The 112th Congress of the United States of America has 535 members, of which 87 are women. An alien lands near the U.S. Capitol and treats members of Congress as a random sample of the human race. He reports to his superiors that a 95% confidence interval for the proportion of the human race that is female has a lower bound of 0.131 and an upper bound of 0.194. What is wrong with the alien's approach to estimating the proportion of the human race that is female?
Q780
Constructing a Confidence Interval about a Population Mean
The Web site fueleconomy.gov allows drivers to report the miles per gallon of their vehicle. The data in Table 3 show the reported miles per gallon of 2007 Ford Taurus automobiles for 13 different owners. Treat the sample as a simple random sample of all 2007 Ford Taurus automobiles. Construct a 95% confidence interval for the mean miles per gallon of a 2007 Ford Taurus. Interpret the interval.
Q781
1. (a) Find thet-value such that the area in the right tail is 0.10 with 25 degrees of freedom.
(b) Find thet-value such that the area in the right tail is 0.05 with 30 degrees of freedom.
(c) Find thet-value such that the area left of thet-value is 0.01 with 18 degrees of freedom.
(d) Find the criticalt-value that corresponds to 90% confidence. Assume 20 degrees of freedom.
2. (a) Find thet-value such that the area in the right tail is 0.02 with 19 degrees of freedom.
(b) Find thet-value such that the area in the right tail is 0.10 with 32 degrees of freedom.
(c) Find thet-value such that the area left of thet-value is 0.05 with 6 degrees of freedom.
(d) Find the criticalt-value that corresponds to 95% confidence. Assume 16 degrees of freedom.
Q782
A simple random sample of sizenis drawn from a population that is normally distributed. The sample mean,x, is found to be 108, and the sample standard deviation,s, is found to be 10.
(a) Construct a 96% confidence interval for µ if the sample size,n, is 25.
(b) Construct a 96% confidence interval for µ if the sample size,n, is 10. How does decreasing the sample size affect the margin of error,E?
(c) Construct a 90% confidence interval for µ if the sample size,n, is 25. Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the size of 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?
Q783
A simple random sample of sizenis drawn from a population that is normally distributed. The sample mean,x, is found to be 50, and the sample standard deviation,s, is found to be 8.
(a) Construct a 98% confidence interval for µ if the sample size,n, is 20.
(b) Construct a 98% confidence interval for µ 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 µ 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?
Q784
A simple random sample of sizenis drawn. The sample mean,x, is found to be 18.4, and the sample standard deviation,s, is found to be 4.5.
(a) Construct a 95% confidence interval for µ if the sample size,n, is 35.
(b) Construct a 95% confidence interval for µ if the sample size,n, is 50. How does increasing the sample size affect the margin of error,E?
(c) Construct a 99% confidence interval for µ if the sample size,n, is 35. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the margin of error,E?
(d) If the sample size isn= 15, what conditions must be satisfied to compute the confidence interval?
Q785
A simple random sample of sizenis drawn. The sample mean,x, is found to be 35.1, and the sample standard deviation,s, is found to be 8.7.
(a) Construct a 90% confidence interval for µ if the sample size,n, is 40.
(b) Construct a 90% confidence interval for µ if the sample size,n, is 100. How does increasing the sample size affect the margin of error,E?
(c) Construct a 98% confidence interval for µ if the sample size,n, is 40. Compare the results to those obtained in part (a). How does increasing the level of confidence affect the margin of error,E?
(d) If the sample size isn= 18, what conditions must be satisfied to compute the confidence interval?
Q786
You Explain It! Hours Worked In a survey conducted by the Gallup organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a 95% confidence interval for mean number of hours worked was lower bound: 42.7 and upper bound: 44.5. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw.
(a) There is a 95% probability the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours.
(b) We are 95% confident that the mean number of hours worked by adult Americans in the previous week was between 42.7 hours and 44.5 hours.
(c) 95% of adult Americans worked between 42.7 hours and 44.5 hours last week.
(d) We are 95% confident that the mean number of hours worked by adults in Idaho in the previous week was between 42.7 hours and 44.5 hours.
Q787
You Explain It! Sleeping 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 full-time college students get during a weekday is between 7.8 hours and 8.8 hours.
Q788
a. You Explain It! Drive-Through Service Time 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?
b. You Explain It! MySpace.com According to Nielsen/ Net Ratings, the mean amount of time spent on MySpace.com per user per month in July 2007 was 171.0 minutes. A 95% confidence interval for the mean amount of time spent on MySpace.com monthly has a lower bound of 151.4 minutes and an upper bound of 190.6 minutes. What does this mean?
c. Hours Worked Revisited For the "Hours Worked" survey conducted by Gallup in Problem 23, provide two recommendations for increasing the precision of the interval.
d. Sleeping Revisited Refer to the "Sleeping" results from Problem 24. What could be done to increase the precision of the confidence interval?
Q789
Blood Alcohol Concentration A random sample of 51 fatal crashes in 2009 in which the driver had a positive blood alcohol concentration (BAC) from the National Highway Traffic Safety Administration results in a mean BAC of 0.167 grams per deciliter (g/dL) with a standard deviation of 0.010 g/dL.
(a) A histogram of blood alcohol concentrations in fatal accidents shows that BACs are highly skewed right. Explain why a large sample size is needed to construct a confidence interval for the mean BAC of fatal crashes with a positive BAC.
(b) In 2009, there were approximately 25,000 fatal crashes in which the driver had a positive BAC. Explain why this, along with the fact that the data were obtained using a simple random sample, satisfies the requirements for constructing a confidence interval.
(c) Determine and interpret a 90% confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC.
(d) All 50 states and the District of Columbia use a BAC of 0.08 g/dL as the legal intoxication level. Is it possible that the mean BAC of all drivers involved in fatal accidents who are found to have positive BAC values is less than the legal intoxication level? Explain.