Consumer Electronics Example

1
Contingency Tables
2
Consumer Electronics Example
Studies are often conducted into consumer
planning for the purchase of durable goods such
as TV sets. Might survey intention of consumers
to purchase large TV in next 12 months, then
follow up to see if actually made purchase.
Suppose survey of 1000 people was made
Joint event
Row marginal
Column marginal
3
Venn diagram
B 100
A 50
200
4
Conditional Probability
Conditional probability distribution
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Statistical Independence
If the survey results were actually like this,
the two variables would be statistically
independent. Note that the marginals are the
same as in the original table.
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c2 Test of Independence

• Shows if a relationship exists between 2
  categorical variables
• One sample is drawn
• Does Not show nature of relationship
• Does Not show causality
• Similar to testing p1 p2 pc
• Used widely in marketing
• Uses contingency table

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c2 Test of Independence
1. Hypotheses H0 Variables Are Independent H1
Variables Are Related (Dependent) 2. Test
Statistic 3. Degrees of Freedom (r - 1)(c - 1)
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Chi-square distribution

• Concentrated on the ve part of the real line.
  It cannot be negative. It is 0 when fe and fo
  are equal in each cell
• It is skewed to the right
• The shape depends on the degrees of freedom
• Mean and standard deviation depend on df mean
  df, s sqrt(2 df)

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Sample size requirements

• The chi-squared distribution is the sampling
  distribution of the chi-squared statistic only if
  the sample size is large. A rough guideline for
  this requirement is that the expected frequency
  should exceed 5 in each cell. Otherwise, the
  chi-squared distribution may poorly approximate
  the actual distribution of the chi-square
  statistic.

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Degrees of freedom

• The interpretation is, given the row and column
  marginal frequencies, the observed frequencies in
  a rectangular block of size
• (r-1)(c-1) within the contingency table determine
  the other cell frequencies. Calculating the
  statistic implies fixing the marginals to
  calculate expected frequencies, so these are not
  free.

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Expected Frequencies

• 1. Statistical independence means joint
  probability equals product of marginal
  probabilities
• P(A and B) P(A)P(B)
• 2. Compute marginal (row column) probabilities
  multiply for joint probability
• 3. Expected frequency is sample size times joint
  probability

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Expected Frequencies Calculation
Expected Frequency (Column Total) (Row
Total)/(Sample Size)
82112160
78112160
7848160
8248160
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Example

• Youre a marketing research analyst. You ask a
  random sample of 286 consumers if they purchase
  Diet Pepsi or Diet Coke. At the 0.05 level, 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
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Expected Frequencies Solution
ü
fe ³ 5 in All Cells
Diet Pepsi
132116286
132154286
No
Yes
Obs.
Exp.
Obs.
Exp.
Diet Coke
Total
No
84
53.5
32
62.5
116
Yes
48
78.5
122
91.5
170
132
132
154
154
286
Total
132170286
154170286
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Test Statistic Solution
Cell
f
f
f
-
f
(
f
-
f
)²
(
f
-
f
)²/
f
o
e
o
e
o
e
o
e
e
1,1
84
53.5
30.5
930.25
17.3879
1,2
32
62.5
-30.5
930.25
14.8840
2,1
48
78.5
-30.5
930.25
11.8503
2,2
122
91.5
30.5
930.25
10.1667
Total
286
286
54.2889
ü
ü
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c2 Test of Independence Solution

• H0 No Relationship
• H1 Relationship
• a .05
• df (2 - 1)(2 - 1) 1
• Critical Value(s)

Test Statistic Decision Conclusion

54
2889
.
Reject at a .05
Reject
a .05
Can reject null hypothesis.
2
c
0
3.841
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Example