DECSCI2 Practice Exercises [One- and Two-Population Tests of Hypotheses]
1.) The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $600 or less. A member of the hotel’s accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of weekend guest bills to test the manager’s claim. a.) Which form of the hypotheses should be used to test the manager’s claim? Explain. b.) What conclusion is appropriate when !! cannot be rejected? c.) What conclusion is appropriate when !! can be rejected? 2.) The manager of an automobile dealership is considering a new bonus plan designed to increase sales volume. Currently, the mean sales volume is 14 automobiles per month. The manager wants to conduct a research study to see whether the new bonus plan increases sales volume. To collect data on the plan, a sample of sales personnel will be allowed to sell under the new bonus plan for a one-month period. a.) Develop the null and alternative hypotheses most appropriate for this research situation. b.) Comment on the conclusion when !! cannot be rejected. c.) Comment on the conclusion when !! cannot be rejected. 3.) Nielsen reported that young men in the US watch 56.2 minutes if prime-time TV daily (The Wall Street Journal Europe, November 18, 2003). A researcher believes that young men in Germany spend more time watching prime-time TV. A sample of German young men will be selected by the researcher and the time they spend watching TV in one day will be recorded. The sample results will be used to test the following null and alternative hypotheses. !! ! ! ! !"!! !! ! ! ! !"!!
a.) What is the Type I error in this situation? What are the consequences of making this error? b.) What is the Type II error in this situation? situation? What are the consequences of making this error? 4.) The label on a 3-quart container of orange juice claims that the orange juice contains an average of 1 gram of fat or less. Suppose that a hypothesis test will be performed to test the validity of the claim on the label. a.) Develop the appropriate null and alternative hypotheses. b.) What is the Type I error in this situation? What are the consequences of making this error? c.) What is the Type II error in this situation? What are the consequences of making this error? 5.) Carpetland salespersons average $8000 per week in sales. The firm’s vice president proposes a compensation plan with new selling incentives. The plan is that the results of a trial selling period will enable the firm to conclude that the compensation plan increases the average sales per salesperson. a.) Develop the appropriate null and alternative hypotheses. b.) What is the Type I error in this situation? What are the consequences of making this error? c.) What is the Type II error in this situation? What are the consequences of making this error?
*For all tests of hypothesis procedures, use both the critical value and the p-value approaches.
6.) Individuals filing federal tax income returns prior to March 31 received an average refund of $1,056. Consider the population of “last-minute” filers who mail their tax income return during the last five days of the income tax period (typically April 10 to April 15). A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. For a sample of 400 individuals who filed a tax return between April 10 and 15, the sample mean refund was $910. Based on prior experience a population standard deviation $1,600 may be assumed. Perform an appropriate test of hypothesis at ! = 0.05 such that the rejection of !! will support the researcher’s contention. 7.) A manufacturing company produces valves in various sizes and shapes. One particular valve plate is supposed to have a tensile strength of 5 lbs/mm. The company tests a random sample of 42 such valve plates from a lot of 650 valve plates. The sample mean is a tensile strength of 5.0611 lbs/mm, and the population standard deviation is .2803 lbs/mm. Use ! = 0.10 and test to determine whether the lot of valve plates has an average tensile strength of 5 lbs/mm. 8.) CCN and ActMedia provided a television channel targeted to individuals waiting in supermarket checkout lines. The channel showed news, short features, and advertisements. The length of the program was based on the assumption that the population mean time a shopper stands in a supermarket checkout line is 8 minutes. Suppose that a sample of 120 shoppers showed a sample mean waiting time of 8.5 minutes. Assume a population standard deviation of 3.2 minutes. Perform an appropriate test of hypothesis at ! = 0.05 to test the assumption and determine whether actual mean waiting time differs from this standard. 9.) According to a report released by CIBC entitled Women Entrepreneurs: Leading the Charge, the average age for Canadian businesswomen in 2008 was 41. In the report, there was some indication that researchers believed that this mean age will increase. Suppose now, a couple of years later, business researchers in Canada want to test to determine if, indeed, the mean age of a Canadian businesswoman has increased. The researchers randomly sample 97 Canadian businesswomen and ascertain that the sample mean age is 43.4. From past experience, it is known that the population standard deviation is 8.95. Test to determine if the mean age of a Canadian businesswoman has increased using a 1% level of significance. 10.) According to HowtoAdvice.com, the average price charged to a customer to have a 12’ by 18’ wall-to-wall carpet shampoo cleaned is about $50. Suppose that a start-up carpet-cleaning company believes that in the region in which they operate, the average price for this service is higher. To test this hypothesis, the carpet-cleaning company randomly contacts 23 customers who have recently had a 12’ by 18’ wall-to-wall carpet shampoo cleaned and asked the customers how much they were charged for the job. Suppose the resulting data are given below and that the population standard deviation price is $3.49. 52
52
56
50
50
51
49
49
54
51
51
56
52
52
53
56
52
52
56
57
48
53
48
Use a 10% level of significance to test their hypothesis. Assume that such prices are normally distributed in the population.
11.) Wall Street securities firms paid out record year-end bonuses of $125,500 per employee for 2005 ( Fortune, February 6, 2006). Suppose we would like to take a sample of employees at the Jones & Ryan securities firm to see whether the mean year-end bonus is different from the reported mean of $125,500 for the population. Suppose a sample of 40 Jones & Ryan employees showed a sample mean year-end bonus of $118,000. Assume a population standard deviation of $30,000. Perform an appropriate test of hypothesis at ! = 0.05 to test whether the year-end bonuses paid by Jones & Ryan were different from the population mean. 12.) For the US, the mean monthly Internet bill is $32.79 per household (CNBC, January 18, 2006). A sample of 50 households in a southern state showed a sample mean of $30.63. Suppose that the population standard deviation is known to be $5.60. Perform an appropriate test of hypothesis at ! = 0.01 to determine whether the sample data support the conclusion that the mean monthly Internet bill in the southern state is less than the national mean of $32.79. 13.) Major cities around the world compete with each other in an effort to attract new businesses. Some of the criteria that businesses use to judge cities as potential locations for their headquarters might include the labor pool; the environment, including work, governmental, and living; the tax structure, the availability of skilled/educated labor, housing, education, medical care; and others. Suppose in a study done several years ago, the city of Atlanta received a mean rating of 3.51 (on a scale of 1 to 5 and assuming an interval level of data) on housing, but that since that time, considerable residential building has occurred in the Atlanta area such that city leaders feel the mean might now be higher. They hire a team of researchers to conduct a survey of businesses around the world to determine how businesses now rate the city on housing (and other variables). Sixty-one businesses take part in the new survey, with a result that Atlanta receives a mean response of 3.72 on housing with a sample standard deviation of 0.65. Assuming that such responses are normally distributed, use a 1% level of significance and these data to test to determine if the mean housing rating for the city of Atlanta by businesses has significantly increased. 14.) According to data released by the World Bank, the mean PM10 (particulate matter) concentration for the city of Kabul, Afghanistan, in 1999 was 46. Suppose that because of efforts to improve air quality in Kabul, increases in modernization, and efforts to establish environmental-friendly businesses, city leaders believe rates of particulate matter in Kabul have decreased. To test this notion, they randomly sample 12 readings over a one-year period with the resulting readings shown below. 31
44
35
53
57
47
32
40
31
38
53
45
Do these data present enough evidence to determine that PM10 readings are significantly less now in Kabul? Assume that particulate readings are normally distributed and that ! = 0.01. 15.) A shareholders’ group, in lodging a protest, claimed that the mean tenure for a chief executive officer (CEO) was at least nine years. A survey of companies reported in The Wall Street Journal found a sample mean tenure of 7.27 years for CEOs with a standard deviation of 6.38 years (The Wall Street Journal , January 2, 2007). Assume that 85 companies were included in the sample. Perform an appropriate test of hypothesis at ! = 0.01 to test the validity of the claim made by the shareholders’ group.
16.) The Employment and Training Administration reported the US mean unemployment insurance benefit of $238 per week (The World Almanac, 2003). A researcher in the state of Virginian anticipated that sample data would show evidence that the mean weekly unemployment insurance benefit in Virginia was below the national level. For a sample of 100 individuals, the sample mean weekly unemployment insurance benefit was $231 with a sample standard deviation of $80. Perform an appropriate test of hypothesis at ! = 0.05 such that the rejection of !! will support the researcher’s contention. 17.) Suppose that in past years the average price per square foot for warehouses in the United States has been $32.28. A national real estate investor wants to determine whether that figure has changed now. The investor hires a researcher who randomly samples 49 warehouses that are for sale across the United States and finds that the mean price per square foot is $31.67, with a standard deviation of $1.29. Assume that prices of warehouse footage are normally distributed in population. If the researcher uses a 5% level of significance, what statistical conclusion can be reached? 18.) AOL Time Warner Inc.’s CNN has been the longtime ratings leader of cable television news. Nielsen Media Research indicated that the mean CNN viewing audience was 600,000 viewers per day during 2002 (The Wall Street Journal , March 10, 2003). Assume that for a sample of 40 days during the first half of 2003, the daily audience was 612,000 viewers with a sample standard deviation of 65,000 viewers. Perform an appropriate test of hypothesis at ! = 0.10 to determine if the sample data indicates an increase in the mean CNN viewership during the first half of 2003. 19.) Annual per capita consumption of milk is 21.6 gallons (Statistical Abstract if the United States: 2006 ). Being from the Midwest, you believe milk consumption is higher there and wish to support your opinion. A sample of 16 individuals from the Midwestern town of Webster City showed a sample mean annual consumption of 24.1 gallons with a standard deviation of 4.8 gallons. Perform an appropriate test of hypothesis at ! = 0.05 to determine whether the mean annual consumption in Webster City is higher than the national mean. 20.) The following data (in pounds), which were selected randomly from a normally distributed population of values, represent measurements of a machine part that is supposed to weigh, on average, 8.3 pounds. 8.1
8.4
8.3
8.2
8.5
8.6
8.4
8.3
8.4
8.2
8.8
8.2
8.2
8.3
8.1
8.3
8.4
8.5
8.5
8.7
Use these data and ! = 0.01 to test the hypothesis that the parts average 8.3 pounds. 21.) A study by Hewitt Associates showed that 79% of companies offer employees flexible scheduling. Suppose a researcher believes that in accounting firms this figure is lower. The researcher randomly selects 415 accounting firms and through interviews determines that 303 of these firms have flexible scheduling. With a 1% level of significance, does the test show enough evidence to conclude that a significantly lower proportion of accounting firms offer employees flexible scheduling?
22.) The National Center for Health Statistics released a report that stated 70% of adults do not exercise regularly (Associated Press, April 7, 2002). A researcher decided to conduct a study to see whether the claim made by the National Center for Health Statistics differed on a state-bystate basis. Suppose that from the state of Wisconsin, 252 from a sample of 350 adults were found to not exercise regularly. Perform an appropriate test of hypothesis at ! = 0.05 assuming the intent of the researcher is to identify states that differ from the 70% reported by the National Center for Health Statistics. 23.) Speaking to a group of analysts in January 2006, a brokerage firm executive claimed that at least 70% of investors are currently confident of meeting their investment objectives. A UBS Investor Optimism Survey, conducted over the period January 2 to January 15 and collected from 300 investors, found that 67% were confident of meeting their investment objectives (CNBC, January 20, 2000). Perform an appropriate test of hypothesis at ! = 0.05 to test the validity of the brokerage firm executive’s claim. 24.) Many investors and financial analysts believe the Dow Jones Industrial Average (DJIA) provides a good barometer of the overall stock market. On January 31, 2006, 9 of the 30 stocks making up the DJIA increased in price (The Wall Street Journal , February 1, 2006). On the basis of this fact, a financial analyst claims we can assume that 30% of the stocks traded on the New York Stock Exchange (NYSE) went up the same day. A sample of 50 stocks traded on the NYSE that day showed that 24 went up. Perform an appropriate test of hypothesis at ! = 0.01 to test the validity of the analyst’s claim. 25.) The Independent Insurance Agents of America conducted a survey of insurance consumers and discovered that 48% of them always re-read their insurance policies, 29% sometimes do, 16% rarely do, and 7% never do. Suppose a large insurance company invests considerable time and money in rewriting policies so that they will be more attractive and easy to read and understand. After using the new policies for a year, company managers want to determine whether re-writing the policies significantly changed the proportion of policyholders who always re-read their insurance policy. They contact 380 of the company’s insurance consumers who purchased a policy in the past year and ask them whether they always re-read their insurance policies. One hundred and sixty-four respond that they always do. Use a 1% level of significance to test the hypothesis. 26.) Eighteen percent of U.S.-based multinational companies provide an allowance for personal long-distance calls for executives living overseas, according to the Institute for International Human Resources and the National Foreign Trade Council. Suppose a researcher thinks that U.S.-based multinational companies are having a more difficult time recruiting executives to live overseas and that an increasing number of these companies are providing an allowance for personal long-distance calls to these executives to ease the burden of living away from home. To test this hypothesis, a new study is conducted by contacting 376 multinational companies. Twenty-two percent of these surveyed companies are providing an allowance for personal long-distance calls to executives living overseas. Does the test show enough evidence to declare that a significantly higher proportion of multinational companies provide a longdistance call allowance? Let ! = 0.01.
27.) Where do CFOs get their money news? According to Robert Half International, 47% get their money news from newspapers, 15% get it from communication/ colleagues, 12% get it from television, 11% from the Internet, 9% from magazines, 5% from radio, and 1% don’t know. Suppose a researcher wants to test these results. She randomly samples 67 CFOs and finds that 40 of them get their money news from newspapers. Does the test show enough evidence to reject the findings of Robert Half International? Let ! = 0.05. 28.) An airline promotion to business travelers is based on the assumption that two-thirds of business travelers use a laptop computer on overnight business trips. Suppose that an American Express sponsored survey have found that 355 of 546 business travelers use a laptop computer on overnight business trips. Perform an appropriate test of hypothesis at ! = 0.05 to test the validity of the airline’s assumption. 29.) In a test of the quality of two television commercials, each commercial was shown in separate test area six times over a one-week period. The following week a telephone survey was conducted to identify individuals who had seen the commercials. Those individuals were asked to state the primary message in the commercials. The following results were recorded. Commercial A
Commercial B
Number Who Saw Commercial
150
200
Number Who Recalled Message
63
60
Use ! = 0.05 to test the hypothesis that there is no difference in the recall proportions for the two commercials. 30.) According to a study conducted for Gateway Computers, 59% of men and 70% of women say that weight is an extremely/very important factor in purchasing a laptop computer. Suppose this survey was conducted using 374 men and 481 women. Do these data show enough evidence to declare that a significantly higher proportion of women than men believe that weight is an extremely/very important factor in purchasing a laptop computer? Use a 5% level of significance. 31.) According to a CCH Unscheduled Absence survey, 9% of small businesses use telecommuting of workers in an effort to reduce unscheduled absenteeism. This proportion compares to 6% for all other businesses. Is there really a significant difference between small businesses and all businesses on this issue? Use these data and an alpha of 0.10 to test this question. Assume that there were 780 small businesses and 915 other businesses in this survey. 32.) Companies that recently developed new products were asked to rate which activities are most difficult to accomplish with new products. Options included such activities as assessing market potential, market testing, finalizing the design, developing a business plan, and the like. A researcher wants to conduct a similar study to compare the results between two industries: the computer hardware industry and the banking industry. He takes a random sample of 56 computer firms and 89 banks. The researcher asks whether market testing is the most difficult activity to accomplish in developing a new product. Some 48% of the sampled computer companies and 56% of the sampled banks respond that it is the most difficult activity. Use a level of significance of 0.20 to test whether there is a significant difference in the responses to the question from these two industries.
33.) During the 2003 Super Bowl, Miller Lite Beer’s commercial referred to as The Miller Lite Girls ranked among the top three most effective advertisements aired during the Super Bowl (USA Today, December 29, 2003). The survey of advertising effectiveness, conducted by the USA Today’s Ad Track poll, reported separate samples by respondent age group to learn about how the Super Bowl advertisement appealed to different age groups. The following sample data apply to The Miller Lite Girls commercial. Age Group
Sample Size
Liked the Ad a Lot
Under 30
100
49
30 to 49
150
54
Perform an appropriate test of hypothesis at ! = 0.05 to determine whether there is a difference between the population proportions for the two age groups regarding the appeal of The Miller Lite Girls commercial. 34.) Chicago O’Hare and Atlanta Hartsfield-Jackson are the two busiest airports in the US. The congestion often leads to delayed flight arrivals as well as delayed flight departures. The Bureau of Transportation tracks the on-time and delayed performance at major airports (Travel & Leisure, November 2006). A flight is considered delayed if it is more than 15 minutes behind schedule. The following sample data show the delayed departures at Chicago O’Hare and Atlanta Hartsfield-Jackson airports. Chicago O’Hare
Atlanta Hartsfield-Jackson
Flights
900
1,200
Delayed Departures
252
312
Perform an appropriate test of hypothesis at ! = 0.05 to determine whether the population proportions of delayed departures differ at these two airports. 35.) A large production facility uses two machines to produce a key part for its main product. Inspectors have expressed concern about the quality of the finished product. Quality control investigation has revealed that the key part made by the two machines is defective at times. The inspectors randomly sampled 35 units of the key part from each machine. Of those produced by machine A, five were defective. Seven of the 35 sampled parts from machine B were defective. Perform an appropriate test of hypothesis at ! = 0.01 to determine if there is a significant difference in the proportions of the populations of parts that are defective between machine A and machine B. 36.) Employee suggestions can provide useful and insightful ideas for management. Some companies solicit and receive employee suggestions more than others, and company culture influences the use of employee suggestions. Suppose a study is conducted to determine whether there is a significant difference in mean number of suggestions a month per employee between the Canon Corporation and the Pioneer Electronic Corporation. The study shows that the average number of suggestions per month is 5.8 at Canon and 5.0 at Pioneer. Suppose these figures were obtained from random samples of 36 and 45 employees, respectively. If the population standard deviations of suggestions per employee are 1.7 and 1.4 for Canon and Pioneer, respectively, is there a significant difference in the population means? Use ! = 0.05.
37.) A company’s auditor believes the per diem cost in Nashville, Tennessee, rose significantly between 1999 and 2009. To test this belief, the auditor samples 51 business trips from the company’s records for 1999; the sample average was $190 per day, with a population standard deviation of $18.50. The auditor selects a second random sample of 47 business trips from the company’s records for 2009; the sample average was $198 per day, with a population standard deviation of $15.60. If he uses a risk of committing a Type I error of 0.01, does the auditor find that the per diem average expense in Nashville has gone up significantly? 38.) During the 2003 season, Major League Baseball took steps to speed up the play of baseball games in order to maintain fan interest ( CNN Headline News, September 30, 2003). The following results come from a sample of 60 games played during the summer of 2002 and a sample of 50 game splayed during the summer of 2003. The sample mean shows the mean duration of the games included in each sample. 2002 Season !! !!
2003 Season
= 60
!!
= 2 hours, 52 minutes
!!
= 50
= 2 hours, 46 minutes
Historical data indicate a population standard deviation of 12 minutes is a reasonable assumption for both years. Perform an appropriate test of hypothesis at ! = 0.05 to test the research hypothesis that the steps taken during the 2003 season have reduced the mean duration of baseball games. 39.) The Trade Show Bureau conducted a survey to determine why people go to trade shows. The respondents were asked to rate a series of reasons on a scale from 1 to 5, with 1 representing little importance and 5 representing great importance. One of the reasons suggested was general curiosity. The following responses for 50 people from the computers/electronics industry and 50 people from the food/beverage industry were recorded for general curiosity. Computers/Electronics
Food/Beverage
1 0 3 3 1 1 2 2 3
2 3 3 2 2 1 1 3 3
1 3 1 2 3 3 4 0 2
3 2 2 2 2 3 1 1 2
2 1 2 2 1 2 4 0 3
3 4 3 4 2 2 3 2 4
3 5 2 3 4 4 5 0 3
2 2 3 3 2 4 3 2 3
4 4 2 3 3 4 3 2 2
3 3 3 3 3 4 2 5 3
2
1
0
2
3
4
3
3
3
2
Use these data and ! = 0.01 to determine whether there is a significant difference between people in these two industries on this question. Assume the variance for the computer/electronics population is 1.0188 and the variance for the food/beverage population is 0.9180.
40.) Arnold Palmer and Tiger Woods are two of the best golfers to ever play the game. To show how these two golfers would compare if both were plating at the top of their game, the following sample data provide the results of 18-hole scores during a PGA tournament competition. Palmer’s scores are from his 1960 season, while Wood’s scores are from his 1999 season (Golf Magazine, February 2000). Arnold Palmer !! !!
Tiger Woods
= 112
!!
= 69.95
!!
= 84
= 69.56
Assume a population standard deviation of 2.5 for both golfers. Use the sample results to test the hypothesis of no difference between the population mean 18-hole scores for the two golfers at ! = 0.01. 41.) FedEx and United Parcel Service (UPS) are the world’s two leading cargo carriers by volume and revenue (The Wall Street Journal , January 27, 2004). According to the Airports Council International, the Memphis International Airport (FedEx) and the Louisville International Airport (UPS) are 2 of the 10 largest cargo airports in the world. The following random samples show the tons of cargo per day handled by these airports. Data are in thousands of tons. Memphis
9.1
15.1
8.8
10.0
7.5
10.5
8.3
9.1
6.0
5.8
12.1
9.3
Louisville
4.7
5.0
4.2
3.3
5.5
2.2
4.1
2.6
3.4
7.0
Suppose that for this data, equality of population variances cannot be assumed. Perform an appropriate test of hypothesis at ! = 0.10 to determine whether the difference between the two population means is significant. 42.) The following data represent the running times (in minutes) of films produced by two motion picture companies. Company 1
103
94
110
87
98
97
82
123
92
175
Company 2
88
118
Suppose that for this data, equality of population variances cannot be assumed. Using a 5% level of significance, determine whether there is a significant difference between the average running times of films produced by the two companies, assuming that the running times for both companies follow normal distributions.
43.) Three-megapixel digital cameras are typically the lightest, most compact, and easiest to use. However, if you plan to enlarge or crop images, you will probably want to spend more for a higher-resolution model. The following shows sample prices of five-megapixel and threemegapixel digital cameras (Consumer Reports Buying Guide, 2004). Five-Megapixel
Three-Megapixel
Model
Price
Model
Nikon 5700 Olympus C-505 SonyDCS-F717 Olympus C-5050 Minolta 7Hi HP 935 Pentax 550 Canon S50 Kyocera TVS
890 620 730 480 1,060 450 540 500 890
Kodak DX4330 Canon A70 Sony DSC P8 Minolta XI Sony DSC P72 Nikon 3100 Panasonic DMC-LC33 Pentax S
Minolta F300
Price
280 290 370 400 310 340 270 380
440
Suppose that for this data, equality of population variances cannot be assumed. Use these data and ! = 0.05 to determine whether there is a significant difference between the mean prices of five-megapixel and three-megapixel digital cameras. 44.) Suppose a Realtor is interested in comparing the asking prices of midrange homes in Peoria, Illinois, and Evansville, Indiana. The Realtor conducts a small telephone survey in the two cities, asking the prices of midrange homes. A random sample of 21 listings in Peoria resulted in a sample average price of $116,900, with a standard deviation of $2,300. A random sample of 26 listings in Evansville resulted in a sample average price of $114,000, with a standard deviation of $1,750. The Realtor assumes prices of midrange homes are normally distributed and the variance in prices in the two cities is about the same. Test whether there is any difference in the mean prices of midrange homes of the two cities for ! = 0.10. 45.) Is there a significant difference in average daily hotel room rates between Minneapolis and New Orleans? Suppose we test this hypothesis by taking hotel rate samples from each city. The data for such a study follow. Minneapolis !! !!
= 22
= $112
!!
= $11
New Orleans !! !!
= 20
= $122
!!
= $12
Use these data to perform the appropriate test of hypothesis at ! = 0.02. Assume the population variances are approximately equal and hotel rates in any given city are normally distributed.
46.) Based on an indication that mean daily car rental rates may be higher for Boston than for Dallas, a survey of eight car rental companies in Boston is taken and the sample mean car rental rate is $47, with a standard deviation of $3. Further, suppose a survey of nine car rental companies in Dallas results in a sample mean of $44 and a standard deviation of $3. Use 5% level of significance to test to determine whether the average daily car rental rates in Boston are significantly higher than those in Dallas. Assume car rental rates are normally distributed and the population variances are equal. 47.) As the prices of heating oil and natural gas increase, consumers become more careful about heating their homes. Researchers want to know how warm homeowners keep their houses in January and how the results from Wisconsin and Tennessee compare. The researchers randomly call 23 Wisconsin households between 7 P.M. and 9 P.M. on January 15 and ask the respondent how warm the house is according to the thermostat (in F). The researchers then call 19 households in Tennessee the same night and ask the same question. The results follow. !
Wisconsin
71 70 75 74 69
71 61 68 68 72
65 67 71 67 67
70
73
72
Tennessee
68 69 73 69 72
73 74 72 74 69
75 73 71 73 70
74 74 69 70 67
71 70 72 72
For ! = 0.01, is the average temperature of a house in Tennessee significantly higher than that of a house in Wisconsin on the evening of January 15? Assume that the house temperatures are normally distributed in each population. [Note: Perform first an appropriate test to compare the population variances to determine if we can assume equal variances or not.] 48.) Injuries to Major League Baseball players have been increasing in recent years. For the period 1992 to 2001, league expansion caused Major League Baseball rosters to increase 15%. However, the number of players being put on the disabled list due to injury increased 32% over the same period (USA Today, July 8, 2002). A research question addressed whether Major League Baseball players being put on the disabled list are on the list longer in 2001 than players put on the disabled list a decade earlier. Assume the following data apply: 2001 Season !!
1992 Season
= 45
!!
!!
= 60 days
!!
!!
= 18 days
!!
= 38
= 51 days
= 15 days
Perform an appropriate test of hypothesis at ! = 0.01 to determine whether the difference between the two population means is significant. Assume that the populations are normally distributed. [Note: Perform first an appropriate test to compare the population variances to determine if we can assume equal variances or not.]
49.) A national grocery store chain wants to test the difference in the average weight of turkeys sold in Detroit and the average weight of turkeys sold in Charlotte. According to the chain’s researcher, a random sample of 20 turkeys sold at the chain’s stores in Detroit yielded a sample mean of 17.53 pounds, with a standard deviation of 3.2 pounds. Her random sample of 24 turkeys sold at the chain’s stores in Charlotte yielded a sample mean of 14.89 pounds, with a standard deviation of 2.7 pounds. Use a 1% level of significance to determine whether there is a difference in the mean weight of turkeys sold in these two cities. Assume that the weights of turkeys sold in the stores are normally distributed. [Note: Perform first an appropriate test to compare the population variances to determine if we can assume equal variances or not.] 50.) Periodically, Merrill Lynch customers are asked to evaluate Merrill Lynch financial consultants and services (2000 Merrill Lynch Client Satisfaction Survey). Higher ratings on the client satisfaction survey indicate better service with 7 maximum service rating. Independent samples of service ratings for two financial consultants are summarized here. Consultant A has 10 years of experience, whereas consultant B has 1 year of experience. Consultant A !!
Consultant B
= 16
!!
!!
= 6.82
!!
!!
= 0.64
!!
= 10
= 6.25
= 0.75
Perform an appropriate test of hypothesis at ! = 0.05 to see whether the consultant with more experience has the higher population mean service rating. Assume that the populations are normally distributed. [Note: Perform first an appropriate test to compare the population variances to determine if we can assume equal variances or not.] 51.) A market research firm used a sample of individuals to rate the purchase potential of a particular product before and after the individuals saw a new television commercial about the product. The purchase potential ratings were based on a 0 to 10 scale, with higher values indicating a higher purchase potential. The null hypothesis stated that the mean rating “after” would be less than or equal to the mean rating “before”. Rejection of this hypothesis would show that the commercial improved the mean purchase potential rating. Use ! = 0.05 and the following data to test the hypothesis. Purchase Rating Individual
1 2 3 4 5 6 7 8
After
Before
6 6 7 4 3 9 7 6
5 4 7 3 5 8 5 6
52.) A manufacturer produces both a deluxe and a standard model of an automatic sander designed for home use. Selling prices obtained from a sample of retail outlets follow. Model Price ($) Retail Outlet
Deluxe
Standard
1 2 3 4 5 6
39 39 45 38 40 39
27 28 35 30 30 34
7
35
29
The manufacturer’s suggested retail prices for the two models show a $10 price differential. Use a 5% level of significance to test that the mean difference between the prices of the two models is $10. 53.) A nationally known supermarket decided to promote its own brand of soft drinks on TV for two weeks. Before the ad campaign, the company randomly selected 21 of its stores across the United States to be part of a study to measure the campaign’s effectiveness. During a specified half-hour period on a certain Monday morning, all the stores in the sample counted the number of cans of its own brand of soft drink sold. After the campaign, a similar count was made. The average difference was an increase of 75 cans, with a standard deviation of difference of 30 cans. Using this information, perform an appropriate test of hypothesis at ! = 0.10 to determine if there is a significant difference in soft drink sales for this company’s brand before and after the ad campaign. Assume the differences in soft drink sales for the company’s brand are normally distributed in the population. 54.) Is there a significant difference in the gasoline mileage of a car for regular unleaded and premium unleaded? To test this question, a researcher randomly selected 15 drivers for a study. They were to drive their cars for one month on regular unleaded and for one month on premium unleaded gasoline. The participants drove their own cars for this experiment. The average sample difference was 2.85 miles per gallon in favor of the premium unleaded, and the sample standard deviation of difference was 1.9 miles per gallon. For ! = 0.01, does the test show enough evidence for the researcher to conclude that there is a significant difference in mileage between regular unleaded and premium unleaded gasoline? Assume the differences in gasoline mileage figures are normally distributed in the population. 55.) An experiment is conducted to compare the starting salaries of male and female college graduates who find jobs. Pairs are formed by choosing a male and a female with the same major and similar GPA. Suppose a random sample of 10 pairs is formed in this manner and the starting annual salary of each person is recorded. It was observed that the differences in the male salaries to that of females is $400 with a standard deviation of $434.61. Compare the mean starting salary for males to mean starting salary for females using an appropriate test of hypothesis at ! = 0.05.
56.) An automotive part must be machined to close tolerances to be acceptable to customers. Production specifications call for a maximum variance in lengths of the parts of 0.0004. Suppose the sample variance for 30 parts turns out to be ! ! = 0.0005. Using ! = 0.05, test to see whether the population variance specification is being violated. 57.) A manufacturer of car batteries claims that his batteries will last, on average, 3 years with a variance of 1 year. If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5, and 4.2 years, perform a test of hypothesis at a 2% significance level and decide if the manufacturer’s claim is valid. Assume the population of battery lives to be approximately normally distributed. 58.) Previous experience shows the variance of a given process to be 14. Researchers are testing to determine whether this value has changed. They gather the following dozen measurements of the process. 52
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51
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38
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Use these data and ! = 0.05 to test the null hypothesis about the variance. Assume the measurements are normally distributed. 59.) A company produces industrial wiring. One batch of wiring is specified to be 2.16 centimeters (cm) thick. A company inspects the wiring in seven locations and determines that, on the average, the wiring is about 2.16 cm thick. However, the measurements vary. It is unacceptable for the variance of the wiring to be more than .04 cm2. The standard deviation of the seven measurements on this batch of wiring is .34 cm. Use ! = 0.01 to determine whether the variance on the sample wiring is too great to meet specifications. Assume wiring thickness is normally distributed. 60.) Media Metrix and Jupiter Communications gathered data on the time adults and the time teens spend online during a month (USA Today, September 14, 2000). The study concluded that on average, adults spend more time online than teens. Assume that a follow-up study sampled 26 adults and 30 teens. The standard deviations of the time online during a month were 94 minutes and 58 minutes, respectively. Do the sample results support the conclusion that adults have a greater variance in online time than teens? Use ! = 0.01.
