Word Problems For each word problem, use a solution sheet to solve the hypothesis test problem. Round expected frequency to two decimal places. A 6-sided die is rolled 120 times. Fill in the expected frequency column. Then, conduct a hypothesis test to determine if the die is fair. The data below are the result of the 120 rolls. Face Value Frequency Expected Frequency 1 15 2 29 3 16 4 15 5 30 6 15

The marital status distribution of the U.S. male population, age 15 and older, is as shown below. (Source: U.S. Census Bureau, Current Population Reports) Marital Status Percent Expected Frequency never married 31.3 married 56.1 widowed 2.5 divorced/separated 10.1

Suppose that a random sample of 400 U.S. young adult males, 18 – 24 years old, yielded the following frequency distribution. We are interested in whether this age group of males fits the distribution of the U.S. adult population. Calculate the frequency one would expect when surveying 400 people. Fill in the above table, rounding to two decimal places. Marital Status Frequency never married 140 married 238 widowed 2 divorced/separated 20

aThe data fits the distribution bThe data does not fit the distribution c3 e19.27 f0.0002 hDecision: Reject Null; Conclusion: Data does not fit the distribution. The next two questions refer to the following information: The real data below are from the California Reinvestment Committee and the California Economic Census. The data concern the percent of loans made by the Small Business Administration for Santa Clara County in recent years. (Source: San Jose Mercury News) Ethnic Group Percent of Loans Percent of Population Percent of Businesses Owned Asian 22.48 16.79 12.17 Black 1.15 3.51 1.61 Latino 6.19 21.00 6.51 White 66.97 58.09 79.70

Perform a goodness-of-fit test to determine whether the percent of businesses owned in Santa Clara County fits the percent of the population, based on ethnicity. Perform a goodness-of-fit test to determine whether the percent of loans fits the percent of the businesses owned in Santa Clara County, based on ethnicity. c3 e10.91 f0.0122 gDecision: Reject null when a = 0 . 05 size 12{a=0 "." "05"} {} ; Conclusion: Percent of loans does not fit the distribution. Decision: Do not reject null when a = 0 . 01 size 12{a=0 "." "01"} {} ; Conclusion Percent of loans fits the distribution. The City of South Lake Tahoe has an Asian population of 1419 people, out of a total population of 23,609 (Source: U.S. Census Bureau, Census 2000). Conduct a goodness of fit test to determine if the self-reported sub-groups of Asians are evenly distributed. Race Frequency Expected Frequency Asian Indian 131 Chinese 118 Filipino 1045 Japanese 80 Korean 12 Vietnamese 9 Other 24

Long Beach is a city in Los Angeles County (L.A.C). The population of Long Beach is 461,522; the population of L.A.C. is 9,519,338 (Source: U.S. Census Bureau, Census 2000). Conduct a goodness of fit test to determine if the racial demographics of Long Beach fit that of L.A.C. Race Percent, L.A.C. Expected #, L.B. Actual #, L.B. American Indian and Alaska Native 0.8 3804 3,881 Asian 11.9 55,591 Black or African American 9.8 68,618 Native Hawaiian and Other Pacific Islander 0.3 5,605 White, including Hispanic/Latino 48.7 208,410 Other 23.5 95,107 Two or more races 4.9 24,310

c6 e27,870 f0 hDecision: Reject null; Conclusion: L.B. does not fit L.A.C. UCLA conducted a survey of more than 263,000 college freshmen from 385 colleges in fall 2005. The results of student expected majors by gender were reported in The Chronicle of Higher Education (2/2/06). Conduct a goodness of fit test to determine if the male distribution fits the female distribution. Major Women Men Arts & Humanities 14.0% 11.4% Biological Sciences 8.4% 6.7% Business 13.1% 22.7% Education 13.0% 5.8% Engineering 2.6% 15.6% Physical Sciences 2.6% 3.6% Professional 18.9% 9.3% Social Sciences 13.0% 7.6% Technical 0.4% 1.8% Other 5.8% 8.2% Undecided 8.0% 6.6%

A recent debate about where in the United States skiers believe the skiing is best prompted the following survey. Test to see if the best ski area is independent of the level of the skier. U.S. Ski Area Beginner Intermediate Advanced Tahoe 20 30 40 Utah 10 30 60 Colorado 10 40 50

c4 e10.53 f0.0324 hDecision: Reject null; Conclusion: Best ski area and level of skier are not independent. Car manufacturers are interested in whether there is a relationship between the size of car an individual drives and the number of people in the driver’s family (that is, whether car size and family size are independent). To test this, suppose that 800 car owners were randomly surveyed with the following results. Conduct a test for independence. Family Size Sub & Compact Mid-size Full-size Van & Truck 1 20 35 40 35 2 20 50 70 80 3 - 4 20 50 100 90 5+ 20 30 70 70

College students may be interested in whether or not their majors have any effect on starting salaries after graduation. Suppose that 300 recent graduates were surveyed as to their majors in college and their starting salaries after graduation. Below are the data. Conduct a test for independence. Major < $30,000 $30,000 - $39,999 $40,000 + English 5 20 5 Engineering 10 30 60 Nursing 10 15 15 Business 10 20 30 Psychology 20 30 20

c8 e33.55 f0 hDecision: Reject null; Conclusion: Major and starting salary are not independent events. Some travel agents claim that honeymoon hot spots vary according to age of the bride and groom. Suppose that 280 East Coast recent brides were interviewed as to where they spent their honeymoons. The information is given below. Conduct a test for independence. Location 20 - 29 30 - 39 40 - 49 50 and over Niagara Falls 15 25 25 20 Poconos 15 25 25 10 Europe 10 25 15 5 Virgin Islands 20 25 15 5

A manager of a sports club keeps information concerning the main sport in which members participate and their ages. To test whether there is a relationship between the age of a member and his or her choice of sport, 643 members of the sports club are randomly selected. Conduct a test for independence. Sport 18 - 25 26 - 30 31 - 40 41 and over racquetball 42 58 30 46 tennis 58 76 38 65 swimming 72 60 65 33

c6 e25.21 f0.0003 hDecision: Reject null A major food manufacturer is concerned that the sales for its skinny French fries have been decreasing. As a part of a feasibility study, the company conducts research into the types of fries sold across the country to determine if the type of fries sold is independent of the area of the country. The results of the study are below. Conduct a test for independence. Type of Fries Northeast South Central West skinny fries 70 50 20 25 curly fries 100 60 15 30 steak fries 20 40 10 10

According to Dan Lenard, an independent insurance agent in the Buffalo, N.Y. area, the following is a breakdown of the amount of life insurance purchased by males in the following age groups. He is interested in whether the age of the male and the amount of life insurance purchased are independent events. Conduct a test for independence. Age of Males None $50,000 - $100,000 $100,001 - $150,000 $150,001 - $200,000 $200,000 + 20 - 29 40 15 40 0 5 30 - 39 35 5 20 20 10 40 - 49 20 0 30 0 30 50 + 40 30 15 15 10

c12 e125.74 f0 hDecision: Reject null Suppose that 600 thirty–year–olds were surveyed to determine whether or not there is a relationship between the level of education an individual has and salary. Conduct a test for independence. Annual Salary Not a high school grad. High school graduate College graduate Masters or doctorate < $30,000 15 25 10 5 $30,000 - $40,000 20 40 70 30 $40,000 - $50,000 10 20 40 55 $50,000 - $60,000 5 10 20 60 $60,000 + 0 5 10 150

A plant manager is concerned her equipment may need recalibrating. It seems that the actual weight of the 15 oz. cereal boxes it fills has been fluctuating. The standard deviation should be at most 12 size 12{ { { size 8{1} } over { size 8{2} } } } {} oz. In order to determine if the machine needs to be recalibrated, 84 randomly selected boxes of cereal from the next day’s production were weighed. The standard deviation of the 84 boxes was 0.54. Does the machine need to be recalibrated? c83 d96.81 e0.1426 gDecision: Do not reject null; Conclusion: The standard deviation is at most 0.5 oz. hIt does not need to be calibrated Consumers may be interested in whether the cost of a particular calculator varies from store to store. Based on surveying 43 stores, which yielded a sample mean of $84 and a sample standard deviation of $12, test the claim that the standard deviation is greater than $15. Isabella, an accomplished Bay to Breakers runner, claims that the standard deviation for her time to run the 7 ½ mile race is at most 3 minutes. To test her claim, Rupinder looks up 5 of her race times. They are 55 minutes, 61 minutes, 58 minutes, 63 minutes, and 57 minutes. c4 d4.52 e0.3402 gDecision: Do not reject null. hNo Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. They are also interested in the variation of the number of babies. Suppose that an airline executive believes the average number of babies on flights is 6 with a variance of 9 at most. The airline conducts a survey. The results of the 18 flights surveyed give a sample average of 6.4 with a sample standard deviation of 3.9. Conduct a hypothesis test of the airline executive’s belief. According to the U.S. Bureau of the Census, United Nations, in 1994 the number of births per woman in China was 1.8. This fertility rate has been attributed to the law passed in 1979 restricting births to one per woman. Suppose that a group of students studied whether or not the standard deviation of births per woman was greater than 0.75. They asked 50 women across China the number of births they had. Below are the results. Does the students’ survey indicate that the standard deviation is greater than 0.75? # of births Frequency 0 5 1 30 2 10 3 5

c49 d54.37 e0.2774 gDecision: Do not reject null; Conclusion: The standard deviation is at most 0.75. hNo According to an avid aquariest, the average number of fish in a 20–gallon tank is 10, with a standard deviation of 2. His friend, also an aquariest, does not believe that the standard deviation is 2. She counts the number of fish in 15 other 20–gallon tanks. Based on the results that follow, do you think that the standard deviation is different from 2? Data: 11; 10; 9; 10; 10; 11; 11; 10; 12; 9; 7; 9; 11; 10; 11 The manager of "Frenchies" is concerned that patrons are not consistently receiving the same amount of French fries with each order. The chef claims that the standard deviation for a 10–ounce order of fries is at most 1.5 oz., but the manager thinks that it may be higher. He randomly weighs 49 orders of fries, which yields: mean of 11 oz., standard deviation of 2 oz. a σ 2 ≤ 1 . 5 2 size 12{σ rSup { size 8{2} } <= left (1 "." 5 right ) rSup { size 8{2} } } {} c48 d85.33 e0.0007 gDecision: Reject null. hYes