The ChiSquare Distribution
Test of Independence
OpenStaxCollege
[latexpage]
Tests of independence involve using a contingency table of observed (data) values.
The test statistic for a test of independence is similar to that of a goodnessoffit test:
where:
 O = observed values
 E = expected values
 i = the number of rows in the table
 j = the number of columns in the table
There are \(i\cdot j\) terms of the form \(\frac{{\left(O–E\right)}^{2}}{E}\).
A test of independence determines whether two factors are independent or not. You first encountered the term independence in Probability Topics. As a review, consider the following example.
The expected value for each cell needs to be at least five in order for you to use this test.
Suppose A = a speeding violation in the last year and B = a cell phone user while driving. If A and B are independent then P(A AND B) = P(A)P(B). A AND B is the event that a driver received a speeding violation last year and also used a cell phone while driving. Suppose, in a study of drivers who received speeding violations in the last year, and who used cell phone while driving, that 755 people were surveyed. Out of the 755, 70 had a speeding violation and 685 did not; 305 used cell phones while driving and 450 did not.
Let y = expected number of drivers who used a cell phone while driving and received speeding violations.
If A and B are independent, then P(A AND B) = P(A)P(B). By substitution,
\(\frac{y}{755}=\left(\frac{70}{755}\right)\left(\frac{305}{755}\right)\)
Solve for y: y = \(\frac{\left(70\right)\left(305\right)}{755}=28.3\)
About 28 people from the sample are expected to use cell phones while driving and to receive speeding violations.
In a test of independence, we state the null and alternative hypotheses in words. Since the contingency table consists of two factors, the null hypothesis states that the factors are independent and the alternative hypothesis states that they are not independent (dependent). If we do a test of independence using the example, then the null hypothesis is:
H_{0}: Being a cell phone user while driving and receiving a speeding violation are independent events.
If the null hypothesis were true, we would expect about 28 people to use cell phones while driving and to receive a speeding violation.
The test of independence is always righttailed because of the calculation of the test statistic. If the expected and observed values are not close together, then the test statistic is very large and way out in the right tail of the chisquare curve, as it is in a goodnessoffit.
The number of degrees of freedom for the test of independence is:
df = (number of columns – 1)(number of rows – 1)
The following formula calculates the expected number (E):
\(E=\frac{\text{(row total)(column total)}}{\text{total number surveyed}}\)
A sample of 300 students is taken. Of the students surveyed, 50 were music students, while 250 were not. Ninetyseven were on the honor roll, while 203 were not. If we assume being a music student and being on the honor roll are independent events, what is the expected number of music students who are also on the honor roll?
About 16 students are expected to be music students and on the honor roll.
In a volunteer group, adults 21 and older volunteer from one to nine hours each week to spend time with a disabled senior citizen. The program recruits among community college students, fouryear college students, and nonstudents. In [link] is a sample of the adult volunteers and the number of hours they volunteer per week.
Type of Volunteer  1–3 Hours  4–6 Hours  7–9 Hours  Row Total 

Community College Students  111  96  48  255 
FourYear College Students  96  133  61  290 
Nonstudents  91  150  53  294 
Column Total  298  379  162  839 
Is the number of hours volunteered independent of the type of volunteer?
The observed table and the question at the end of the problem, “Is the number of hours volunteered independent of the type of volunteer?” tell you this is a test of independence. The two factors are number of hours volunteered and type of volunteer. This test is always righttailed.
H_{0}: The number of hours volunteered is independent of the type of volunteer.
H_{a}: The number of hours volunteered is dependent on the type of volunteer.
The expected result are in [link].
Type of Volunteer  13 Hours  46 Hours  79 Hours 

Community College Students  90.57  115.19  49.24 
FourYear College Students  103.00  131.00  56.00 
Nonstudents  104.42  132.81  56.77 
For example, the calculation for the expected frequency for the top left cell is
\(E=\frac{\left(\text{row total}\right)\left(\text{column total}\right)}{\text{total number surveyed}}=\frac{\left(255\right)\left(298\right)}{839}=90.57\)
Calculate the test statistic:χ^{2} = 12.99 (calculator or computer)
Distribution for the test:\({\chi }_{4}^{2}\)
df = (3 columns – 1)(3 rows – 1) = (2)(2) = 4
Graph:
Probability statement:pvalue=P(χ^{2} > 12.99) = 0.0113
Compare α and the pvalue: Since no α is given, assume α = 0.05. pvalue = 0.0113. α > pvalue.
Make a decision: Since α > pvalue, reject H_{0}. This means that the factors are not independent.
Conclusion: At a 5% level of significance, from the data, there is sufficient evidence to conclude that the number of hours volunteered and the type of volunteer are dependent on one another.
For the example in [link], if there had been another type of volunteer, teenagers, what would the degrees of freedom be?
Press the MATRX
key and arrow over to EDIT
. Press 1:[A]
. Press 3 ENTER 3 ENTER
. Enter the table values by row from [link]. Press ENTER
after each. Press 2nd QUIT
. Press STAT
and arrow over to TESTS
. Arrow down to C:χ2TEST
. Press ENTER
. You should see Observed:[A] and Expected:[B]
. Arrow down to Calculate
. Press ENTER
. The test statistic is 12.9909 and the pvalue = 0.0113. Do the procedure a second time, but arrow down to Draw
instead of calculate
.
The Bureau of Labor Statistics gathers data about employment in the United States. A sample is taken to calculate the number of U.S. citizens working in one of several industry sectors over time. [link] shows the results:
Industry Sector  2000  2010  2020  Total 

Nonagriculture wage and salary  13,243  13,044  15,018  41,305 
Goodsproducing, excluding agriculture  2,457  1,771  1,950  6,178 
Servicesproviding  10,786  11,273  13,068  35,127 
Agriculture, forestry, fishing, and hunting  240  214  201  655 
Nonagriculture selfemployed and unpaid family worker  931  894  972  2,797 
Secondary wage and salary jobs in agriculture and private household industries  14  11  11  36 
Secondary jobs as a selfemployed or unpaid family worker  196  144  152  492 
Total  27,867  27,351  31,372  86,590 
We want to know if the change in the number of jobs is independent of the change in years. State the null and alternative hypotheses and the degrees of freedom.
H_{0} : The number of jobs is independent of the year.
H_{a} : The number of jobs is dependent on the year.
df = 12
Press the MATRX
key and arrow over to EDIT
. Press 1:[A]
. Press 3 ENTER 3 ENTER
. Enter the table values by row. Press ENTER
after each. Press 2nd QUIT
. Press STAT
and arrow over to TESTS
. Arrow down to C:χ2TEST
. Press ENTER
. You should see Observed:[A] and Expected:[B]
. Arrow down to Calculate
. Press ENTER
. The test statistic is 227.73 and the p−value = 5.90E – 42 = 0. Do the procedure a second time but arrow down to Draw
instead of calculate
.
De Anza College is interested in the relationship between anxiety level and the need to succeed in school. A random sample of 400 students took a test that measured anxiety level and need to succeed in school. [link] shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events.
Need to Succeed in School  High
Anxiety 
Medhigh
Anxiety 
Medium
Anxiety 
Medlow
Anxiety 
Low
Anxiety 
Row Total 

High Need  35  42  53  15  10  155 
Medium Need  18  48  63  33  31  193 
Low Need  4  5  11  15  17  52 
Column Total  57  95  127  63  58  400 
a. How many high anxiety level students are expected to have a high need to succeed in school?
a. The column total for a high anxiety level is 57. The row total for high need to succeed in school is 155. The sample size or total surveyed is 400.
\(E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}=\frac{155\cdot 57}{400}=22.09\)
The expected number of students who have a high anxiety level and a high need to succeed in school is about 22.
b. If the two variables are independent, how many students do you expect to have a low need to succeed in school and a medlow level of anxiety?
b. The column total for a medlow anxiety level is 63. The row total for a low need to succeed in school is 52. The sample size or total surveyed is 400.
c. \(E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}\) = ________
c. \(E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}=8.19\)
d. The expected number of students who have a medlow anxiety level and a low need to
succeed in school is about ________.
d. 8
Refer back to the information in [link]. How many service providing jobs are there expected to be in 2020? How many nonagriculture wage and salary jobs are there expected to be in 2020?
12,727, 14,965
References
DiCamilo, Mark, Mervin Field, “Most Californians See a Direct Linkage between Obesity and Sugary Sodas. Two in Three Voters Support Taxing SugarSweetened Beverages If Proceeds are Tied to Improving School Nutrition and Physical Activity Programs.” The Field Poll, released Feb. 14, 2013. Available online at http://field.com/fieldpollonline/subscribers/Rls2436.pdf (accessed May 24, 2013).
Harris Interactive, “Favorite Flavor of Ice Cream.” Available online at http://www.statisticbrain.com/favoriteflavoroficecream (accessed May 24, 2013)
“Youngest Online Entrepreneurs List.” Available online at http://www.statisticbrain.com/youngestonlineentrepreneurlist (accessed May 24, 2013).
Chapter Review
To assess whether two factors are independent or not, you can apply the test of independence that uses the chisquare distribution. The null hypothesis for this test states that the two factors are independent. The test compares observed values to expected values. The test is righttailed. Each observation or cell category must have an expected value of at least 5.
Formula Review
 The number of degrees of freedom is equal to (number of columns – 1)(number of rows – 1).
 The test statistic is
\(\underset{\left(i\cdot j\right)}{\Sigma }\frac{{\left(O–E\right)}^{2}}{E}\) where
O = observed values,
E = expected values,
i = the number of rows in the
table, and
j = the number of columns in
the table.  If the null hypothesis is true, the expected number
\(E=\frac{\text{(row total)(column total)}}{\text{total surveyed}}\).
Determine the appropriate test to be used in the next three exercises.
A pharmaceutical company is interested in the relationship between age and presentation of symptoms for a common viral infection. A random sample is taken of 500 people with the infection across different age groups.
a test of independence
The owner of a baseball team is interested in the relationship between player salaries and team winning percentage. He takes a random sample of 100 players from different organizations.
A marathon runner is interested in the relationship between the brand of shoes runners wear and their run times. She takes a random sample of 50 runners and records their run times as well as the brand of shoes they were wearing.
a test of independence
Use the following information to answer the next seven exercises: Transit Railroads is interested in the relationship between travel distance and the ticket class purchased. A random sample of 200 passengers is taken. [link] shows the results. The railroad wants to know if a passenger’s choice in ticket class is independent of the distance they must travel.
Traveling Distance  Third class  Second class  First class  Total 

1–100 miles  21  14  6  41 
101–200 miles  18  16  8  42 
201–300 miles  16  17  15  48 
301–400 miles  12  14  21  47 
401–500 miles  6  6  10  22 
Total  73  67  60  200 
State the hypotheses.
H_{0}: _______
H_{a}: _______
df = _______
8
How many passengers are expected to travel between 201 and 300 miles and purchase secondclass tickets?
How many passengers are expected to travel between 401 and 500 miles and purchase firstclass tickets?
6.6
What is the test statistic?
What is the pvalue?
0.0435
What can you conclude at the 5% level of significance?
Use the following information to answer the next eight exercises: An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the selfreported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites.
Complete the table.
Smoking Level Per Day  African American  Native Hawaiian  Latino  Japanese Americans  White  TOTALS 

110  
1120  
2130  
31+  
TOTALS 
Smoking Level Per Day  African American  Native Hawaiian  Latino  Japanese Americans  White  Totals 

110  9,886  2,745  12,831  8,378  7,650  41,490 
1120  6,514  3,062  4,932  10,680  9,877  35,065 
2130  1,671  1,419  1,406  4,715  6,062  15,273 
31+  759  788  800  2,305  3,970  8,622 
Totals  18,830  8,014  19,969  26,078  27,559  10,0450 
State the hypotheses.
H_{0}: _______
H_{a}: _______
Enter expected values in [link]. Round to two decimal places.
Calculate the following values:
Smoking Level Per Day  African American  Native Hawaiian  Latino  Japanese Americans  White 

110  7777.57  3310.11  8248.02  10771.29  11383.01 
1120  6573.16  2797.52  6970.76  9103.29  9620.27 
2130  2863.02  1218.49  3036.20  3965.05  4190.23 
31+  1616.25  687.87  1714.01  2238.37  2365.49 
df = _______
\({\chi }^{\text{2}}\) test statistic = ______
10,301.8
pvalue = ______
Is this a righttailed, lefttailed, or twotailed test? Explain why.
right
Graph the situation. Label and scale the horizontal axis. Mark the mean and test statistic. Shade in the region corresponding to the pvalue.
State the decision and conclusion (in a complete sentence) for the following preconceived levels of α.
α = 0.05
 Decision: ___________________
 Reason for the decision: ___________________
 Conclusion (write out in a complete sentence): ___________________
 Reject the null hypothesis.
 pvalue < alpha
 There is sufficient evidence to conclude that smoking level is dependent on ethnic group.
α = 0.01
 Decision: ___________________
 Reason for the decision: ___________________
 Conclusion (write out in a complete sentence): ___________________
Homework
For each problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chisquare solution sheet. Round expected frequency to two decimal places.
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 
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 [link]. Conduct a test of independence.
Family Size  Sub & Compact  Midsize  Fullsize  Van & Truck 

1  20  35  40  35 
2  20  50  70  80 
3–4  20  50  100  90 
5+  20  30  70  70 
 H_{0}: Car size is independent of family size.
 H_{a}: Car size is dependent on family size.
 df = 9
 chisquare distribution with df = 9
 test statistic = 15.8284
 pvalue = 0.0706
 Check student’s solution.

 Alpha: 0.05
 Decision: Do not reject the null hypothesis.
 Reason for decision: pvalue > alpha
 Conclusion: At the 5% significance level, there is insufficient evidence to conclude that car size and family size are dependent.
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. [link] 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 hot spots 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 [link]. Conduct a test of 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 
 H_{0}: Honeymoon locations are independent of bride’s age.
 H_{a}: Honeymoon locations are dependent on bride’s age.
 df = 9
 chisquare distribution with df = 9
 test statistic = 15.7027
 pvalue = 0.0734
 Check student’s solution.

 Alpha: 0.05
 Decision: Do not reject the null hypothesis.
 Reason for decision: pvalue > alpha
 Conclusion: At the 5% significance level, there is insufficient evidence to conclude that honeymoon location and bride age are dependent.
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 and over 

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 [link]. 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 
 H_{0}: The types of fries sold are independent of the location.
 H_{a}: The types of fries sold are dependent on the location.
 df = 6
 chisquare distribution with df = 6
 test statistic =18.8369
 pvalue = 0.0044
 Check student’s solution.

 Alpha: 0.05
 Decision: Reject the null hypothesis.
 Reason for decision: pvalue < alpha
 Conclusion: At the 5% significance level, There is sufficient evidence that types of fries and location are dependent.
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  < 💲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 thirtyyearolds were surveyed to determine whether or not 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 
 H_{0}: Salary is independent of level of education.
 H_{a}: Salary is dependent on level of education.
 df = 12
 chisquare distribution with df = 12
 test statistic = 255.7704
 pvalue = 0
 Check student’s solution.

Alpha: 0.05
Decision: Reject the null hypothesis.
Reason for decision: pvalue < alpha
Conclusion: At the 5% significance level, there is sufficient evidence to conclude that salary and level of education are dependent.
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.
true
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.
true
An ice cream maker performs a nationwide survey about favorite flavors of ice cream in different geographic areas of the U.S. Based on [link], do the numbers suggest that geographic location is independent of favorite ice cream flavors? Test at the 5% 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 
[link] provides 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% significance level.
Age Group\ Net Worth Value (in millions of US 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 
 H_{0}: Age is independent of the youngest online entrepreneurs’ net worth.
 H_{a}: Age is dependent on the net worth of the youngest online entrepreneurs.
 df = 2
 chisquare distribution with df = 2
 test statistic = 1.76
 pvalue 0.4144
 Check student’s solution.

 Alpha: 0.05
 Decision: Do not reject the null hypothesis.
 Reason for decision: pvalue > alpha
 Conclusion: At the 5% significance level, there is insufficient evidence to conclude that age and net worth for the youngest online entrepreneurs are dependent.
A 2013 poll in California surveyed people about taxing sugarsweetened beverages. The results are presented in [link], 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% significance level.
Opinion/Ethnicity  AsianAmerican  White/NonHispanic  AfricanAmerican  Latino  Row Total 

Against tax  48  433  41  160  628 
In Favor of tax  54  234  24  147  459 
No opinion  16  43  16  19  84 
Column Total  118  710  71  272  1171 
Glossary
 Contingency Table
 a table that displays sample values for two different factors that may be dependent or contingent on one another; it facilitates determining conditional probabilities.