1. t(ea) for Two:
Test between the Means of Different Groups
When you want to know if there is a ‘difference’
between the two groups in the mean
Use “t-test”.
Why can’t we just use the “difference” in score?
Because we have to take the ‘variability’ into
account.
T = difference between group means
sampling variability
2. One-Sample T Test
Evaluates whether the mean on a test
variable is significantly different from a
constant (test value).
Test value typically represents a neutral
point. (e.g. midpoint on the test variable,
the average value of the test variable
based on past research)
3. Example of One-sample T-test
Is the starting salary of company A
($17,016.09) the same as the average
of the starting salary of the national
average ($20,000)?
Null Hypothesis:
Starting salary of company A = National average
Alternative Hypothesis:
Starting salary of company A = National average
4. SPSS demo (“employee data”)
Review:
Standard deviation: Measure of dispersion or
spread of scores in a distribution of scores.
Standard error of the mean: Standard deviation
of sampling distribution. How much the mean
would be expected to vary if the differences
were due only to error variance.
Significance test: Statistical test to determine
how likely it is that the observed
characteristics of the samples have occurred
by chance alone in the population from which
the samples were selected.
5. z and t
Z score : standardized scores
Z distribution : normal curve with mean value
z=0
95% of the people in the given sample (or
population) have
z-scores between –1.96 and 1.96.
T distribution is adjustment of z distribution for
sample size (smaller sampling distribution
has flatter shape with small samples).
T = difference between group means
sampling variability
6. Confidence Interval
A range of values of a sample statistic that
is likely (at a given level of probability, i.e.
confidence level) to contain a population
parameter.
The interval that will include that
population parameter a certain percentage
(= confidence level) of the time.
7. Confidence Interval for difference
and Hypothesis Test
When the value 0 is not included in the
interval, that means 0 (no difference) is not a
plausible population value.
It appears unlikely that the true difference
between Company A’s salary average and
the national salary average is 0.
Therefore, Company A’s salary average is
significantly different from the national salary
average.
8. Independent-Sample T test
Evaluates the difference between the
means of two independent groups.
Also called “Between Groups T test”
Ho: 1= 2
H1: 1= 2
9. Paired-Sample T test
Evaluates whether the mean of the difference
between the paired variables is significantly
different than zero.
Applicable to 1) repeated measures and 2)
matched subjects.
Also called “Within Subject T test” “Repeated
Measures T test”.
Ho: d= 0
H1: d= 0
11. Analysis of Variance (ANOVA)
An inferential statistical procedure used
to test the null hypothesis that the
means of two or more populations are
equal to each other.
The test statistic for ANOVA is the F-
test (named for R. A. Fisher, the creator
of the statistic).
12. T test vs. ANOVA
T-test
Compare two groups
Test the null hypothesis that two populations
has the same average.
ANOVA:
Compare more than two groups
Test the null hypothesis that two populations
among several numbers of populations has the
same average.
13. ANOVA example
Example: Curricula A, B, C.
You want to know what the average score on the
test of computer operations would have been
if the entire population of the 4th
graders in the school
system had been taught using Curriculum A;
What the population average would have been had
they been taught using Curriculum B;
What the population average would have been had they
been taught using Curriculum C.
Null Hypothesis: The population averages would have
been identical regardless of the curriculum used.
Alternative Hypothesis: The population averages differ for
at least one pair of the population.
14. ANOVA: F-ratio
The variation in the averages of these samples, from one sample to
the next, will be compared to the variation among individual
observations within each of the samples.
Statistic termed an F-ratio will be computed. It will summarize the
variation among sample averages, compared to the variation among
individual observations within samples.
This F-statistic will be compared to tabulated critical values that
correspond to selected alpha levels.
If the computed value of the F-statistic is larger than the critical
value, the null hypothesis of equal population averages will be
rejected in favor of the alternative that the population averages differ.
15. Interpreting Significance
p<.05
The probability of observing an F-statistic
at least this large, given that the null
hypothesis was true, is less than .05.
16. Logic of ANOVA
If 2 or more populations have identical averages,
the averages of random samples selected from
those populations ought to be fairly similar as well.
Sample statistics vary from one sample to the next,
however, large differences among the sample
averages would cause us to question the hypothesis
that the samples were selected from populations
with identical averages.
17. Logic of ANOVA cont.
How much should the sample averages differ before we
conclude that the null hypothesis of equal population
averages should be rejected.
In ANOVA, the answer to this question is obtained by
comparing the variation among the sample averages to the
variation among observations within each of the samples.
Only if variation among sample averages is substantially
larger than the variation within the samples, do we
conclude that the populations must have had different
averages.
19. Sources of Variation
Three sources of variation:
1) Total, 2) Between groups, 3) Within groups
Sum of Squares (SS): Reflects variation. Depend on sample size.
Degrees of freedom (df): Number of population averages being
compared.
Mean Square (MS): SS adjusted by df. MS can be compared
with each other. (SS/df)
F statistic: used to determine whether the population averages
are significantly different. If the computed F static is larger than
the critical value that corresponds to a selected alpha level, the
null hypothesis is rejected.
20. Computing F-ratio
SS Total: Total variation in the data
df total: Total sample size (N) -1
MS total: SS total/ df total
SS between: Variation among the groups compared.
df between: Number of groups -1
MS between : SS between/df between
SS within: Variation among the scores who are in the same
group.
df within: Total sample size - number of groups -1
MS within: SS within/df within
F ratio = MS between / MS within
21. Formula for One-way ANOVA
Formula Name How To
Sum of Square Total Subtract each of the scores from
the mean of the entire sample.
Square each of those deviations.
Add those up for each group,
then add the two groups
together.
Sum of Squares Among Each group mean is subtracted
from the overall sample mean,
squared, multiplied by how
many are in that group, then
those are summed up. For two
groups, we just sum together
two numbers.
Sum of Squares Within Here's a shortcut. Just find the
SST and the SSA and find the
difference. What's left over is the
SSW.
22. Alpha inflation
Conducting multiple ANOVAs, will incur a large risk
that at least one of them would be statistically
significant just by chance.
The risk of committee Type I error is very large for the
entire set of ANOVAs.
Example: 2 tests .05 Alpha
Probability of not having Type I error .95
.95x.95 = .9025
Probability of at least one Type I error is
1-9025= .0975. Close to 10 %.
Use more stringent criteria. e.g. .001
23. Relation between t-test and F-test
When two groups are compared both t-test
and F-test will lead to the same answer.
t2
= F.
So by squaring t you’ll get F
(or square root of t is F)
24. Follow-up test
Conducted to see specifically which means are
different from which other means.
Instead of repeating t-test for each combination
(which can lead to an alpha inflation) there are some
modified versions of t-test that adjusts for the alpha
inflation.
Most recommended: Tukey HSD test
Other popular tests: Bonferroni test , Scheffe test
25. Within-Subject (Repeated
Measures) ANOVA
SS tr : Sum of Squares Treatment
SS block : Sum of Squares Block
SS error = SS total - SS block - SS tr
MS tr = SS tr/k-1
MSE = SS error/(n-1)(k-1)
F = MS tr/MSE
26. Within-Subject (Repeated
Measures) ANOVA
Examine differences on a dependent
variable that has been measured at more
than two time points for one or more
independent categorical variables.
27. Within-Subject (Repeated
Measures) ANOVA
Formula Name Description
Sum of Squares
Treatment
Represents variation
due to treatment
effect
Sum of Squares Block Represents variation
within an individual
(within block)
Sum of Squares Error Represents error
variation
Sum of Squares Total Represents total
variation
28. Factorial ANOVA
T-test and One way ANOVA
1 independent variable (e.g. Gender), 1
dependent variable (e.g. Test score)
Two-way ANOVA (Factorial ANOVA)
2 (or more) independent variables (e.g.
Gender and Academic Standing), 1
dependent variable (e.g. Test score)
30. Main Effects and
Interaction Effects
Main Effects
The effects for each independent variable on the dependent
variable.
Differences between the group means for each independent
variable on the dependent variable.
Interaction Effect
When the relationship between the dependent variable and one
independent variable differs according to the level of a second
independent variable.
When the effect of one independent variable on the dependent
variable differs at various levels of second independent variable.
31. T-distribution
A family of theoretical probability distributions used in
hypothesis testing.
As with normal distributions (or z-distributions), t distributions
are unimodal, symmetrical and bell shaped.
Important for interpreting data gather on small samples when the
population variance is unknown.
The larger the sample, the more closely the t approximates the
normal distribution. For sample greater than 120, they are
practically equivalent.
