Nonparametric Methods:

1
Chapter 15

• Nonparametric Methods
• Chi-Square Applications

2
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!

3
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

4
Chi-square (?2) Applications

• Testing Method where we dont 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

5
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

6
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

7
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
• a .01 or .05

8
Hypothesis Testing

• Step 3 Identify the test statistic (Chi Square
  ?2) and draw curve with critical value
• Use a and df to look up critical value in
  appendix B
• k number of categories
• (k 1) degrees of freedom

9
Hypothesis Testing

• Step 4 Formulate a decision rule
• If our calculated test statistic is greater than
  18.307, we reject Ho and accept H1, otherwise we
  fail to reject Ho

10
Hypothesis Testing

• 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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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

12
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

13
C2 Distribution
14
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 gt 5
• If there are more than two cells
• 20 of fe cells contain values less than 5