statistical analyses: chi-square test
DESCRIPTION
Statistical Analyses: Chi-square test. Psych 250 Winter 2013. Types of Measures / Variables. Nominal / categorical Gender, major, blood type, eye color Ordinal Rank-order of favorite films; Likert scales? Interval / scale Time, money, age, GPA. Main Analysis Techniques. Question. - PowerPoint PPT PresentationTRANSCRIPT
Statistical Analyses:
Chi-square test
Psych 250Winter 2013
Types of Measures / Variables
• Nominal / categorical– Gender, major, blood type, eye color
• Ordinal– Rank-order of favorite films; Likert scales?
• Interval / scale– Time, money, age, GPA
Variable Type Example Commonly-used Statistical
MethodNominal by Nominal blood type by
genderChi-square
Scale by Nominal GPA by gender
GPA by major
t-test
Analysis of Variance
Scale by Scale weight by heightGPA by SAT
RegressionCorrelation
Main Analysis Techniques
Question
Do men and women differ in the % that choose jail time vs. probation only?
Variable Type Example Commonly-used Statistical
MethodNominal by Nominal(categorical by categorical)
blood type by gender
Chi-square
Scale by Nominal GPA by gender
GPA by major
t-test
Analysis of Variance
Scale by Scale weight by heightGPA by SAT
RegressionCorrelation
Main Analysis Techniques
Stat Analysis / Hypothesis Testing
1. Form of the relationship
2. Statistical significance
Variables:Categorical by Categorical
• Form of the relationship: Cross-tab= two-way table
• Statistical Significance: Chi Square[ if n very small Fisher’s exact test ]
Example: cross-tab
Probationn = 24
Jailn = 16
Males n = 20
16 4
Females n = 20
8 12
Example: cross-tab
Probationn = 24
Jailn = 16
Males n = 20
16 80%
4 20%
Females n = 20
8 40%
12 60%
Example
• Men more likely to choose probation in the sample
• Can we infer men in general more likely to choose probation?
Statistical Significance
Statistical Significance
• Q: Is this a “statistically significant” difference?
• Can the “null hypothesis” be rejected?
Null hypothesis: there are NO differences between men and women in sentencing
Universen = ∞
M: ?% probationF: ?% probation
Samplen = 40
M: 80% probationF: 40% probation
sample
inference
Universen = ∞
Null Hypothesis:M% = F%
Samplen = 40
M: 80% probationF: 40% probation
sample
inference
Logic of Statistical Inference
1. If the Null Hypothesis is True…
… what are the expected frequencies for Men and Women in any sample?
2. Do the frequencies in my sample (n = 40) differ from the expected frequencies?
Testing Null Hypothesis:Expected Frequencies
Probationn = 24
Jailn = 16
Males n = 20
exp: 12 exp: 8
Females n = 20
exp: 12 exp: 8
Probationn = 24
Jailn = 16
Males n = 20
16 exp: 12
4 exp: 8
Females n = 20
8 exp: 12
12 exp: 8
Observed & Expected Frequencies
Logic of Statistical Inference• What is the probability of drawing the
observed sample (M = 16 probation vs. F = 8 probation) from a universe with no differences?
• If probability very low, then differences in sample likely reflect differences in universe
• Then null hypothesis can be rejected; difference in sample is statistically significant
Statistical Significance
• If probability of obtaining my sample is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe.
p < .05
• The finding from the sample is statistically significant
Strategy
• Draw an infinite number of samples of n = 40, and graph the distribution of their male vs. female probation %-s
Null Hyp:M = 60% probationF = 60% probation
M: 60%F: 50%
Samples of n = 40 Universe n = ∞
M: 80%F: 40%
M: 70%F: 70%
M: 50%F: 65%
Chi Square Distribution
2.5% of area M % > F %
2.5% of area F % > M %
M % = F %
Statistical Significance
• If probability of obtaining my sample is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe.
p < .05
• The finding from the sample is statistically significant
Testing Null Hypothesis:Sample with small difference
Probationn = 24
Jailn = 16
Males n = 20
13 exp: 12
7 exp: 8
Females n = 20
11 exp: 12
9 exp: 8
UniverseN = ∞
M = 60% probationF = 60% probation
SampleN = 40
M = 65%F = 55%
sample
p < .05 ?
Chi Square p = .519
Probationn = 24
Jailn = 16
Males n = 20
13 exp: 12
7 exp: 8
Females n = 20
11 exp: 12
9 exp: 8
Small Difference• p = .519
• Over 50% of samples drawn from null hypothesis universe will have differences this large (65% vs. 55%)
• Difference is not statistically significant
Probationn = 24
Jailn = 16
Males n = 20
16 exp: 12
4 exp: 8
Females n = 20
8 exp: 12
12 exp: 8
Testing Null Hypothesis:Sample with large differences
UniverseN = ∞
M = 60% probationF = 60% probation
SampleN = 40
M = 80%F = 40%
sample
p < .05 ?
Chi Square p = .010
Probationn = 24
Jailn = 16
Males n = 20
16 exp: 12
4 exp: 8
Females n = 20
8 exp: 12
12 exp: 8
Report Findings
• “Men were found to choose probation more frequently than women: 80% of the time vs. 40% of the time (df = 1, χ2 = 6.67, p. < ,05).”
• “Men chose probation 80% of the time, and women only 40% of the time, a difference which was statistically significant (df = 1, χ2 = 6.67, p. < ,05).”