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Determine whether we can safely use a \(t^{*}\) critical value to calculate a confidence interval for the population mean in each of the following settings. (a) We collect data from a random sample of adult residents in a state. Our goal is to estimate the overall percent of adults in the state who are college graduates. (b) The coach of a college men's basketball team records the resting heart rates of the 15 team members. We use these data to construct a confidence interval for the mean resting heart rate of all male students at this college. (c) Do teens text more than they call? To find out, an \(\mathrm{AP}^{8}\) Statistics class at a large high school collected data on the number of text messages and calls sent or received by each of 25 randomly selected students. The Fathom boxplot below displays the difference (texts - calls) for each student.

A medical study finds that \(\bar{x}=114.9\) and \(s_{x}=9.3\) for the seated systolic blood pressure of the 27 members of one treatment group. What is the standard error of the mean? Interpret this value in context.

When two lights close together blink alternately, we "see" one light moving back and forth if the time between blinks is short. What is the longest interval of time between blinks that preserves the illusion of motion? Ask subjects to turn a knob that slows the blinking until they "see" two lights rather than one light moving. A report gives the results in the form "mean plus or minus the standard error of the mean." \({ }^{22}\) Data for 12 subjects are summarized as \(251 \pm 45\) (in milliseconds). (a) Find the sample standard deviation \(s_{x}\) for these measurements. Show your work. (b) A hasty reader believes that the interval given in the report is a \(95 \%\) confidence interval for the population mean. Find the actual confidence level for the given interval.

The body mass index (BMI) of all American young women is believed to follow a Normal distribution with a standard deviation of about 7.5. How large a sample would be needed to estimate the mean BMI \(\mu\) in this population to within ±1 with \(99 \%\) confidence? Show your work.