91Ó°ÊÓ

Construct a confidence interval of the population proportion at the given level of confidence. \(x=80, n=200,98 \%\) confidence

A simple random sample of size \(n\) is drawn from a population that is normally distributed. The sample mean, \(\bar{x},\) is found to be \(50,\) and the sample standard deviation, \(s,\) is found to be \(8 .\) (a) Construct a \(98 \%\) confidence interval for \(\mu\) if the sample size, \(n,\) is 20 (b) Construct a \(98 \%\) confidence interval for \(\mu\) if the sample size, \(n\), is \(15 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(95 \%\) confidence interval for \(\mu\) if the sample size, \(n\), is 20. Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, \(E\) ? (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?

Clayton Kershaw of the Los Angeles Dodgers is one of the premier pitchers in baseball. His most popular pitch is a four-seam fastball. The data in the next column represent the pitch speed (in miles per hour) for a random sample of 18 of his four-seam fastball pitches. $$ \begin{array}{llllll} \hline 93.63 & 93.83 & 94.18 & 94.71 & 95.52 & 95.07 \\ \hline 95.12 & 95.35 & 94.15 & 94.62 & 96.08 & 93.86 \\ \hline 94.75 & 94.70 & 95.28 & 95.49 & 95.77 & 93.34 \\ \hline \end{array} $$ (a) Is "pitch speed" a quantitative or qualitative variable? Why is it important to know this when determining the type of confidence interval you may construct? (b) Draw a normal probability plot to verify that "pitch speed" could come from a population that is normally distributed. (c) Draw a boxplot to verify the data set has no outliers. (d) Are the requirements for constructing a confidence interval for the mean pitch speed of a Clayton Kershaw four-seam fastball satisfied? (e) Construct and interpret a \(95 \%\) confidence interval for the mean pitch speed of a Clayton Kershaw four-seam fastball. (f) Do you believe that a \(95 \%\) confidence interval for the mean pitch speed of all major league pitchers' four-seam fastbal would be narrower or wider? Why?

Construct a confidence interval of the population proportion at the given level of confidence. \(x=540, n=900,96 \%\) confidence

Sleep apnea is a disorder in which you have one or more pauses in breathing or shallow breaths while you sleep. In a cross-sectional study of 320 individuals who suffer from sleep apnea, it was found that 192 had gum disease. Note: In the general population, about \(17.5 \%\) of individuals have gum disease. (a) What does it mean for this study to be cross-sectional? (b) What is the variable of interest in this study? Is it qualitative or quantitative? Explain. (c) Estimate the proportion of individuals who suffer from sleep apnea who have gum disease with \(95 \%\) confidence. Interpret your result.

A \(90 \%\) confidence interval for the number of hours that full-time college students sleep during a weekday is lower bound: 7.8 hours and upper bound: 8.8 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. (a) \(90 \%\) of full-time college students sleep between 7.8 hours and 8.8 hours. (b) We are \(90 \%\) confident that the mean number of hours of sleep that full- time college students get any day of the week is between 7.8 hours and 8.8 hours. (c) There is a \(90 \%\) probability that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours. (d) We are \(90 \%\) confident that the mean hours of sleep that fulltime college students get during a weekday is between 7.8 hours and 8.8 hours.

A USA Today/Gallup poll asked 1006 adult Americans how much it would bother them to stay in a room on the 13 th floor of a hotel. Interestingly, \(13 \%\) said it would bother them. The margin of error was 3 percentage points with \(95 \%\) confidence. Which of the following represents a reasonable interpretation of the survey results? For those not reasonable, explain the flaw. (a) We are \(95 \%\) confident that the proportion of adult Americans who would be bothered to stay in a room on the 13th floor is between 0.10 and 0.16 . (b) We are between \(92 \%\) and \(98 \%\) confident that \(13 \%\) of adult Americans would be bothered to stay in a room on the 13th floor. (c) In \(95 \%\) of samples of adult Americans, the proportion who would be bothered to stay in a room on the 13 th floor is between 0.10 and 0.16 . (d) We are \(95 \%\) confident that \(13 \%\) of adult Americans would be bothered to stay in a room on the 13 th floor.

The trade magazine QSR routinely checks the drive-through service times of fast-food restaurants. A \(90 \%\) confidence interval that results from examining 607 customers in Taco Bell's drive-through has a lower bound of 161.5 seconds and an upper bound of 164.7 seconds. What does this mean?

A Rasmussen Reports national survey of 1000 adult Americans found that \(18 \%\) dreaded Valentine's Day. The margin of error for the survey was 4.5 percentage points with \(95 \%\) confidence. Explain what this means.

A survey of 2306 adult Americans aged 18 and older conducted by Harris Interactive found that 417 have donated blood in the past two years. (a) Obtain a point estimate for the population proportion of adult Americans aged 18 and older who have donated blood in the past two years. (b) Verify that the requirements for constructing a confidence interval about \(p\) are satisfied. (c) Construct a \(90 \%\) confidence interval for the population proportion of adult Americans who have donated blood in the past two years. (d) Interpret the interval.