7-1. Exercises
1. BASIC
Ex 1. A random sample is drawn from a population of known standard deviation 11.3. Construct a 90% confidence interval for the population mean based on the information given (not all of the information given need be used).
Ex 2. A random sample is drawn from a population of known standard deviation 22.1. Construct a 95% confidence interval for the population mean based on the information given (not all of the information given need be used).
Ex 3. A random sample is drawn from a population of unknown standard deviation. Construct a 99% confidence interval for the population mean based on the information given.
Ex 4. A random sample is drawn from a population of unknown standard deviation. Construct a 98% confidence interval for the population mean based on the information given.
Ex 5. A random sample of size 144 is drawn from a population whose distribution, mean, and standard deviation are all unknown. The summary statistics are and .
Construct an 80% confidence interval for the population mean μ.
Construct a 90% confidence interval for the population mean μ.
Comment on why one interval is longer than the other.
Ex 6. A random sample of size 256 is drawn from a population whose distribution, mean, and standard deviation are all unknown. The summary statistics are and .
Construct a 90% confidence interval for the population mean μ.
Construct a 99% confidence interval for the population mean μ.
Comment on why one interval is longer than the other.
2. APPLICATIONS
Ex 7. A government agency was charged by the legislature with estimating the length of time it takes citizens to fill out various forms. Two hundred randomly selected adults were timed as they filled out a particular form. The times required had mean 12.8 minutes with standard deviation 1.7 minutes. Construct a 90% confidence interval for the mean time taken for all adults to fill out this form.
Ex 8. Four hundred randomly selected working adults in a certain state, including those who worked at home, were asked the distance from their home to their workplace. The average distance was 8.84 miles with standard deviation 2.70 miles. Construct a 99% confidence interval for the mean distance from home to work for all residents of this state.
Ex 9. On every passenger vehicle that it tests an automotive magazine measures, at true speed 55 mph, the difference between the true speed of the vehicle and the speed indicated by the speedometer. For 36 vehicles tested the mean difference was −1.2 mph with standard deviation 0.2 mph. Construct a 90% confidence interval for the mean difference between true speed and indicated speed for all vehicles.
Ex 10. A corporation monitors time spent by office workers browsing the web on their computers instead of working. In a sample of computer records of 50 workers, the average amount of time spent browsing in an eight-hour work day was 27.8 minutes with standard deviation 8.2 minutes. Construct a 99.5% confidence interval for the mean time spent by all office workers in browsing the web in an eight-hour day.
Ex 11. A sample of 250 workers aged 16 and older produced an average length of time with the current employer (“job tenure”) of 4.4 years with standard deviation 3.8 years. Construct a 99.9% confidence interval for the mean job tenure of all workers aged 16 or older.
Ex 12. The amount of a particular biochemical substance related to bone breakdown was measured in 30 healthy women. The sample mean and standard deviation were 3.3 nanograms per milliliter (ng/mL) and 1.4 ng/mL. Construct an 80% confidence interval for the mean level of this substance in all healthy women.
Ex 13. A corporation that owns apartment complexes wishes to estimate the average length of time residents remain in the same apartment before moving out. A sample of 150 rental contracts gave a mean length of occupancy of 3.7 years with standard deviation 1.2 years. Construct a 95% confidence interval for the mean length of occupancy of apartments owned by this corporation.
Ex 14. The designer of a garbage truck that lifts roll-out containers must estimate the mean weight the truck will lift at each collection point. A random sample of 325 containers of garbage on current collection routes yielded , . Construct a 99.8% confidence interval for the mean weight the trucks must lift each time.
Ex 15. In order to estimate the mean amount of damage sustained by vehicles when a deer is struck, an insurance company examined the records of 50 such occurrences, and obtained a sample mean of $2,785 with sample standard deviation $221. Construct a 95% confidence interval for the mean amount of damage in all such accidents.
Ex 16. In order to estimate the mean FICO credit score of its members, a credit union samples the scores of 95 members, and obtains a sample mean of 738.2 with sample standard deviation 64.2. Construct a 99% confidence interval for the mean FICO score of all of its members.
3. ADDITIONAL EXERCISES
Ex 17. For all settings a packing machine delivers a precise amount of liquid; the amount dispensed always has standard deviation 0.07 ounce. To calibrate the machine its setting is fixed and it is operated 50 times. The mean amount delivered is 6.02 ounces with sample standard deviation 0.04 ounce. Construct a 99.5% confidence interval for the mean amount delivered at this setting. Hint: Not all the information provided is needed.
Ex 18. A power wrench used on an assembly line applies a precise, preset amount of torque; the torque applied has standard deviation 0.73 foot-pound at every torque setting. To check that the wrench is operating within specifications it is used to tighten 100 fasteners. The mean torque applied is 36.95 foot-pounds with sample standard deviation 0.62 foot-pound. Construct a 99.9% confidence interval for the mean amount of torque applied by the wrench at this setting. Hint: Not all the information provided is needed.
Ex 19. The number of trips to a grocery store per week was recorded for a randomly selected collection of households, with the results shown in the table. 2 2 2 1 4 2 3 2 5 4 2 3 5 0 3 2 3 1 4 3 3 2 1 6 2 3 3 2 4 4 Construct a 95% confidence interval for the average number of trips to a grocery store per week of all households.
Ex 20. For each of 40 high school students in one county the number of days absent from school in the previous year were counted, with the results shown in the frequency table. x 0 1 2 3 4 5 f 24 7 5 2 1 1
Construct a 90% confidence interval for the average number of days absent from school of all students in the county.
Ex 21. A town council commissioned a random sample of 85 households to estimate the number of four-wheel vehicles per household in the town. The results are shown in the following frequency table. x 0 1 2 3 4 5 f 1 16 28 22 12 6
Construct a 98% confidence interval for the average number of four-wheel vehicles per household in the town.
Ex 22. The number of hours per day that a television set was operating was recorded for a randomly selected collection of households, with the results shown in the table. 3.7 4.2 1.5 3.6 5.9 4.7 8.2 3.9 2.5 4.4 2.1 3.6 1.1 7.3 4.2 3.0 3.8 2.2 4.2 3.8 4.3 2.1 2.4 6.0 3.7 2.5 1.3 2.8 3.0 5.6 Construct a 99.8% confidence interval for the mean number of hours that a television set is in operation in all households.
4. LARGE DATA SET EXERCISES
Ex 23. Large Data Set 1 records the SAT scores of 1,000 students. Regarding it as a random sample of all high school students, use it to construct a 99% confidence interval for the mean SAT score of all students.
Ex 24. Large Data Set 1 records the GPAs of 1,000 college students. Regarding it as a random sample of all college students, use it to construct a 95% confidence interval for the mean GPA of all students.
Ex 25. Large Data Set 1 lists the SAT scores of 1,000 students.
Regard the data as arising from a census of all students at a high school, in which the SAT score of every student was measured. Compute the population mean .
Regard the first 36 students as a random sample and use it to construct a 99% confidence for the mean μ of all 1,000 SAT scores. Does it actually capture the mean ?
Ex 26. Large Data Set 1 lists the GPAs of 1,000 students.
Regard the data as arising from a census of all freshman at a small college at the end of their first academic year of college study, in which the GPA of every such person was measured. Compute the population mean .
Regard the first 36 students as a random sample and use it to construct a 95% confidence for the mean μ of all 1,000 GPAs. Does it actually capture the mean ?
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