Test of means

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

10. SPSS Demo

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.

18. Three types of ANOVA
 One-way ANOVA
 Within-subjects ANOVA (Repeated
measures, randomized complete block)
 Factorial ANOVA (Two-way ANOVA)

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
S
um of S
quare Total S
ubtract each of the scoresfrom
the mean of the entire sample.
S
quare each of those deviations.
Add those up for each group,
then add the two groups
together.
S
um of S
quaresAmong Each group mean issubtracted
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.
S
um of S
quaresWithin Here'sa shortcut. Just find the
S
S
T and the S
S
A and find the
difference. What'sleft over isthe
S
S
W.

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
Representsvariation
due to treatment
effect
Sum of SquaresBlock Representsvariation
within an individual
(within block)
Sum of SquaresError Representserror
variation
Sum of SquaresTotal Representstotal
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)

29. (End of Analytic Method I)

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.