Goodness of Fit Test for Proportions of Multinomial Population

1
Goodness of Fit Test for Proportions of
Multinomial Population

• Chi-square distribution
• Hypotheses test/Goodness of fit test

2
Chi-square distribution
With 2 degrees of freedom
With 5 degrees of freedom
With 10 degrees of freedom
0
3
Chi-Square Distribution

• We will use the notation to denote the
  value for the chi-square distribution that
  provides an area of a to the right of the stated
  value.

• For example, there is a .95 probability of
  obtaining a c2 (chi-square) value such that

4
For 9 d.f. and a .975
Selected Values from the Chi-Square Distribution
Table
5
.025
Area in Upper Tail .975
?2
0
2.700
6
For 9 d.f. and a .025
Selected Values from the Chi-Square Distribution
Table
7
Area in Upper Tail .025
?2
0
19.023
8
For 9 d.f. and a .10
Selected Values from the Chi-Square Distribution
Table
9
Area in Upper Tail .10
?2
14.684
0
10

• For 9 d.f. and 16.919, .05
• For 8 d.f. and 3.49, .90
• For 6 d.f. and 16.812, .01
• For 10 d.f. and 18.9, between .05
  and .025

11
Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
This is simply a hypothesis test to see if the
hypothesized population proportions agree with
the observed population proportions from our
sample.
1. Set up the null and alternative hypotheses.
2. Select a random sample and record the
observed frequency, fi , for each of the k
categories.
3. Assuming H0 is true, compute the expected
frequency, ei , in each category by
multiplying the category probability by the
sample size.
12
Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
4. Compute the value of the test statistic.
where
fi observed frequency for category i
ei expected frequency for category i
k number of categories
Note The test statistic has a chi-square
distribution with k 1 df provided that the
expected frequencies are 5 or more for all
categories.
13
Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
5. Rejection rule
p-value approach
Reject H0 if p-value lt a
Critical value approach
where ? is the significance level and there are
k - 1 degrees of freedom
14
Multinomial Distribution Goodness of Fit Test

• Example Finger Lakes Homes (A)

Finger Lakes Homes manufactures four
models of prefabricated homes, a two-story
colonial, a log cabin, a split-level, and an
A-frame. To help in production planning,
management would like to determine if
previous customer purchases indicate that
there is a preference in the style selected.
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Multinomial Distribution Goodness of Fit Test

• Example Finger Lakes Homes (A)

The number of homes sold of each model for
100 sales over the past two years is shown below.
Split-
A- Model Colonial Log Level Frame
Sold 30 20 35 15
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Multinomial Distribution Goodness of Fit Test

• Hypotheses

H0 pC pL pS pA .25
Ha The population proportions are not
pC .25, pL .25, pS .25, and pA .25
where pC population proportion that
purchase a colonial pL population
proportion that purchase a log cabin pS
population proportion that purchase a
split-level pA population proportion that
purchase an A-frame
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Multinomial Distribution Goodness of Fit Test

• Rejection Rule

Reject H0 if p-value lt .05 or c2 gt 7.815.
With ? .05 and k - 1 4 - 1 3
degrees of freedom
Do Not Reject H0
Reject H0
?2
7.815
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Multinomial Distribution Goodness of Fit Test

• Expected Frequencies
• Test Statistic

• e1 .25(100) 25 e2 .25(100) 25
• e3 .25(100) 25 e4 .25(100) 25

1 1 4 4 10
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Multinomial Distribution Goodness of Fit Test

• Conclusion Using the p-Value Approach

Area in Upper Tail .10 .05 .025
.01 .005
c2 Value (df 3) 6.251 7.815 9.348
11.345 12.838
Because c2 10 is between 9.348 and 11.345,
the area in the upper tail of the distribution
is between .025 and .01.
The p-value lt a . We can reject the null
hypothesis.
20
Multinomial Distribution Goodness of Fit Test

• Conclusion Using the Critical Value Approach

c2 10 gt 7.815
We reject, at the .05 level of
significance, the assumption that there is no
home style preference.