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Statistics for Business and Economics

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Title: Chap. 9: The Chi-Square Test & The Analysis of Contingency Tables Subject: Statistics, 10/e, by McClave, Benson & Sincich Author: John J. McGill/Lyn Noble – PowerPoint PPT presentation

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Title: Statistics for Business and Economics


1
Statistics for Business and Economics
  • Chapter 9
  • Categorical Data Analysis

2
Learning Objectives
  • Explain ?2 Test for Proportions
  • Explain ?2 Test of Independence
  • Solve Hypothesis Testing Problems
  • More Than Two Population Proportions
  • Independence

3
Data Types
4
Qualitative Data
  • Qualitative random variables yield responses that
    classify
  • Example gender (male, female)
  • Measurement reflects number in category
  • Nominal or ordinal scale
  • Examples
  • What make of car do you drive?
  • Do you live on-campus or off-campus?

5
Hypothesis Tests Qualitative Data
6
Chi-Square (?2) Test for k Proportions
7
Hypothesis Tests Qualitative Data
8
Multinomial Experiment
  • n identical trials
  • k outcomes to each trial
  • Constant outcome probability, pk
  • Independent trials
  • Random variable is count, nk
  • Example ask 100 people (n) which of 3 candidates
    (k) they will vote for

9
One-Way Contingency Table
  • Shows number of observations in k independent
    groups (outcomes or variable levels)

Outcomes (k 3)
Candidate
Tom
Bill
Mary
Total
35
20
45
100
Number of responses
10
  • ????????????
  • H Prof (Tom) Prob (Mary)Prof(Bill)1/3
  • ??????????,????
  • H1 Prof (Tom) 1/3
  • H2 Prof (Mary) 1/3
  • H3 Prof (Bill) 1/3

11
  • Calculate the probability of incorrectly
    rejecting the null using the common sense test
    based on the three individual t-statistics.
  • To simplify the calculation, suppose that ,
    and are independently distributed. Let
    t1 and t2 be the t-statistics.
  • The common sense test is reject
    if t1gt1.96 and/or t2 gt
    1.96 and/or t3 gt 1.96 . What is the probability
    that this common sense test rejects H0 when H0
    is actually true? (It should be 5.)

12
  • Probability of incorrectly rejecting the null

13
  • which is not the desired 5.

14
  • The size of a test is the actual rejection rate
    under the null hypothesis.
  • The size of the common sense test isnt 5.
  • Its size actually depends on the correlation
    between t1 t2 and t3(and thus on the correlation
    between and ).
  • Two Solutions.
  • Use a different critical value in this procedure
    - not 1.96 (this is the Bonferroni method).
    This is rarely used in practice.
  • Use a different test statistic that test at once

15
Chi-Square (?2) Test for k Proportions
  • Tests equality () of proportions only
  • Example p1 .2, p2.3, p3 .5
  • One variable with several levels
  • Uses one-way contingency table

16
Conditions Required for a Valid Test One-way
Table
  1. A multinomial experiment has been conducted
  2. The sample size n is large E(ni) is greater than
    or equal to 5 for every cell

17
?2 Test for k Proportions Hypotheses Statistic
18
?2 Test Basic Idea
  • Compares observed count to expected count
    assuming null hypothesis is true
  • Closer observed count is to expected count, the
    more likely the H0 is true
  • Measured by squared difference relative to
    expected count
  • Reject large values

19
Finding Critical Value Example
What is the critical ?2 value if k 3, and ?
.05?
df k - 1 2
20
?2 Test for k Proportions Example
  • As personnel director, you want to test the
    perception of fairness of three methods of
    performance evaluation. Of 180 employees, 63
    rated Method 1 as fair, 45 rated Method 2 as
    fair, 72 rated Method 3 as fair. At the .05
    level of significance, is there a difference in
    perceptions?

21
?2 Test for k Proportions Solution
  • H0
  • Ha
  • ?
  • n1 n2 n3
  • Critical Value(s)

Test Statistic Decision Conclusion
22
?2 Test for k Proportions Solution
23
?2 Test for k Proportions Solution
Test Statistic Decision Conclusion
  • H0
  • Ha
  • ?
  • n1 n2 n3
  • Critical Value(s)

?2 6.3
Reject at ? .05
There is evidence of a difference in proportions
24
Hypothesis Tests Qualitative Data
25
Contingency Table Example
  • Left-Handed vs. Gender
  • Dominant Hand Left vs. Right
  • Gender Male vs. Female
  • 2 categories for each variable, so
  • called a 2 x 2 table
  • Suppose we examine a sample of
  • 300 children

26
Contingency Table Example
(continued)
  • Sample results organized in a contingency table

Gender Hand Preference Hand Preference
Gender Left Right
Female 12 108 120
Male 24 156 180
36 264 300
sample size n 300
120 Females, 12 were left handed 180 Males, 24
were left handed
27
Contingency Table Example Solution
  • H0
  • Ha
  • ?
  • n1 n2
  • Critical Value(s)

Test Statistic Decision Conclusion
28
Contingency Table Example Solution
If the two proportions are equal, then
P(Left Handed Female) P(Left Handed Male)
.12 i.e., we would expect (.12)(120) 14.4
females to be left handed (.12)(180) 21.6
males to be left handed
29
Contingency Table Example Solution
  • H0
  • Ha
  • ?
  • n1 n2
  • Critical Value(s)

Test Statistic Decision Conclusion
?2 0.7576
Reject at ? .05
There is evidence of a difference in proportions
30
?2 Test of Independence
31
Hypothesis Tests Qualitative Data
32
?2 Test of Independence
  • Shows if a relationship exists between two
    qualitative variables
  • One sample is drawn
  • Does not show causality
  • Uses two-way contingency table

33
?2 Test of Independence Contingency Table
  • Shows number of observations from 1 sample
    jointly in 2 qualitative variables

34
Conditions Required for a Valid ?2 Test
Independence
  1. Multinomial experiment has been conducted
  2. The sample size, n, is large Eij is greater than
    or equal to 5 for every cell

35
?2 Test of Independence Hypotheses Statistic
  • Hypotheses
  • H0 Variables are independent
  • Ha Variables are related (dependent)

36
?2 Test of Independence Expected Counts
  1. Statistical independence means joint probability
    equals product of marginal probabilities
  2. Compute marginal probabilities and multiply for
    joint probability
  3. Expected count is sample size times joint
    probability

37
Expected Count Example
38
Expected Count Example
39
Expected Count Example
78 160
Marginal probability
40
Expected Count Calculation

House Location



Urban

Rural


House Style

Obs.

Exp.

Obs.

Exp.

Total

Split
-
Level

63


49



112

Ranch

15


33



48

Total

78

78

82

82


160


41
?2 Test of Independence Example
  • As a realtor you want to determine if house style
    and house location are related. At the .05 level
    of significance, is there evidence of a
    relationship?

42
?2 Test of Independence Solution
  • H0
  • Ha
  • ?
  • df
  • Critical Value(s)

Test Statistic Decision Conclusion
43
?2 Test of Independence Solution
?
Eij ? 5 in all cells
11282 160
11278 160
4878 160
4882 160
44
?2 Test of Independence Solution
45
?2 Test of Independence Solution
Test Statistic Decision Conclusion
  • H0
  • Ha
  • ?
  • df
  • Critical Value(s)

?2 8.41
Reject at ? .05
There is evidence of a relationship
46
?2 Test of Independence Thinking Challenge
  • Youre a marketing research analyst. You ask a
    random sample of 286 consumers if they purchase
    Diet Pepsi or Diet Coke. At the .05 level of
    significance, is there evidence of a relationship?

Diet Pepsi
Diet Coke
No
Yes
Total
No
84
32
116
Yes
48
122
170
Total
132
154
286
47
?2 Test of Independence Solution
  • H0
  • Ha
  • ?
  • df
  • Critical Value(s)

Test Statistic Decision Conclusion
48
?2 Test of Independence Solution
?
Eij ? 5 in all cells
116132 286
154132 286
170132 286
170154 286
49
?2 Test of Independence Solution
50
?2 Test of Independence Solution
Test Statistic Decision Conclusion
  • H0
  • Ha
  • ?
  • df
  • Critical Value(s)

?2 54.29
Reject at ? .05
There is evidence of a relationship
51
Example
  • The meal plan selected by 200 students is shown
    below

Class Standing Number of meals per week Number of meals per week Number of meals per week Total
Class Standing 20/week 10/week none Total
Fresh. 24 32 14 70
Soph. 22 26 12 60
Junior 10 14 6 30
Senior 14 16 10 40
Total 70 88 42 200
52
Example
(continued)
  • The hypothesis to be tested is

H0 Meal plan and class standing are
independent (i.e., there is no relationship
between them) H1 Meal plan and class standing
are dependent (i.e., there is a relationship
between them)
53
Example Expected Cell Frequencies
(continued)
Observed
Class Standing Number of meals per week Number of meals per week Number of meals per week Total
Class Standing 20/wk 10/wk none Total
Fresh. 24 32 14 70
Soph. 22 26 12 60
Junior 10 14 6 30
Senior 14 16 10 40
Total 70 88 42 200
Expected cell frequencies if H0 is true
Class Standing Number of meals per week Number of meals per week Number of meals per week Total
Class Standing 20/wk 10/wk none Total
Fresh. 24.5 30.8 14.7 70
Soph. 21.0 26.4 12.6 60
Junior 10.5 13.2 6.3 30
Senior 14.0 17.6 8.4 40
Total 70 88 42 200
Example for one cell
54
Example The Test Statistic
(continued)
  • The test statistic value is

12.592 from the chi-squared
distribution with (4 1)(3 1) 6 degrees of
freedom
55
Example Decision and Interpretation
(continued)
Decision Rule If gt 12.592, reject
H0, otherwise, do not reject H0
Here, 0.709 lt 12.592,
so do not reject H0 Conclusion there is not
sufficient evidence that meal plan and class
standing are related at ? 0.05
0.05
0
?2
Reject H0
Do not reject H0
?20.0512.592
56
Conclusion
  • Explained ?2 Test for Proportions
  • Explained ?2 Test of Independence
  • Solved Hypothesis Testing Problems
  • More Than Two Population Proportions
  • Independence
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