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The Chi-Square Distribution: Homework

Module by: Dr. Barbara Illowsky, Susan Dean

Summary: This module provides homework on Chi-Square Distribution as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Exercise 1

  • a. Explain why the “goodness of fit” test and the “test for independence” are generally right tailed tests.
  • b. If you did a left-tailed test, what would you be testing?

Word Problems

For each word problem, use a solution sheet to solve the hypothesis test problem. Round expected frequency to two decimal places.

Exercise 2

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  

Exercise 3

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

Solution 3

  • a. The data fits the distribution
  • b. The data does not fit the distribution
  • c. 3
  • e. 19.27
  • f. 0.0002
  • h. Decision: 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

Exercise 4

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.

Exercise 5

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.

Solution 5

  • c. 3
  • e. 10.91
  • f. 0.0122
  • g. Decision: Reject null when a = 0 . 05 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 a = 0 . 01 size 12{a=0 "." "01"} {} ; Conclusion Percent of loans fits the distribution.

Exercise 6

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  

Exercise 7

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

Solution 7

  • c. 6
  • e. 27,870
  • f. 0
  • h. Decision: Reject null; Conclusion: L.B. does not fit L.A.C.

Exercise 8

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%

Exercise 9

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

Solution 9

  • c. 4
  • e. 10.53
  • f. 0.0324
  • h. Decision: Reject null; Conclusion: Best ski area and level of skier are not independent.

Exercise 10

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

Exercise 11

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

Solution 11

  • c. 8
  • e. 33.55
  • f. 0
  • h. Decision: Reject null; Conclusion: Major and starting salary are not independent events.

Exercise 12

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

Exercise 13

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

Solution 13

  • c. 6
  • e. 25.21
  • f. 0.0003
  • h. Decision: Reject null

Exercise 14

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

Exercise 15

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

Solution 15

  • c. 12
  • e. 125.74
  • f. 0
  • h. Decision: Reject null

Exercise 16

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

Exercise 17

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 1212 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?

Solution 17

  • c. 83
  • d. 96.81
  • e. 0.1426
  • g. Decision: Do not reject null; Conclusion: The standard deviation is at most 0.5 oz.
  • h. It does not need to be calibrated

Exercise 18

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.

Exercise 19

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.

Solution 19

  • c. 4
  • d. 4.52
  • e. 0.3402
  • g. Decision: Do not reject null.
  • h. No

Exercise 20

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.

Exercise 21

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

Solution 21

  • c. 49
  • d. 54.37
  • e. 0.2774
  • g. Decision: Do not reject null; Conclusion: The standard deviation is at most 0.75.
  • h. No

Exercise 22

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

Exercise 23

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.

Solution 23

  • a. σ 2 ≤ 1 . 5 2 σ 2 ≤ 1 . 5 2 size 12{σ rSup { size 8{2} } <= left (1 "." 5 right ) rSup { size 8{2} } } {}
  • c. 48
  • d. 85.33
  • e. 0.0007
  • g. Decision: Reject null.
  • h. Yes

Try these true/false questions.

Exercise 24

As the degrees of freedom increase, the graph of the chi-square distribution looks more and more symmetrical.

Solution 24

True

Exercise 25

The standard deviation of the chi-square distribution is twice the mean.

Solution 25

False

Exercise 26

The mean and the median of the chi-square distribution are the same if df=24df=24 size 12{ ital "df"="24"} {}.

Solution 26

False

Exercise 27

In a Goodness-of-Fit test, the expected values are the values we would expect if the null hypothesis were true.

Solution 27

True

Exercise 28

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.

Solution 28

True

Exercise 29

The degrees of freedom for a Test for Independence are equal to the sample size minus 1.

Solution 29

False

Exercise 30

Use a Goodness-of-Fit test to determine if high school principals believe that students are absent equally during the week or not.

Solution 30

True

Exercise 31

The Test for Independence uses tables of observed and expected data values.

Solution 31

True

Exercise 32

The test to use when determining if the college or university a student chooses to attend is related to his/her socioeconomic status is a Test for Independence.

Solution 32

True

Exercise 33

The test to use to determine if a coin is fair is a Goodness-of-Fit test.

Solution 33

True

Exercise 34

In a Test of Independence, the expected number is equal to the row total multiplied by the column total divided by the total surveyed.

Solution 34

True

Exercise 35

In a Goodness-of Fit test, if the p-value is 0.0113, in general, do not reject the null hypothesis.

Solution 35

False

Exercise 36

For a Chi-Square distribution with degrees of freedom of 17, the probability that a value is greater than 20 is 0.7258.

Solution 36

False

Exercise 37

If df=2df=2 size 12{ ital "df"=2} {}, the chi-square distribution has a shape that reminds us of the exponential.

Solution 37

True

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