Testing Differences in Proportions for Independent Samples

1
Testing Differences in Proportions for
Independent Samples
2
The number of people littering or not depending
on the amount of litter already on the ground.
3
Parametric Tests

• Test specific population parameters (e.g. ? or ?1
  - ?2).
• Make assumptions about the shape of the
  population distribution.
• Scores are interval or ratio data.

4
Non-parametric Tests

• Hypothesis not stated in terms of population
  parameters.
• Make few if any assumptions about the population
  distribution (i.e. distribution-free tests).
• Well-suited for data measured on ordinal or
  nominal scales.
• Less sensitive than parametric tests

5
Some Non-Parametric Tests

• Mann-Whitney U (in place of independent t)
• Wilcoxon Signed-Ranks Test (in place of RM t
  test)
• Kruskal-Wallis Test (in place of Independent
  Measures ANOVA)
• Friedman Test (in place of RM ANOVA)

6
Chi-square Test for Goodness of Fit

• The chi-square test for goodness of fit uses
  sample data to test hypotheses about the shape or
  proportions of a population distribution. The
  test determines how well the obtained sample
  proportions fit the population proportions
  specified by the null hypothesis.

7
Frequency distribution 1
4
3
Frequency
2
1
1
2
3
4
5
6
7
Scores
8
Frequency Distribution 2
20
15
Frequency
10
5
A
B
C
Personality Type
9
Frequency Distribution of Eye Color
20
15
f
10
5
Blue
Brown
Green
Other
Eye Color
10
Table - Men vs. Women
11
Generally H0 falls into 2 categories

• No preference

2. No difference from a comparison population
(Proportions for the California pop. are not
different from the Colorado pop.)
12
Example
n 40 Sample
13
Observed Frequency

• The observed frequency is the number of
  individuals from the sample who are classified in
  a particular category. Each individual is
  counted in one and only one category.

14
Expected Frequency Example
Expected frequency fe pn
15
Expected Frequency

• The expected frequency for each category is the
  frequency value that is predicted from the null
  hypothesis and the sample size (n). The expected
  frequencies define an ideal, hypothetical sample
  distribution that would be obtained if the sample
  proportions were in perfect agreement with the
  proportions specified in the null hypothesis.

16
Chi-square Statistic
17
Chi-square Distribution
18
Chi-square distributions with differing dfs
19
Significance table for Chi-square
20
A researcher is interested in the factors
involved in course selection. A sample of 50
students is asked Which of the following
factors is most important to you when selecting a
course? Students must choose one and only one
of the following alternatives

• Interest in course topic
• Ease of passing the course
• Instructor for the course
• Time of day course is offered

21
Introvert vs. Extrovert data table
22
The Null Hypothesis

• Version 1
• H0 For the general population of students,
  there is no relationship between color preference
  and personality.
• Version 2
• H0 In the general population, the distribution
  of color preferences has the same shape (same
  proportions) for both categories of personality
  (introverts and extroverts).

23
Empty data table
24
Data table of expected frequencies
Expected Frequencies
25
Formula for Determining Expected Frequencies
fc frequency total for column fr frequency
total for row fe expected frequency for a
cell n number in entire sample
26
Chi-square Test of Independence
27
Half-filled data table
df (R - 1)(C - 1)
28
A researcher is investigating the relationship
between academic performance and self-esteem. A
sample (n 150) of 10-year old children is
obtained and each child is classified by level of
academic performance and self-esteem. The
observed frequencies for this sample are
presented in the following table
29
Empty data table
30
Expected frequencies table
Expected Frequencies
31
Assumptions and Restrictions for Chi-square Tests

• Independence of Observations
• Size of expected frequencies (fe)
• Dont use ?2 when fe for any cell is less than 5

32
Original data