confidence interval would be narrower is because a lower degree of confidence means we accept more uncertainty about where the true value is and thus results in a narrower range. Secondly, the Zα/2 score for a 90% confidence interval is lower (1.645) compared to the Zα/2 score for a 95% confidence interval (1.96), which directly reduces the margin of error and hence results in a narrower confidence interval.