11.1 Facts About the Chi-Square Distribution
Decide whether the following statements are true or false:
As the number of degrees of freedom increases, the graph of the chi-square distribution looks more and more symmetrical.
The standard deviation of the chi-square distribution is twice the mean.
The mean and the median of the chi-square distribution are the same if df = 24.
11.2 Goodness-of-Fit Test
For each problem, use a solution sheet to solve the hypothesis test problem. Go to Table C3 for the chi-square solution sheet. Round expected frequency to two decimal places.
A six-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 in Table 11.34 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, ages 15 and older, is as shown in Table 11.35.
Marital Status | % | 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. males, 18 to 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 Table 11.35, rounding to two decimal places.
Marital Status | Frequency |
---|---|
Never Married | 140 |
Married | 238 |
Widowed | 2 |
Divorced/Separated | 20 |
Use the following information to answer the next two exercises: The columns in Table 11.37 contain the Race/Ethnicity of U.S. Public Schools for a recent year, the percentages for the Advanced Placement Examinee Population for that class, and the Overall Student Population. Suppose the right column contains the results of a survey of 1,000 local students from that year who took an AP exam.
Race/Ethnicity | AP Examinee Population | Overall Student Population | Survey Frequency |
---|---|---|---|
Asian, Asian American, or Pacific Islander | 10.2% | 5.4% | 113 |
Black or African American | 8.2% | 14.5% | 94 |
Hispanic or Latino | 15.5% | 15.9% | 136 |
American Indian or Alaska Native | 0.6% | 1.2% | 10 |
White | 59.4% | 61.6% | 604 |
Not Reported/Other | 6.1% | 1.4% | 43 |
Perform a goodness-of-fit test to determine whether the local results follow the distribution of the U.S. overall student population based on ethnicity.
Perform a goodness-of-fit test to determine whether the local results follow the distribution of U.S. AP examinee population, based on ethnicity.
The city of South Lake Tahoe, California, has an Asian population of 1,419 out of a total population of 23,609. Suppose that a survey of 1,419 self-reported Asians in the borough of Manhattan in the New York City area yielded the data in Table 11.38. Conduct a goodness-of-fit test to determine if the self-reported subgroups of Asians in Manhattan fit that of the South Lake Tahoe area.
Race | South Lake Tahoe Frequency | Manhattan Frequency |
---|---|---|
Asian Indian | 131 | 174 |
Chinese | 118 | 557 |
Filipino | 1,045 | 518 |
Japanese | 80 | 54 |
Korean | 12 | 29 |
Vietnamese | 9 | 21 |
Other | 24 | 66 |
Use the following information to answer the next two exercises: UCLA conducted a survey of more than 263,000 college freshmen from 385 colleges in fall 2005. The results of students’ expected majors by gender were reported in The Chronicle of Higher Education (2/2/2006). Suppose a survey of 5,000 graduating females and 5,000 graduating males was done as a follow-up last year to determine what their actual majors were. The results are shown in the tables for Exercise 11.77 and Exercise 11.78. The second column in each table does not add to 100 percent because of rounding.
Conduct a goodness-of-fit test to determine if the actual college majors of graduating females fit the distribution of their expected majors.
Major | Females—Expected Major | Females—Actual Major |
---|---|---|
Arts & Humanities | 14% | 670 |
Biological Sciences | 8.4% | 410 |
Business | 13.1% | 685 |
Education | 13% | 650 |
Engineering | 2.6% | 145 |
Physical Sciences | 2.6% | 125 |
Professional | 18.9% | 975 |
Social Sciences | 13% | 605 |
Technical | 0.4% | 15 |
Other | 5.8% | 300 |
Undecided | 8% | 420 |
Conduct a goodness-of-fit test to determine if the actual college majors of graduating males fit the distribution of their expected majors.
Major | Males—Expected Major | Males—Actual Major |
---|---|---|
Arts & Humanities | 11% | 600 |
Biological Sciences | 6.7% | 330 |
Business | 22.7% | 1,130 |
Education | 5.8% | 305 |
Engineering | 15.6% | 800 |
Physical Sciences | 3.6% | 175 |
Professional | 9.3% | 460 |
Social Sciences | 7.6% | 370 |
Technical | 1.8% | 90 |
Other | 8.2% | 400 |
Undecided | 6.6% | 340 |
Read the statement and decide whether it is true or false.
In a goodness-of-fit test, the expected values are the values we would expect if the null hypothesis were true.
In general, if the observed values and expected values of a goodness-of-fit test are not close together, then the test statistic can get very large and on a graph will be way out in the right tail.
Use a goodness-of-fit test to determine if high school principals believe that students are absent equally during the week.
The test to use to determine if a six-sided die is fair is a goodness-of-fit test.
In a goodness-of-fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.
A sample of 212 commercial businesses was surveyed for recycling one commodity; a commodity here means any one type of recyclable material such as plastic or aluminum. Table 11.41 shows the business categories in the survey, the sample size of each category, and the number of businesses in each category that recycle one commodity. Based on the study, on average half of the businesses were expected to be recycling one commodity. As a result, the last column shows the expected number of businesses in each category that recycle one commodity. At the 5 percent significance level, perform a hypothesis test to determine if the observed number of businesses that recycle one commodity follows the uniform distribution of the expected values.
Business Type | Number in Class | Observed Number that Recycle One Commodity | Expected Number that Recycle One Commodity |
---|---|---|---|
Office | 35 | 19 | 17.5 |
Retail/Wholesale | 48 | 27 | 24 |
Food/Restaurants | 53 | 35 | 26.5 |
Manufacturing/Medical | 52 | 21 | 26 |
Hotel/Mixed | 24 | 9 | 12 |
Table 11.42 contains information from a survey of 499 participants classified according to their age groups. The researchers making the survey wanted to find out how many people were diagnosed with a particular disease within the last year. The second column shows the percentage of people with the disease per age class among the study participants. The last column comes from a different study at the national level that shows the corresponding percentages of people with the disease in the same age classes in the United States. Perform a hypothesis test at the 5 percent significance level to determine whether the survey participants are a representative sample of the people with the disease nationwide.
Age Class (years) | % of People Diagnosed | % of Expected U.S. Average |
---|---|---|
20–30 | 75% | 32.6% |
31–40 | 26.5% | 32.6% |
41–50 | 13.6% | 36.6% |
51–60 | 21.9% | 36.6% |
61–70 | 21% | 39.7% |
11.3 Test of Independence
For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected frequency to two decimal places.
A recent debate about where in the U.S. 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 |
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 results in Table 11.44. Conduct a test of 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 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. Table 11.45 shows the data. Conduct a test of independence.
Major | < $50,000 | $50,000–$68,999 | $69,000 + |
---|---|---|---|
English | 5 | 20 | 5 |
Engineering | 10 | 30 | 60 |
Nursing | 10 | 15 | 15 |
Business | 10 | 20 | 30 |
Psychology | 20 | 30 | 20 |
Some travel agents claim that honeymoon hotspots vary according to age of the bride. Suppose that 280 recent brides were interviewed as to where they spent their honeymoons. The information is given in Table 11.46. Conduct a test of independence.
Location | 20–29 | 30–39 | 40–49 | 50+ |
---|---|---|---|---|
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 of independence.
Sport | 18–25 | 26–30 | 31–40 | 41+ |
---|---|---|---|---|
Racquetball | 42 | 58 | 30 | 46 |
Tennis | 58 | 76 | 38 | 65 |
Swimming | 72 | 60 | 65 | 33 |
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 shown in Table 11.48. Conduct a test of 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 Leonard, an independent insurance agent in the Buffalo, New York 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 | < $200,000 | $200,000–$400,000 | $401,001–$1,000,000 | $1,000,001+ |
---|---|---|---|---|---|
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 |
Suppose that 600 thirty-year-olds were surveyed to determine whether there is a relationship between the level of education an individual has and salary. Conduct a test of independence.
Annual Salary | Not a High School Graduate | 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 |
Read the statement and decide whether it is true or false.
The number of degrees of freedom for a test of independence is equal to the sample size minus one.
The test for independence uses tables of observed and expected data values.
The test to use when determining if the college or university a student chooses to attend is related to his or her socioeconomic status is a test for independence.
In a test of independence, the expected number is equal to the row total multiplied by the column total divided by the total surveyed.
An ice cream maker performs a nationwide survey about favorite flavors of ice cream in different geographic areas of the United States. Based on Table 11.51, do the numbers suggest that geographic location is independent of favorite ice cream flavors? Test at the 5 percent significance level.
U.S. Region/Flavor | Strawberry | Chocolate | Vanilla | Rocky Road | Mint Chocolate Chip | Pistachio | Row Total |
---|---|---|---|---|---|---|---|
West | 12 | 21 | 22 | 19 | 15 | 8 | 97 |
Midwest | 10 | 32 | 22 | 11 | 15 | 6 | 96 |
East | 8 | 31 | 27 | 8 | 15 | 7 | 96 |
South | 15 | 28 | 30 | 8 | 15 | 6 | 102 |
Column Total | 45 | 112 | 101 | 46 | 60 | 27 | 391 |
Table 11.52 provides results of a recent survey of the youngest online entrepreneurs whose net worth is estimated at one million dollars or more. Their ages range from 17 to 30. Each cell in the table illustrates the number of entrepreneurs who correspond to the specific age group and their net worth. Are the ages and net worth independent? Perform a test of independence at the 5 percent significance level.
Age Group/Net Worth Value (in millions of U.S. dollars) | 1–5 | 6–24 | ≥25 | Row Total |
---|---|---|---|---|
17–25 | 8 | 7 | 5 | 20 |
26–30 | 6 | 5 | 9 | 20 |
Column Total | 14 | 12 | 14 | 40 |
A 2013 poll in California surveyed people about a new tax. The results are presented in Table 11.53 and are classified by ethnic group and response type. Are the poll responses independent of the participants’ ethnic group? Conduct a test of independence at the 5 percent significance level.
Opinion/Ethnicity | Asian American | White/Non-Hispanic | African American | Latino | Row Total |
---|---|---|---|---|---|
Against Tax | 48 | 433 | 41 | 160 | 682 |
In Favor of Tax | 54 | 234 | 24 | 147 | 459 |
No Opinion | 16 | 43 | 16 | 19 | 94 |
Column Total | 118 | 710 | 81 | 326 | 1,235 |
11.4 Test for Homogeneity
For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Table C3 for the chi-square solution sheet. Round expected frequency to two decimal places.
A psychologist is interested in testing whether there is a difference in the distribution of personality types for business majors and social science majors. The results of the study are shown in Table 11.54. Conduct a test of homogeneity. Test at a 5 percent level of significance.
Open | Conscientious | Extrovert | Agreeable | Neurotic | |
Business | 41 | 52 | 46 | 61 | 58 |
Social Science | 72 | 75 | 63 | 80 | 65 |
Do men and women select different breakfasts? The breakfasts ordered by randomly selected men and women at a popular breakfast place are shown in Table 11.55. Conduct a test for homogeneity at a 5 percent level of significance.
French Toast | Pancakes | Waffles | Omelettes | |
Men | 47 | 35 | 28 | 53 |
Women | 65 | 59 | 55 | 60 |
A fisherman is interested in whether the distribution of fish caught in Green Valley Lake is the same as the distribution of fish caught in Echo Lake. Of the 191 randomly selected fish caught in Green Valley Lake, 105 were rainbow trout, 27 were other trout, 35 were bass, and 24 were catfish. Of the 293 randomly selected fish caught in Echo Lake, 115 were rainbow trout, 58 were other trout, 67 were bass, and 53 were catfish. Perform a test for homogeneity at a 5 percent level of significance.
In 2007, the United States had 1.5 million homeschooled students, according to the U.S. National Center for Education Statistics. In Table 11.56, you can see that parents decide to homeschool their children for different reasons, and some reasons are ranked by parents as more important than others. According to the survey results shown in the table, is the distribution of applicable reasons the same as the distribution of the most important reason? Provide your assessment at the 5 percent significance level. Did you expect the result you obtained?
Reasons for Homeschooling | Applicable Reason (in thousands of respondents) | Most Important Reason (in thousands of respondents) | Row Total |
---|---|---|---|
Concern About the Environment of Other Schools | 1,321 | 309 | 1,630 |
Dissatisfaction with Academic Instruction at Other Schools | 1,096 | 258 | 1,354 |
To Provide Religious or Moral Instruction | 1,257 | 540 | 1,797 |
Child Has Special Needs, Other Than Physical or Mental | 315 | 55 | 370 |
Nontraditional Approach to Child’s Education | 984 | 99 | 1,083 |
Other Reasons (e.g., finances, travel, family time, etc.) | 485 | 216 | 701 |
Column Total | 5,458 | 1,477 | 6,935 |
When looking at energy consumption, we are often interested in detecting trends over time and how they correlate among different countries. The information in Table 11.57 shows the average energy use in units of kg of oil equivalent per capita in the United States and the joint European Union countries (EU) for the six-year period 2005 to 2010. Do the energy use values in these two areas come from the same distribution? Perform the analysis at the 5 percent significance level.
Year | European Union | United States | Row Total |
---|---|---|---|
2010 | 3,413 | 7,164 | 10,557 |
2009 | 3,302 | 7,057 | 10,359 |
2008 | 3,505 | 7,488 | 10,993 |
2007 | 3,537 | 7,758 | 11,295 |
2006 | 3,595 | 7,697 | 11,292 |
2005 | 3,613 | 7,847 | 11,460 |
Column Total | 20,965 | 45,011 | 65,976 |
The Insurance Institute for Highway Safety collects safety information about all types of cars every year and publishes a report of top safety picks among all cars, makes, and models. Table 11.58 presents the number of top safety picks in six car categories for the two years 2009 and 2013. Analyze the table data to conclude whether the distribution of cars that earned the top safety picks safety award has remained the same between 2009 and 2013. Derive your results at the 5 percent significance level.
Year/Car Type | Small | Mid-Size | Large | Small SUV | Mid-Size SUV | Large SUV | Row Total |
---|---|---|---|---|---|---|---|
2009 | 12 | 22 | 10 | 10 | 27 | 6 | 87 |
2013 | 31 | 30 | 19 | 11 | 29 | 4 | 124 |
Column Total | 43 | 52 | 29 | 21 | 56 | 10 | 211 |
11.5 Comparison of the Chi-Square Tests
For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Table C3 for the chi-square solution sheet. Round expected frequency to two decimal places.
Is there a difference between the distribution of community college statistics students and the distribution of university statistics students in what technology they use on their homework? Of some randomly selected community college students, 43 used a computer, 102 used a calculator with built-in statistics functions, and 65 used a table from the textbook. Of some randomly selected university students, 28 used a computer, 33 used a calculator with built-in statistics functions, and 40 used a table from the textbook. Conduct an appropriate hypothesis test using a 0.05 level of significance.
Read the statement and decide whether it is true or false.
If df = 2, the chi-square distribution has a shape that reminds us of the exponential.
11.6 Test of a Single Variance
Use the following information to answer the next 12 exercises: Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes.
Is the traveler disputing the claim about the average or about the variance?
A sample standard deviation of 15 minutes is the same as a sample variance of __________ minutes.
Is this a right-tailed, left-tailed, or two-tailed test?
H_{0}: __________
df = ________
chi-square test statistic = ________
p-value = ________
Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade the p-value.
Let α = 0.05
How did you know to test the variance instead of the mean?
If an additional test were done on the claim of the average delay, which distribution would you use?
If an additional test were done on the claim of the average delay, but 45 flights were surveyed, which distribution would you use?
For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Table C3 for the chi-square solution sheet. Round expected frequency to two decimal places.
A plant manager is concerned her equipment may need recalibrating. It seems that the actual weight of the 15-ounce cereal boxes it fills has been fluctuating. The standard deviation should be at most 0.5 ounces. 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?
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 7.5 mile race is at most 3 minutes. To test her claim, Isabella looks up five of her race times. They are 55 minutes, 61 minutes, 58 minutes, 63 minutes, and 57 minutes.
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 six with a variance of nine 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.
The number of births per woman in China is 1.6, down from 5.91 in 1966. 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 the standard deviation of births per woman was greater than 0.75. They asked 50 women across China the number of births they had. The results are shown in Table 11.59. 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 |
According to an avid aquarist, the average number of fish in a 20-gallon tank is 10, with a standard deviation of two. His friend, also an aquarist, does not believe that the standard deviation is two. 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 two? Data: 11, 10, 9, 10, 10, 11, 11, 10, 12, 9, 7, 9, 11, 10, and 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 ounces, but the manager thinks that it may be higher. He randomly weighs 49 orders of fries, which yields a mean of 11 ounces and a standard deviation of 2 ounces.
You want to buy a specific computer. A sales representative of the manufacturer claims that retail stores sell this computer at an average price of $1,249 with a very narrow standard deviation of $25. You find a website that has a price comparison for the same computer at a series of stores as follows: $1,299; $1,229.99; $1,193.08; $1,279; $1,224.95; $1,229.99; $1,269.95; and $1,249. Can you argue that pricing has a larger standard deviation than claimed by the manufacturer? Use the 5 percent significance level. As a potential buyer, what would be the practical conclusion from your analysis?
A company packages apples by weight. One of the weight grades is Class A apples. Class A apples have a mean weight of 150 grams, and there is a maximum allowed weight tolerance of 5 percent above or below the mean for apples in the same consumer package. A batch of apples is selected to be included in a Class A apple package. Given the following apple weights of the batch, does the fruit comply with the Class A grade weight tolerance requirements? Conduct an appropriate hypothesis test.
a. At the 5 percent significance level
b. At the 1 percent significance level
Weights in selected apple batch (in grams): 158, 167, 149, 169, 164, 139, 154, 150, 157, 171, 152, 161, 141, 166, and 172.