Response errors are also nonsampling errors. They occur when people do not know, will not say, or overstate. Virtually no statistical method is available to measure or control for nonsampling errors. The statistical techniques presented in this text are based on the assumption that none of these nonsampling errors were committed. The researcher must eliminate these errors through carefully planning and executing the research study.
1) For each of the following research projects, list three variables for stratification of the
sample.
a. A nationwide study of motels and hotels is being conducted. An attempt will be made to determine the extent of the availability of online links for customers. A sample of motels and hotels will be taken.
b. A consumer panel is to be formed by sampling people in Michigan. Members of the panel will be interviewed periodically in an effort to understand current consumer attitudes and behaviors.
c. A large soft drink company wants to study the characteristics of the U.S. bottlers of its products, but the company does not want to conduct a census.
d. The business research bureau of a large university is conducting a project in which the bureau will sample paper-manufacturing companies.
2) If a company employs 3,500 people and if a random sample of 175 of these employees has been taken by systematic sampling, what is the value of k? The researcher would start the sample selection between what two values? Where could the researcher obtain a frame for this study?
3) The Statistical Abstract of the United States published by the U.S. Census Bureau reports that the average annual consumption of fresh fruit per person is 99.9 pounds. The standard deviation of fresh fruit consumption is about 30 pounds. Suppose a researcher took a random sample of 38 people and had them keep a record of the fresh fruit they ate for one year.
a. What is the probability that the sample average would be less than 90 pounds?
b. What is the probability that the sample average would be between 98 and 105 pounds?
c. What is the probability that the sample average would be less than 112 pounds?
d. What is the probability that the sample average would be between 93 and 96 pounds?
4) Suppose the average checkout tab at a large supermarket is $65.12, with a standard deviation of $21.45. Twenty-three percent of the time when a random sample of 45 customer tabs is examined, the sample average should exceed what value?
5) A given population proportion is .25. For the given value of n, what is the probability of getting each of the following sample proportions?
a. n = 110 and pn … .21
b. n = 33 and pn 7 .24
c. n = 59 and .24 … pn 6 .27
d. n = 80 and pn 6 .30
e. n = 800 and pn 6 .30
6) Suppose a population proportion is .40, and 80% of the time when you draw a random sample from this population you get a sample proportion of .35 or more. How large a sample were you taking?
7) According to a study by Decision Analyst, 21% of the people who have credit cards are very close to the total limit on the card(s). Suppose a random sample of 600 credit card users is taken.What is the probability that more than 150 credit card users are very close to the total limit on their card(s)?
8) A Travel Weekly International Air Transport Association survey asked business travelers about the purpose for their most recent business trip. Nineteen percent responded that it was for an internal company visit. Suppose 950 business travelers are randomly selected.
a. What is the probability that more than 25% of the business travelers say that the reason for their most recent business trip was an internal company visit?
b. What is the probability that between 15% and 20% of the business travelers say that the reason for their most recent business trip was an internal company visit?
c. What is the probability that between 133 and 171 of the business travelers say that the reason for their most recent business trip was an internal company visit?
9) According to the U.S. Bureau of Labor Statistics, 20% of all people 16 years of age or older do volunteer work. In this age group, women volunteer slightly more than men, with 22% of women volunteering and 19% of men volunteering.What is the probability of randomly sampling 140 women 16 years of age or older and getting 35 or more who do volunteer work? What is the probability of getting 21 or fewer from this group? Suppose a sample further that a random telephone sample of 32 lawyers in Iowa is taken and that the sample average charge per hour for out-of-court work is $110. If the population variance is $525, what is the probability of getting a sample mean of $110 or larger? What is the probability of getting a sample mean larger than $135 per hour? What is the probability of getting a sample mean of between $120 and $130 per hour?
Using the Computer to Construct
z Confidence Intervals for the Mean
It is possible to construct a z confidence interval for the mean with either Excel or Minitab.
Excel yields the ; error portion of the confidence interval that must be placed with the sample mean to construct the complete confidence interval. Minitab constructs the complete confidence interval. Figure 8.6 shows both the Excel output and the Minitab output
for the cellular telephone example.
A random sample of size 70 is taken from a population that has a variance of 49. The sample mean is 90.4 What is the point estimate of Construct a 94% confidence interval for .
10) A random sample of size 70 is taken from a population that has a variance of 49. The sample mean is 90.4 What is the point estimate of ? Construct a 94% confidence interval for.
11) The average total dollar purchase at a convenience store is less than that at a supermarket. Despite smaller-ticket purchases, convenience stores can still be profitable because of the size of operation, volume of business, and the markup. A researcher is interested in estimating the average purchase amount for convenience stores in suburban Long Island. To do so, she randomly sampled 24 purchases from several convenience stores in suburban Long Island and tabulated the amounts to the nearest dollar. Use the following data to construct a 90% confidence interval for the population average amount of purchases. Assume that the population standard deviation is 3.23 and the population is normally distributed.
$2 $11 $8 $7 $9 $3
5 4 2 1 10 8
14 7 6 3 7 2
4 1 3 6 8 4
12) A meat-processing company in the Midwest produces and markets a package of eight small sausage sandwiches. The product is nationally distributed, and the company is interested in knowing the average retail price charged for this item in stores across the country. The company cannot justify a national census to generate this information. Based on the company information system’s list of all retailers who carry the product, a researcher for the company contacts 36 of these retailers and ascertains the selling prices for the product. Use the following price data and a population standard deviation of 0.113 to determine a point estimate for the national retail price of the product. Construct a 90% confidence interval to estimate this price.
$2.23 $2.11 $2.12 $2.20 $2.17 $2.10
2.16 2.31 1.98 2.17 2.14 1.82
2.12 2.07 2.17 2.30 2.29 2.19
2.01 2.24 2.18 2.18 2.32 2.02
1.99 1.87 2.09 2.22 2.15 2.19
2.23 2.10 2.08 2.05 2.16 2.26
13) A random sample of 15 items is taken, producing a sample mean of 2.364 with a sample variance of .81. Assume x is normally distributed and construct a 90% confidence interval for the population mean.
14) Some fast-food chains offer a lower-priced combination meal in an effort to attract budget-conscious customers. One chain test-marketed a burger, fries, and a drink combination for $1.71. The weekly sales volume for these meals was impressive. Suppose the chain wants to estimate the average amount its customers spent on a meal at their restaurant while this combination offer was in effect. An analyst gathers data from 28 randomly selected customers. The following data represent the sample meal totals.
$3.21 5.40 3.50 4.39 5.60 8.65 5.02 4.20 1.25 7.64
3.28 5.57 3.26 3.80 5.46 9.87 4.67 5.86 3.73 4.08
5.47 4.49 5.19 5.82 7.62 4.83 8.42 9.10
Use these data to construct a 90% confidence interval to estimate the population mean value. Assume the amounts spent are normally distributed.
15) The Universal Music Group is the music industry leader worldwide in sales according to the company Web site. Suppose a researcher wants to determine what market share the company holds in the city of St. Louis by randomly selecting 1,003 peoplem who purchased a CD last month. In addition, suppose 25.5% of the purchases made by these people were for products manufactured and distributed by the Universal Music Group.
a. Based on these data, construct a 99% confidence interval to estimate the proportion of the CD sales market in St. Louis that is held by the Universal Music Group.
b. Suppose that the survey had been taken with 10,000 people. Recompute the confidence interval and compare your results with the first confidence interval. How did they differ? What might you conclude from this about sample size and confidence intervals?
16) What proportion of commercial airline pilots are more than 40 years of age? Suppose a researcher has access to a list of all pilots who are members of the Commercial Airline Pilots Association. If this list is used as a frame for the study, she can randomly select a sample of pilots, contact them, and ascertain their ages. From 89 of these pilots so selected, she learns that 48 are more than 40 years of age.
Construct an 85% confidence interval to estimate the population proportion of commercial airline pilots who are more than 40 years of age.
17) A bank officer wants to determine the amount of the average total monthly deposits per customer at the bank. He believes an estimate of this average amount using a confidence interval is sufficient. How large a sample should he take to be within $200 of the actual average with 99% confidence? He assumes the standard deviation of total monthly deposits for all customers is
about $1,000.
18) A group of investors wants to develop a chain of fast-food restaurants. In determining potential costs for each facility, they must consider, among other expenses, the average monthly electric bill. They decide to sample some fast-food restaurants currently operating to estimate the monthly cost of electricity. They want to be 90% confident of their results and want the error of the interval estimate to be no more than $100. They estimate that such bills range from $600 to $2,500. How large a sample should they take?
19) What proportion of secretaries of Fortune 500 companies has a personal computer at his or her workstation? You want to answer this question by conducting a random survey. How large a sample should you take if you want to be 95% confident of the results and you want the error of the confidence interval to be no more than .05? Assume no one has any idea of what the proportion actually is.
20) Suppose you want to estimate the proportion of cars that are sport utility vehicles (SUVs) being driven in Kansas City, Missouri, at rush hour by standing on the corner of I-70 and I-470 and counting SUVs.You believe the figure is no higher than .40. If you want the error of the confidence interval to be no greater than .03, how many cars should you randomly sample? Use a 90% level of confidence.
21) According to a survey by Topaz Enterprises, a travel auditing company, the average error by travel agents is $128. Suppose this figure was obtained from a random sample of 41 travel agents and the sample standard deviation is $21. What is the point estimate of the national average error for all travel agents? Compute a 98% confidence interval for the national average error based on these sample results. Assume the travel agent errors are normally distributed in the population. How wide is the interval? Interpret the interval.
22) According to a survey by Runzheimer International, the average cost of a fast-food meal (quarter-pound cheeseburger, large fries, medium soft drink, excluding taxes) in Seattle is $4.82. Suppose this figure was based on a sample of 27 different establishments and the standard deviation was $0.37.Construct a 95% confidence interval for the population mean cost for all fast-food meals in Seattle. Assume the costs of a fast-food meal in Seattle are normally distributed. Using the interval as a guide, is it likely that the population mean is really $4.50? Why or why not?
23) A regional survey of 560 companies asked the vice president of operations how satisfied he or she was with the software support received from the computer staff of the company. Suppose 33% of the 560 vice presidents said they were satisfied. Construct a 99% confidence interval for the proportion of the population of vice presidents who would have said they were satisfied with the software
support if a census had been taken.