Transcript Applications of the Chi

Applications of the
Chi-Square Statistic
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
The Binomial Situation
• The experiment consists of n repetitions(trials).
• The trials are independent.
• Each trial has two possible outcomes (success,
failure).
• The probability of a success for each trial is p.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Ho: p = po versus Ha: p  po
Lot Sampling Example
Ho: p1 = .04 p2 = .96
Ha: p1  .04 p2  .96
2
(O  E) 2

E
(O1  E1 ) 2 (O2  E 2 ) 2


E1
E2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Reject Ho if 2 > 2.05,1 = 3.84
Lot Sampling Example
2
(13 6) 2 (137144) 2
*

6
144
 8.17 .34 8.51
8.51 > 3.84 therefore we reject Ho
We conclude that the proportion of defectives is not .04
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
The Multinomial Situation
Assumptions
• The experiment consists of n independent
repetitions.
• Each trial outcome falls in exactly one of k
categories.
• The probabilities of the k outcomes are
denoted by p1, p2, ..., pk and remain the same
on each trial. Further: p1+ p2,+ ...+ pk = 1
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Hypothesis Testing for the
Multinomial Situation
2
(O

E)
2  
E
Introduc
tion to
Busines
s
Statistic
s, 5e
Kvanli/
Guynes/
Pavur
(c)2000
SouthWestern
College
Publishi
ng
Where:
1. The summations is across all categories
2. The O’s are the observed frequencies in each
category using the sample.
3. The E’s are the expected frequencies in each
category if Ho is true.
4. The df for the chi-square statistic are k-1, where
k is the number of categories.
Chi-Square
Test of Independence
Null and Alternative Hypothesis
Ho: the classifications are independent
Ha: the classifications are dependent
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Chi-Square
Test of Independence
Estimating the Expected Frequencies
( row total for this cell)  (column total for this cell)
ˆ
E
n
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Expected Frequencies
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Figure 13.4
Chi-Square
Test of Independence
Estimating the Expected Frequencies Example
Sex
Male
Female
Total
<30
60
40
100
Age
30-45
20
30
50
>45
40 (30)
10
50
Estimate for Male and Over 45
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
120 50 (120)(50)
ˆ
E  200


30
200 200
200
Total
120
80
200
Chi-Square
Test of Independence
The Testing Procedure
Ho: the row and column classifications are independent
Ha: the row and column classifications are dependent
Reject Ho if 2 > 2.,df
where df = (r-1)(c-1)
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Chi-Square Test of Independence
The Testing Procedure
The test statistic is
Introduc
tion to
Busines
s
Statistic
s, 5e
Kvanli/
Guynes/
Pavur
(c)2000
SouthWestern
College
Publishi
ng
2
(O

E)
2  
E
Where:
1. The summation is over all cells of the contingency table
2. O is the observed frequency
3. E is the expected frequency
( row total for this cell)  (column total for this cell)
ˆ
E
n
4. The degrees of freedom are df = (r-1)(c-1)