1. Ch 15 - Chi-square Nonparametric Methods: Chi-Square Applications
Chapter 15
Nonparametric Methods:
Chi-Square Applications
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2. Nonparametric One-Look.Com Definition:
Ch 15 - Chi-square
Nonparametric
One-Look.Com Definition:
adjective:   not involving an estimation of the parameters of a statistic
adjective: not requiring knowledge of underlying distribution: used to describe or relating to statistical methods that do not require assumptions about the form of the underlying distribution
You mean we can test without assuming a normal curve?
Yes!
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3. Ch 15 - Chi-square
Goals
Conduct a test of hypothesis comparing an observed set of frequencies to an expected set of frequencies
Goodness-of-fit tests:
Equal Expected Frequencies
Unequal Expected Frequencies
List the characteristics of the Chi-square distribution
We can test a
hypothesis with
assuming
data distribution
is normal!
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4. Chi-square (2) Applications
Ch 15 - Chi-square
Chi-square (2) Applications
Testing Method where we don’t need assumptions about the shape of the data
Testing methods for Nominal data
Data with no natural order
Examples:
Gender
Brand preference
Color
There will be two difference from earlier tests when we do our hypothesis testing:
Look up critical value of Chi-square in appendix B
Use new formula for Calculated Test Statistic
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5. Goodness-of-fit tests:
Ch 15 - Chi-square
Conduct A Test Of Hypothesis Comparing An Observed Set Of Frequencies To An Expected Set Of Frequencies
Goodness-of-fit tests:
Equal Expected Frequencies
Unequal Expected Frequencies
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6. Purpose Of Goodness-of-fit Tests:
Ch 15 - Chi-square
Purpose Of Goodness-of-fit Tests:
Compare an observed distribution (sample) to an expected distribution (population)
We will ask the question:
Is the difference between the observed values and the expected values:
Due to chance (sampling error):
The observed distribution is the same as the expected distribution
Not due to chance:
The observed distribution is not the same as the expected distribution
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7. Hypothesis Testing: Equal Expected Frequencies
Ch 15 - Chi-square
Hypothesis Testing: Equal Expected Frequencies
Step 1: State null and alternate hypotheses
Ho : There is no significant difference between the set of observed frequencies and the set of expected frequencies
H1 : There is a difference between the observed and expected frequencies
Step 2: Select a level of significance
α = .01 or .05…
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8. Ch 15 - Chi-square
Hypothesis Testing
Step 3: Identify the test statistic (Chi Square = 2) and draw curve with critical value
Use α and df to look up critical value in appendix B
k = number of categories
(k – 1) = degrees of freedom
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9. Hypothesis Testing Step 4: Formulate a decision rule
Ch 15 - Chi-square
Hypothesis Testing
Step 4: Formulate a decision rule
If our calculated test statistic is greater than , we reject Ho and accept H1, otherwise we fail to reject Ho
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10. Ch 15 - Chi-square
Hypothesis Testing
Equal
Expected
Frequencies
Unequal
Expected
Frequencies
Step 5: Take a random sample, compute the calculated test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses fe
will be
given or
n*% for cell
1st
2nd
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11. Hypothesis Testing Step 5: Conclude: There is either:
Ch 15 - Chi-square
Hypothesis Testing
Step 5: Conclude:
There is either:
The sample evidence suggests that there is not a difference between the observed and expected frequencies
The observed distribution is the same as the expected distribution
The sample evidence suggests that there is a difference between the observed and expected frequencies
The observed distribution is not the same as the expected distribution
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12. List The Characteristics Of The Chi-square Distribution
Ch 15 - Chi-square
List The Characteristics Of The Chi-square Distribution
It is positively skewed
However, as the degrees of freedom increase, the curve approaches normal
It is non-negative
Because (fo – fe)2 is never negative
There is a family of chi-square distributions
df determines which curve to use
df = k – 1
k = # of categories
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13. df = 3 df = 5 df = 10 c2 C2 Distribution Ch 15 - Chi-square
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14. Limitations Of Chi-Square
Ch 15 - Chi-square
Limitations Of Chi-Square
Because fe is used in the denominator, very small fe could result in very large calculated test statistic
In General, avoid using Chi-Square when:
If there are only two cells:
fe >= 5
If there are more than two cells
20% of fe cells contain values less than 5
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