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The Chi-Square Distribution: Homework

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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. Go to The Table of Contents 14. Appendix for the chi-square solution sheet. 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.

Table 1
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)

Table 2
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.

Table 3
Marital Status Frequency
never married 140
married 238
widowed 2
divorced/separated 20

Solution

  • 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 columns in the chart below 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. (Source: http://www.collegeboard.com). Suppose the right column contains the result of a survey of 1000 local students from that year who took an AP Exam.

Table 4
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

Exercise 4

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.

Exercise 5

Perform a goodness-of-fit test to determine whether the local results follow the distribution of U. S. AP Examinee Population, based on ethnicity.

Solution

  • c. 5
  • e. 13.4
  • f. 0.0199
  • g. Decision: Reject null when a = 0 . 05 a = 0 . 05 size 12{a=0 "." "05"} {} ; Conclusion: Local data do not fit the AP Examinee Distribution. Decision: Do not reject null when a = 0 . 01 a = 0 . 01 size 12{a=0 "." "01"} {} ; Conclusion: There is insufficient evidence to conclude that Local data do not fit the AP Examinee Distribution.

Exercise 6

The City of South Lake Tahoe, CA, has an Asian population of 1419 people, out of a total population of 23,609 (Source: U.S. Census Bureau). Suppose that a survey of 1419 self-reported Asians in Manhattan, NY, area yielded the data in the table below. Conduct a goodness of fit test to determine if the self-reported sub-groups of Asians in the Manhattan area fit that of the Lake Tahoe area.

Table 5
Race Lake Tahoe Frequency Manhattan Frequency
Asian Indian 131174  
Chinese 118557  
Filipino 1045518  
Japanese 8054  
Korean 1229  
Vietnamese 921  
Other 2466  

The next two questions refer to the following information: 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 (2/2/2006). Suppose a survey of 5000 graduating females and 5000 graduating males was done as a follow-up last year to determine what their actual major was. The results are shown in the tables for Exercises 7 and 8. The second column in each table does not add to 100% because of rounding.

Exercise 7

Conduct a hypothesis test to determine if the actual college major of graduating females fits the distribution of their expected majors.

Table 6
Major Women - Expected Major Women - Actual Major
Arts & Humanities 14.0% 670
Biological Sciences 8.4% 410
Business 13.1% 685
Education 13.0% 650
Engineering 2.6% 145
Physical Sciences 2.6% 125
Professional 18.9% 975
Social Sciences 13.0% 605
Technical 0.4% 15
Other 5.8% 300
Undecided 8.0% 420

Solution

  • c. 10
  • e. 11.48
  • f. 0.3214
  • h. Decision: Do not reject null when a = 0 . 05 a = 0 . 05 size 12{a=0 "." "05"} {} and a = 0 . 01 a = 0 . 01 size 12{a=0 "." "05"} {} ; Conclusion: There is insufficient evidence to conclude that the distribution of majors by graduating females does not fit the distribution of expected majors.

Exercise 8

Conduct a hypothesis test to determine if the actual college major of graduating males fits the distribution of their expected majors.

Table 7
Major Men - Expected Major Men - Actual Major
Arts & Humanities 11.0% 600
Biological Sciences 6.7% 330
Business 22.7% 1130
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

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.

Table 8
U.S. Ski Area Beginner Intermediate Advanced
Tahoe 20 30 40
Utah 10 30 60
Colorado 10 40 50

Solution

  • 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.

Table 9
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.

Table 10
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

Solution

  • 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.

Table 11
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.

Table 12
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

  • 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.

Table 13
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.

Table 14
Age of Males None < $200,000 $200,000 - $400,000 $401,001 - $1,000,000 $1,000,000 +
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

  • 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.

Table 15
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

Exercise 17

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 below. Conduct a Test of Homogeneity. Test at a 5% level of significance.

Table 16
  Open Conscientious Extrovert Agreeable Neurotic
Business 41 52 46 61 58
Social Science 72 75 63 80 65

Solution

  • c: 4
  • d: Chi-Square with df = 4
  • e: 3.01
  • f: p-value = 0.5568
  • h:
    ii. Do not reject the null hypothesis.
    iv. There is insufficient evidence to conclude that the distribution of personality types is different for business and social science majors.

Exercise 18

Do men and women select different breakfasts? The breakfast ordered by randomly selected men and women at a popular breakfast place is shown below. Conduct a test of homogeneity. Test at a 5% level of significance

Table 17
  French Toast Pancakes Waffles Omelettes
Men 47 35 28 53
Women 65 59 55 60

Solution

  • c: 3
  • e: 4.01
  • f: p-value = 0.2601
  • h:
    ii. Do not reject the null hypothesis.
    iv. There is insufficient evidence to conclude that the distribution of breakfast ordered is different for men and women.

Exercise 19

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 the 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 the 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.

Solution

  • c: 2
  • e: 7.05
  • f: p-value = 0.0294
  • h:
    ii. Reject the null hypothesis.
    iv. There is sufficient evidence to conclude that the distribution of technology use for statistics homework is not the same for statistics students at community colleges and at universities.

Exercise 20

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 the hypothesis test at a 5% level of significance.

Solution

  • c: 3
  • d: Chi-Square with df = 3
  • e: 11.75
  • f: p-value = 0.0083
  • h:
    ii. Reject the null hypothesis.
    iv. There is sufficient evidence to conclude that the distribution of fish in Green Valley Lake is not the same as the distribution of fish in Echo Lake.

Exercise 21

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

  • c. 83
  • d. Chi-Square with df = 83
  • e. 96.81
  • f. p-value = 0.1426; There is a 0.1426 probability that the sample standard deviation is 0.54 or more.
  • h. Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the standard deviation is more than 0.5 oz. It cannot be determined whether the equipment needs to be recalibrated or not.

Exercise 22

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 23

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

  • c. 4
  • d. Chi-Square with df = 4
  • e. 4.52
  • f. 0.3402
  • h. Decision: Do not reject null.

Exercise 24

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 25

The number of births per woman in China is 1.6 down from 5.91 in 1966 (SourceWorld Bank, 6/5/12). 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?

Table 18
# of births Frequency
0 5
1 30
2 10
3 5

Solution

  • c. 49
  • d. Chi-Square with df = 49
  • e. 54.37
  • f. p-value = 0.2774; If the null hypothesis is true, there is a 0.2774 probability that the sample standard deviation is 0.79 or more.
  • h. Decision: Do not reject null; Conclusion: There is insufficient evidence to conclude that the standard deviation is more than 0.75. It cannot be determined if the standard deviation is greater than 0.75 or not.

Exercise 26

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 27

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 a mean of 11 oz. and a standard deviation of 2 oz.

Solution

  • 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. Chi-Square with df = 48
  • e. 85.33
  • f. 0.0007
  • h. Decision: Reject null.

Try these true/false questions.

Exercise 28

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

Solution

True

Exercise 29

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

Solution

False

Exercise 30

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

Solution

False

Exercise 31

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

Solution

True

Exercise 32

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

True

Exercise 33

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

Solution

False

Exercise 34

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

Solution

True

Exercise 35

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

Solution

True

Exercise 36

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

True

Exercise 37

The test to use to determine if a six-sided die is fair is a Goodness-of-Fit test.

Solution

True

Exercise 38

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

True

Exercise 39

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

Solution

False

Exercise 40

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

Solution

False

Exercise 41

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

Solution

True

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