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# Exercises

Exercises for Sections  7.1 and  7.2

1.
There are two basic forms of sleep: slow wave sleep (SWS) and rapid eye movement (REM) sleep. Infants spend about 50% of their sleep time in SWS and 50% in REM sleep. Adults below age 60 spend about 20% of their sleep time in REM and 80% in SWS sleep. In a study of sleep patterns, data was collected on 13 elderly males over age 60. The percentage of total sleep time spent in REM sleep is presented below.
21, 20, 22, 7, 9, 14, 23, 9, 10, 25, 15, 17, 11
(a)
Calculate the sample average and standard deviation.
(b)
Calculate the population average and standard deviation, if possible.
(c)
Do you think the sample average equals the population average? If not, by how much do you expect it to miss?
(d)
Calculate the standard error (SE) of the sample average.
(e)
With 95% confidence, the estimate should miss the population average by less than _________.
(f)
The population average should be between _________ and _________ with 95% confidence.
(g)
If the level of confidence above is reduced to 90%, will the new interval be shorter or longer?
(h)
Calculate a 90% confidence interval for the population average.

2.
Twenty-five factory workers total were asked how many vacation days they take a year. The average of the sample was 22.85 days, and the standard deviation of the sample was 5.80.
(a)
Do you think the population average equals 22.85? If not, do you think the population average is higher or lower than 22.85 days? By how much do you expect it to miss?
(b)
Calculate the standard error for the sample average.
(c)
Construct a 95% confidence interval for the average number of vacation days per year taken by factory workers.

3.
A new breakfast cereal Frosted Corn is test marketed for one month. The total sales for the first 9 quarters (3-month periods) indicate an average of $8350, with a standard deviation of$1840.
(a)
Treating the first 9 quarters as a sample of size 9, do you think the sample average equals the 'population' average of future sales? If not, can you calculate a value which estimates how much you expect it to miss?
(b)
Is it reasonable to assume that the first 9 quarters is a random sample from, say, the first 40 quarters (a 10-year period)?
(c)
Calculate a standard error for the sample average.
(d)
Construct a 90% confidence interval for the 'population' average of future quarterly sales.

Exercises for Section  7.3

4.
A Michigan automobile insurance company would like to estimate the average size of an accident claim. A random sample of 20 accident claims yielded an average of $1600 with a standard deviation of$800.
(a)
What is the standard error of the $1600 estimate? (b) What is the 95% margin of error of the estimate? (c) Estimate the true average accident claim with a 95% interval estimate. (d) Suppose the confidence interval above is considered too wide. What sample size is necessary to reduce the width of the confidence interval to one-third of its original size? (e) What sample size is necessary to reduce the 95% margin of error to$50?

5.
In order to test a new production method, 15 employees were selected randomly to try the new method. The mean production rate for the sample was 80 parts per hour, and the standard deviation was calculated to be 10 parts per hour.
(a)
Give a 95% confidence interval for the true mean production rate for the new method.
(b)
Suppose the confidence interval in (a) is considered too wide. What sample size is necessary to reduce the width of the confidence interval to one half of its original size?
(c)
What sample size is necessary to reduce the standard error of the sample mean to 1.5 parts per hour?
(d)
What is the 95% margin of error of the estimate?
(e)
What sample size is necessary to reduce the 95% margin of error to 1.5 parts per hour?

6.
Safe Lumber Company hires an independent contractor to monitor the groundwater within a one mile radius of its factory to ensure that no toxic levels of arsenic are being released. The contractor takes a sample of groundwater from selected sites within the one mile radius. Safe Lumber will need to pay for an expensive clean-up if a confidence interval of the average arsenic level in groundwater lies completely above 50 parts per billion (for example,if the confidence interval is (75,95)).
(a)
If Safe Lumber wanted to avoid an expensive clean-up, should it prefer a 90% or 95% confidence interval?
(b)
If Safe Lumber wanted to avoid an expensive clean-up, should it prefer a sample size of 9 or 25?
(c)
Suppose that data from 9 sites had arsenic levels averaging 75 ppb with a sample SD of 30 ppb. Construct 90% and 95% confidence intervals for the average arsenic level.
(d)
Suppose that data from 25 sites resulted in the same average of 75 ppb and SD of 30 ppb. Construct 90% and 95% confidence intervals for the average arsenic level.
(e)
If Safe Lumber wanted to avoid an expensive clean-up, would it prefer to sample from high ground areas or low ground areas?
(f)
If Safe Lumber wanted to avoid an expensive clean-up, would it prefer to sample closer or farther from the factory?

7.
A public broadcasting station began a week-long on-air promotional telethon to get viewers to pledge money to help support the station. The first day yielded 55 callers with an average pledge of $48 and a standard deviation of$19.
(a)
If these 55 callers are treated as a random sample from the population of callers yet to come, determine a 95% interval estimate for the average amount of all pledges.
(b)
How many callers are needed before the population average pledge may be estimated to within $2 with 95% confidence? 8. In developing patient appointment schedules, a medical center desires to estimate the mean time that a staff member spends with each patient. How large a sample should be taken if the precision of the estimate is to be within minutes at a 95% level of confidence? Assume the population standard deviation is estimated to be around 8 minutes. 9. A quarter barrel'' keg of beer is supposed to contain 8 gallons. The quality inspector at a local brewing company wants to estimate the average volume in the kegs to within 1/16 gallon (i.e. 1/2 pint) with 95% confidence. (a) Assuming the volume of beer in the kegs has standard deviation 1/4 gallons, what sample size is needed? (b) If the standard deviation estimate is changed from 1/4 to 1/8 gallon, will the required sample size be bigger or smaller? (c) If the margin of error is increased from 1/16 to 1/8 gallon, will the required sample size be bigger or smaller? (d) if the confidence level is changed from 95% to 90%, will the required sample size be bigger or smaller? Exercises for Section 7.4 10. A personnel agency is comparing salaries of uncertified accountants versus CPA's. Among recent accountants that they placed, 4 were uncertified while 6 were CPA's. The annual salaries (in dollars) are summarized in the following table:  Uncertified CPA Average:$48,500 $56,600 SD:$8,700 \$9,100 Sample size: 4 6

(a)
On the average, CPA certification is worth an estimated __________ dollars per year according to the data.
(b)
Calculate a standard error for the estimate in (a).
(c)
Calculate a 95% margin of error for the estimate in (a).
(d)
Calculate a 95% confidence interval for the difference between average salaries of uncertified accountants and CPA's.
(e)
What assumptions did you need for your confidence interval?

11.
Do credit cards with no annual fee charge higher interest rates than cards that have annual fees? Among 29 cards surveyed, 17 had no annual fees while 12 charged an annual fee. Among the cards with no annual fee, the average interest rate was 19% (SD=8%). Among cards with an annual fee, the average interest rate was 17% (SD=3%).
(a)
On the average, cards with no annual fee charge an estimated _______ percent higher APR than cards with annual fees.
(b)
Calculate a standard error for the estimate in (a).
(c)
Calculate a 95% margin of error for the estimate in (a).
(d)
Construct a 95% interval estimate for the difference in average interest rates.
(e)
What assumptions did you need for your confidence interval?

Exercises for Section  7.5

12.
A new gasoline additive is supposed to make gas burn more cleanly and increase gas mileage in the process. Consumer Protection Anonymous conducted a mileage test to confirm this. They took seven of their cars, filled it with regular gas, and drove it on I-94 until it was empty. They repeated the process using the same cars, but using the gas additive. The recorded gas mileage follows:
 Car 1 2 3 4 5 6 7 Without Additive 22 15 18 28 12 25 18 With Additive 26 19 17 34 17 25 22
(a)
For each of the 7 cars, calculate the mileage difference (With-Without), between the two fuel types.
(b)
Calculate the average and SD for the differences in (a).
(c)
Fill in the blanks: On the average, the gas additive increases mileage by _______ mpg, give or take _______ mpg or so (Reminder: The SE, not the SD, goes in the second blank.)
(d)
Calculate a 95% margin of error.
(e)
Construct a 95% confidence interval for the 'population' mileage effect of the additive.

13.
Some stock market analysts have speculated that parts of West Michigan Telecom might be worth more than the whole. For example, the company's communication systems in Ann Arbor and Detroit could be sold to other communications companies. Suppose that a stock market analyst chose 9 acquisition experts and asked each to predict the return (in percent) on investment in the company held to the year 2003 if (i) it does business as usual, or (ii) if it breaks up its communication systems and sells all its parts. Their predictions follow:
 Expert 1 2 3 4 5 6 7 8 9 Not Break Up 12 21 8 20 16 5 18 21 10 Break Up 15 25 12 17 17 10 21 28 15
(a)
On the average, experts believe breaking up the company will earn an estimated _______ percent higher return.
(b)
Calculate a standard error for the estimate in (a).
(c)
Calculate a 95% margin of error for the estimate in (a).
(d)
Construct a 95% interval estimate for the difference in average predicted return.
(e)
What assumptions did you need for your confidence interval above?

Exercises for Section  7.6

14.
A survey conducted by USA Today showed that 118 of 250 investors owned some real estate.
(a)
What proportion of the surveyed investors owned some real estate?
(b)
Do you think the proportion for the population equals your estimate in (a)? If not, by how much do you expect it to miss?
(c)
Calculate a standard error (SE) for your estimate in (a).
(d)
Find two numbers that satisfy the following statement: "The population proportion should be between _________ and _________ with 95% confidence."
(e)
If the level of confidence above is reduced to 90%, will the new interval be shorter or longer?
(f)
Calculate a 90% confidence interval for the population proportion.

15.
A survey conducted by USA Today showed that 118 of 250 investors owned some real estate.
(a)
What percentage of the surveyed investors owned some real estate?
(b)
Do you think the percentage for the population equals your estimate in (a)? If not, by how much do you expect it to miss?
(c)
Calculate a standard error (SE) for your estimate in (a).
(d)
Find two numbers that satisfy the following statement: "The population percentage should be between _________ and _________ with 95% confidence."
(e)
If the level of confidence above is reduced to 90%, will the new interval be shorter or longer?
(f)
Calculate a 90% confidence interval for the population percentage.

16.
An appliance manufacturer offers maintenance contracts on its major appliances. A manager wants to know what fraction of buyers of the company's convection ovens are also buying the maintenance contract with the oven. From a random sample of 120 sales slips, 31 of the oven buyers opted for the contract.
(a)
The proportion of customers who buy the contract along with their oven is estimated as _________.
(b)
Calculate a standard error for the estimate in (a).
(c)
Calculate a 95% interval estimate for the true proportion of customers who buy the contract along with their oven.

Exercises for Section  7.7

17.
Researchers who were concerned if doctors were consistently adjusting dosages for weight of elderly patients studied 2000 prescriptions. They found that for 600 of the prescriptions, the doctors failed to adjust the dosages.

(a)
Doctors fail to adjust dosage for an estimated _________ percent of prescriptions.
(b)
Calculate a standard error for the percentage in (a).
(c)
Calculate a 95% margin of error for the percentage in (a).
(d)
Calculate a 95% interval estimate for the true proportion
(e)
Calculate a 95% confidence interval for the true percentage of prescriptions where doctors fail to adjust dosages.

18.
The Tourism Bureau of South Carolina plans to sample visitors at major beaches throughout the state to estimate the proportion of beach goers who are not South Carolina residents. Preliminary estimates reveal that 55% are not South Carolina residents. How large a sample must be taken to estimate the proportion of out-of-state visitors with a .03 margin of error? Use a 95% confidence level.

19.
The production manager for a large city newspaper wants to determine the proportion of newspapers printed that have a defect, such as excessive ruboff, missing pages, duplicate pages, and so on. A random sample of 20 newspapers resulted in 2 containing some kind of defect.
(a)
The percentage of defective newspapers is estimated as _________.
(b)
What is the standard error of the above estimate? .
(c)
Construct a 95% confidence interval for the defect percentage.
(d)
What is the 95% margin of error of the above estimate? .
(e)
How many newspapers need to be sampled to reduce the 95% margin of error to 5%?
(f)
How many newspapers need to be sampled to have a 90% margin of error equal to 5%?

Exercises for Section  7.8

20.
According to a survey by Town Plaza Hotels, 12% of 150 men who were travelling on business brought a friend or spouse. In contrast, 28% of 101 women in the survey who were travelling on business brought a friend or spouse.
(a)
According to the data, there is an estimated difference of _________ between the the percentage of men and women who brought a friend or spouse along on business travel.
(b)
Calculate a standard error for the estimate in (a).
(c)
Calculate a 95% margin of error for the estimate in (a).
(d)
Construct a 95% interval estimate for the difference between the two percentages.

21.
Borrowers Bank classifies credit cardholders as "prime" whenever they pay interest charges for at least 6 of the last 12 billing cycles. The table below cross classifies 2080 cardholders according to whether they are (i) prime or not, and (ii) below age 30 or not

 Below 30 30 and Over Prime 475 510 Not Prime 376 719

(a)
Construct a 95% confidence interval for the percentage of prime cardholders among those who are less than 30 years old
(b)
Construct a 95% confidence interval for the percentage of prime cardholders among those who are 30 years or older.
(c)
Construct a 95% confidence interval for the difference between the two percentages in (a) and (b)

Next: Testing Hypotheses Up: Confidence Intervals Previous: Computing a Confidence Interval p p

2003-09-08