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Harris Announces a Margin of Error. Exercise \(22.25\) describes a Harris Poll survey of smokers in which 848 of a sample of 1010 smokers agreed that smoking would probably shorten their lives. Harris announces a margin of error of \(\pm 3\) percentage points for all samples of about this size. Opinion polls announce the margin of error for \(95 \%\) confidence. a. What is the actual margin of error (in percent) for the large-sample confidence interval from this sample? b. The margin of error is largest when \(\hat{p}=0.5\). What would the margin of error (in percent) be if the sample had resulted in \(\widehat{p}=0.5\) ? c. Why do you think that Harris announces a \(\pm 3 \%\) margin of error for all samples of about this size?

Running Red Lights. A random digit dialing telephone survey of 880 drivers asked, "Recalling the last 10 traffic lights you drove through, how many of them were red when you entered the intersections?" Of the 880 respondents, 171 admitted that at least one light had been red. 25 a. Give a \(95 \%\) confidence interval for the proportion of all drivers who ran one or more of the last 10 red lights they met. b. Nonresponse is a practical problem for this survey: only \(21.6 \%\) of calls that reached a live person were completed. Another practical problem is that people may not give truthful answers. What is the likely direction of the bias? Do you think more or fewer than 171 of the 880 respondents really ran a red light? Why?