Chi-Square Distributions


Right-Skewed distributions with minimum value of 0.
Specific Chi-Square distribution indicated by a parameter called a degrees of freedom.
Chi-Square Goodness-of-Fit Test
p1 = hypothesized proportion for category 1
H 0:
.
.
.
p k = hypothesized proportion for category k
H a:
H0 is not true, so at least one of the category proportions differs from the corresponding hypothesized value.
Test Statistic:  2 =
(observed cell count - expected cell count) 2

expected cell count
Rejection Region: Reject H0 if  2   2  , k -1
Assumptions:
1. A random sample is selected from the population.
2. All expected counts are greater than or equal to 1.
3. No more than 20% of the expected cell counts are less than 5.
Example M&M’s plain chocolate candies come in six different colors: brown, yellow, red, orange, green, and tan.
According to the manufacturer (Mars, Inc.), the color ratio in each large production batch is 30% brown, 20% yellow,
20% red, 10% orange, 10% green, and 10% tan. To test this claim, a professor at Carleton College (Minnesota) had
students count the colors of M&M’s found in “fun size” bags of candy (Teaching Statistics, Spring 1993). The results for
the 370 M&M’s are shown in the table. [Note: In 1995, Mars, Inc. added a seventh color - blue - to bags of M&M’s.]
Color
# M&M’s
Brown
84
Yellow
79
Red
75
Orange
49
Green
36
Tan
47
Total
370
Conduct a test to determine whether the true percentages of the colors produced differ from the manufacturer’s stated
percentages. Use =. 05.
Example A statistics department at a state university maintains a tutoring service for students in its introductory service
courses. The service has been staffed with the expectation that 40% of its students would be from the business statistics
course, 30% from engineering statistics, 20% from the statistics course for social science students, and the other 10% from
the course for agriculture students. A random sample of n=120 students revealed 50, 40, 18, and 12 from the four courses.
Does this data suggest that the percentages on which staffing was based are not correct? Conduct hypothesis using
  .05.
Chi-Square Test of Independence
H0: The two variables are independent
Ha: The two variables are dependent(related)
(observed cell count - expected cell count) 2
Test Statistic:  = 
expected cell count
all
2
cells
Where expected cell count = (row total  column total)/total sample size
Rejection Region: Reject Ho if  2   2  , (r -1 )(c-1 )
Assumptions:
1. A random sample is selected from the population.
2. All expected counts are greater than or equal to 1.
3. No more than 20% of the expected cell counts are less than 5.
Example Opinion polls often provide information on how different groups’ opinions vary on controversial issues. A
random sample of 102 registered voters was taken from the Supervisor of Election’s roll. Each of the registered voters
was asked the following two questions:
1. What is your political party affiliation?
2. Are you in favor of increased arms spending?
The results are summarized in the table below.
Opinion
Favor
No favor
Democrat
16
24
Party
Republican
21
17
None
11
13
Conduct test to determine if the opinions of individuals concerning military spending are related to party affiliation.
Expected counts are printed below observed counts
Chi-Square contributions are printed below expected counts
1
2
C1
16
18.82
0.424
C2
21
17.88
0.544
C3
11
11.29
0.008
Total
48
24
21.18
0.376
17
20.12
0.483
13
12.71
0.007
54
Total
40
38
24
102
ChiSq = 1.841, DF = 2, P-Value = 0.398
Example Are the educational aspirations of students related to family income? This question was investigated in the
article “Aspirations and Expectations of High School Youth” (Int. J. of Comp. Soc. (1975): 25). The accompanying 4 X 3
table resulted from classifying 273 students according to expected level of education and family income. Does the data
indicate that education aspirations and family income are not independent? Conduct hypothesis test using = .05.
Income
Aspired Level
Some High School
High School Graduate
Some College
College Graduate
Low
9
44
13
10
Middle
11
52
23
22
High
9
41
12
27