Transcript Definitions

Difference Between Means Test (“t” statistic)
Analysis of Variance (“F” statistic)
Review of test statistics
Procedure
Level of
Measurement
Statistic
Interpretation
Correlation
All variables
continuous
r
Range -1 to +1, with 0 meaning no relationship.
For example, .35 denotes a moderately strong
positive relationship
Regression
All variables
continuous
r2, R2
Proportion of change in the dependent variable
accounted for by change in the independent
variable. R2 denotes cumulative effect of multiple
independent variables.
Unit change in the dependent variable caused by
a one-unit change in the independent variable
b
Logistic
regression
DV nominal &
dichotomous,
IV’s nominal or
continuous
b
exp(B)
Don’t try
Odds that DV will change if IV changes one unit,
or, if IV is dichotomous, if it changes its state.
Range 0 to infinity; 1 denotes even odds, or no
relationship. Higher than 1 means positive
relationship, lower negative relationship. Use
percentage to describe likelihood of effect.
Chi-Square
All variables
categorical, not
ordinal
X2
Reflects difference between Observed and
Expected frequencies. Use table to determine if
coefficient is sufficiently large to reject null
hypothesis
Difference
between
means
IV dichotomous, DV
continuous
t
Reflects magnitude of difference. Use table to
determine if coefficient is sufficiently large to
reject null hypothesis.
Difference Between Means Test
•
Used to test hypotheses with categorical independent variable and continuous
dependent variable
– Males are taller than females (height in inches)
– Female officers are less cynical than male officers (cynicism on a 1-5 scale)
•
To determine if variables are associated compare the means of two randomly
drawn samples
•
Null hypothesis: More than 5 chances in 100 (p> .05) that the difference between
means was produced by chance
– To overcome the null we need a difference sufficiently large so that the
probability it was produced by chance is less than five in one-hundred (p< .05)
– In other words, the t coefficient exceeds sampling error
– In the t this sampling error is the “standard error of the difference between
means” – the difference between all possible pairs of means, due to chance
alone
Major advantage: Remember our concern that in large datasets, weak real-life
effects can produce statistically significant results?
– When comparing means, we know their actual values. This lets us recognize
situations when, statistical significance or not, the actual differences are trivial.
•
Class exercise
H1: Male officers more cynical than females (1 - tailed)
H2: Officer gender determines cynicism (2 - tailed)
1.
2.
3.
4.
5.
Review: What is the null hypothesis?
Draw one sample of male officers, one of females
Compute the Standard Error of the Difference Between Means
Calculate the t coefficient
Use the t table to check for significance (no more than 5 chances in 100
that the null hypo. is correct)
Note One-tailed hypotheses (direction of the effect on the dependent variable
is predicted) require a smaller t to reach statistical significance than twotailed hypotheses, where only an effect is predicted, not its direction
Why? Because we’re using only one side of the probability distribution
Calculating t
1. Obtain the “pooled sample variance” Sp2
(Simplified method – midpoint between the two sample variances)
2. Compute the S.E. of the Diff. Between Means
x1 -x2
3. Compute the statistic
t=
x
1
-x2
x
1
Sp 2 =
s2 1 + s2 2
2
- x2 =  Sp2 (
1
n1
+
1
n2
)
Actual (“obtained”) difference between means
Predicted difference due to sampling error
•
The result is a ratio: the smaller the predicted error, and the greater the obtained
difference between means, the larger the t coefficient
•
The larger the t, the more likely we are to reject the null hypothesis, that any
difference between means is due to chance alone
•
We use a table to determine whether the t is large enough to reject the null
hypothesis (see next slide). We can reject the null if the probability that the
difference between means is due to chance is less than five in one-hundred (p< .05).
•
If the probability that the difference between the means is due to chance is greater
than five in one-hundred (p> .05), the null hypothesis is true.
1. Is hypothesis one-tailed (direction of change
in the DV predicted) or two-tailed (direction not
predicted)?
H1: Males more cynical than females.
This is one-tailed, so use the top row.
H2: Males and females differ in cynicism.
This is two-tailed, so use the second row.
2. df, “Degrees of Freedom” represents sample
size – add the numbers of cases in both samples,
then subtract two: df = (n1 + n2) – 2
3. To call a t “significant” (thus reject the null
hypothesis) the coefficient must be as large or
larger than what is required at the .05 level; that
is, we cannot take more than 5 chances in 100
that the difference between means is due to
chance.
•
For a one-tailed test, use the top row, then
slide over to the .05 column. For a two-tailed
test, use the second row, then slide to .05
column. If the t is smaller than the number at
the intersection of the .05 column and the
appropriate df row, it is non-significant.
•
If the t is that size or larger, it is significant.
Slide to the right to see if it is large enough
to be significant at a more stringent level.
More complex mean
comparisons:
Analysis of Variance
When there are more than two groups:
Analysis of Variance
Independent variables: categorical
Dependent variable: continuous
Example: does officer professionalism vary between cities? (scale 1-10)
City
L.A.
S.F.
S.D.
Mean
8
5
3
Calculate the “F” statistic, look up the table. An “F” statistic that is sufficiently
large can overcome the null hypothesis that the differences between the
means are due to chance.
“Two-way” Analysis of Variance
•
Stratified independent variable(s)
•
City
L.A.
S.F.
S.D.
Mean – M
10
7
5
Mean - F
6
3
2
Within
Between
F statistic is a ratio of “between-group” to “within” group differences. To
overcome the null hypothesis, the differences in scores between groups
(between cities and, overall, between genders) should be much greater than
the differences in scores within cities
Between group variance (error + systematic effects of ind. variable)
Within group variance (how scores disperse within each city)
Parking lot exercise
Homework
Homework assignment
Two random samples of 10 patrol officers from the XYZ
Police Department, each officer tested for cynicism
(continuous variable, scale 1-5)
Sample 1 scores: 3 3 3 3 3 3 3 1 2 5 -- Variance = .99
Sample 2 scores: 2 1 1 2 3 3 3 3 4 2 -- Variance = .93
Pooled sample variance Sp2
Simplified method: midpoint between the two sample variances
2
Sp =
s2 1 + s2 2
2
Standard error of the difference between means
x
1
-x2 =  Sp2 (
1
n1
x1
-x2
1
+n )
2
T-Test for significance of the difference between means
x1 -x2
t = -------------x -x
1
2
CALCULATIONS
Pooled sample variance: .96
Standard error of the difference between means: .44
t statistic: 1.14
df – degrees of freedom: (n1 + n2) – 2 = 18
Would you use a ONE-tailed t-test OR a TWO-tailed ttest?
Depends on the hypothesis
Two-tailed (does not predict direction of the change):
Gender  cynicism
One-tailed (predicts direction of the change): Males
more cynical than females
Can you reject the NULL hypothesis? (probability
that the t coefficient could have been produced by
chance must be less than five in a hundred)
NO – For a ONE-tailed test need a t of 1.734 or higher
NO – For a TWO-tailed test need a t of 2.101 or higher
Final exam practice
•
You will be given scores and variances for two samples and asked to decide whether their
means are significantly different.
•
You will be asked to state the null hypothesis. You will then compute the t statistic. You be
given formulas, but should know the methods by heart. Please refer to week 15 slide show.
•
To compute the t you will compute the pooled sample variance and the standard error of the
difference between means.
•
You will then compute the degrees of freedom (adjusted sample size) and use the t table to
determine whether the coefficient is sufficiently large to reject the null hypothesis.
– Print and bring to class:
http://www.sagepub.com/fitzgerald/study/materials/appendices/app_f.pdf
– Use the one-tailed test if the direction of the effect is specified, or two-tailed if not
•
You will be asked to express using words what the t-table conveys about the significance (or
non-significance) of the t coefficient
•
Sample question: Are male CJ majors significantly more cynical than female CJ majors? We
randomly sampled five males and five females. Males: 4, 5, 5, 3, 4 Females: 4, 3, 4, 4, 5
– Null hypothesis: No significant difference between cynicism of males and females
– Variance for males (provided): 0.7 Variance for females (provided): 0.5
– Pooled sample variance = .6 SE of the difference between means = .49 t = .41 df = 8
– Check the “t” table. Can you reject the null hypothesis? NO
– Describe conclusion using words: The t must be at least 1.86 (one-tailed test) to reject
the null hypothesis of no significant difference in cynicism, with only five chances in
100 that it is true.