Anova and T-Test

The T-Test
- Independent Sample T-Test
- Paired Sample T-Test
- One Sample T-Test
- Test of Significance
The One-Way ANOVA
- Post Hoc Comparisons
- Contrasts
Descriptive Statistics

What is a T-Test
T-Test is a procedure used for comparing Sample
Means to see if there is sufficient evidence to infer that
the means of the corresponding population
distributions also differ. The important things are;
1. Two (t-test always compare two different means)
2. Some variable of interest
Ref: Statistics- A first course by John E Freund and Benjamin M.Perler, 7th
Edition

INDEPENDENT SAMPLE T-TEST
PAIRED SAMPLE T-TEST
ONE SAMPLE T-TEST
INDEPENDENT SAMPLE T-TEST
PAIRED SAMPLE T-TEST
ONE SAMPLE T-TEST
ONE TAIL TESTS
TWO TAIL TESTS
ONE TAIL TESTS
TWO TAIL TESTS
SPSS PROVIDES THREE
DIFFERENT TYPES OF T-TESTS
SPSS PROVIDES THREE
DIFFERENT TYPES OF T-TESTS
SPSS PRODUCES TWO-TAILED
SIGNIFICANCE VALUE BY
DEFAULT
SPSS PRODUCES TWO-TAILED
SIGNIFICANCE VALUE BY
DEFAULT

• The two samples share some
variable of interest in
common, but there is no
overlap between
membership of two groups.
• Compare the running speeds
of horses and zebra would be
an independent design as
there is no sensible way to
pair off each horse with each
zebra.
• Usually based on the group of
individuals who experience
both conditions of the
variable of interest.
• Also called a Repeated
Measure Design or a Paired
Design.
• Compare the running speed
of horses for a week of eating
one type of feed with the
same horses for a week on a
different type of feed would
be a paired design as you can
pair off measurements from
the same horse
Difference Between Independent Sample T-Test
and Paired Sample T-Test
Independent Sample T-Test Paired Sample T-Test

MALEMALE FEMALEFEMALEEXAM SCORE
Independent Sample T-Test
Paired Sample T-Test
RM
Students
RM
Students
RM
Students
RM
Students
QUIZ
”1”
QUIZ
”2”
QUIZ
”1”
QUIZ
”2”
RM
Students
RM
Students

One Sample T-Test
It is designed to test whether the mean of a
distribution differs significantly from some present
value e.g.
Performance
of RM Class
Performance
of RM Class
Professor
goal is 82%
average
Professor
goal is 82%
average
Differ Significantly

Textile
Mills
Textile
Mills
Steel
Mills
Steel
Mills
PRODUCTIVITY
Can You identify a T-Test
MarriedMarried
UnmarriedUnmarried
LIFE
SATISFACTION
SCORE
SAT GRE
SET OF
STUDENTS
SET OF
STUDENTS

A Typical T-Table

Test of Significance
Test of Significance can be one-tailed or two tailed test;
Two tailed test examines whether the mean of one
distribution differs significantly from the mean of other
distribution. (Regardless of the direction +ve or –ve)
The one tailed test measures only whether the second
distribution differs in a particular direction from the
first.
Ref: Statistics- A first course by John E Freund and Benjamin M.Perler, 7th
Edition

If you have stated your experimental hypothesis with care, it will tell
you which type of effect you are looking for.
For example, the hypothesis that "Coffee improves memory" is
_______tailed test.
The hypothesis, "Men weigh a different amount from women"
suggests a _________tailed test.
So remember, don't be vague with your hypothesis if you are looking
for a specific effect! Be careful with the null hypothesis too - avoid "A
does not effect B" if you really mean "A does not improve B".
One Tailed OR Two Tailed
Ref: http://www.socialresearchmethods.net/kb/stat_t.php

How close a sample mean might be to
the mean of the population from which
the sample came.
Ref: Student Edition Statistics (Tutorial and Softare) for Behavioral Sciences by Joseph D.Allen
and David .J. Pittenger, 2nd
Edition

Levene’s Test determines whether
the variability from two groups is
significantly different. If this were
significant, one might consider using
the t-test for un-equal variances
Levene’s Test determines whether
the variability from two groups is
significantly different. If this were
significant, one might consider using
the t-test for un-equal variances
P-ValueP-Value
The degrees of freedom (df) of an
estimate is the number of
independent pieces of information
on which the estimate is based.
Edition
Ref: SPSS for psychologists by Nicola Brace, Richard Kemp and Rosemary Snelgar

So far we have learned the following things about a t-test;
1. The t-test produces a single value, t, which grows larger as the difference
between the means of two samples grows larger;
2. t does not cover a fixed range such as 0 to 1 like probabilities do;
3. You can convert a t-value into a probability, called a p-value;
4. The p-value is always between 0 and 1 and it tells you the probability of the
difference in your data being due to sampling error;
5. The p-value should be lower than a chosen significance level (0.05 for example)
before you can reject your null hypothesis.
This value is called the critical value. The final thing to do is compare this value
with your value of t;
If your t-value is greater than or equal to this value, then t is significant and you
have found a difference
If your t-value is less than this value, then t is not significant.

The ‘Mean Difference’ is the
difference between the two group
means.
The ‘Mean Difference’ is the
difference between the two group
means.
The standard error of the difference
is the function of two groups
individual standard errors;
The standard error of the difference
is the function of two groups
individual standard errors;
Edition

T-Test
Ref: http://www.socialresearchmethods.net/kb/stat_t.php

CASE OF EQUAL VARINACES
ASSUMED
CASE OF EQUAL VARINACES NOT
ASSUMED
n1+n2-2n1+n2-2
Standard Error of the
Difference
Standard Error of the
Difference

Paired Sample T-Test
Edition

One Sample T-Test
The Mean Difference is the difference
between the sample mean and the
user-specified test value.
The Mean Difference is the difference
between the sample mean and the
user-specified test value.
Edition

ONE WAY ANALYSIS OF VARIANCE
(ONE WAY ANOVA)

• Analysis of variance is a procedure used for comparing
sample means to see if there is sufficient evidence to infer
that the means of the corresponding population
distributions also differ.
• Where t-test compare only two distributions, analysis of
variance is able to compare many.
• What does the one-way part mean? It is one dependent
variable (always continuous) and exactly one independent
variable (always categorical). A single independent variable
can have many levels.
ANOVA
Ref: Business Research Methods by Alan Bryman $ Emma Bell

ANOVA
MALEMALE FEMALEFEMALEEXAM SCORE
Ethnic
Group
Ethnic
Group EXAM SCORE
- Asian
- Black
- White
- Hispanic
- Asian
- Black
- White
- Hispanic
One-way ANOVA will generate a significance value indicating whether there are significant
differences within the comparisons being made. This significance value does not indicate
where the difference is or what the differences are; but a ‘Test’ can identify which groups
differ significantly from each other.
One-way ANOVA will generate a significance value indicating whether there are significant
differences within the comparisons being made. This significance value does not indicate
where the difference is or what the differences are; but a ‘Test’ can identify which groups
differ significantly from each other.

One Way ANOVAOne Way ANOVA
Ref: SPSS Software

The sum of squares is the estimate of the variance
multiplied by the degrees of freedom. It can be
calculated in three ways;
-SST -SSW -SSB
The advantage of sum of squares is that it can deal
with unequal subgroup sizes and they add up like;
SST = SSB +SSW
The sum of squares is the estimate of the variance
multiplied by the degrees of freedom. It can be
calculated in three ways;
-SST -SSW -SSB
The advantage of sum of squares is that it can deal
with unequal subgroup sizes and they add up like;
SST = SSB +SSW
Edition

When “F” ratio is close to “1”, the estimates will be said to be similar (no indication
of detectable differences between subgroup means). When F-ratio is large, the
estimates are dissimilar (indication of differences between subgroup means).
SPSS calculated that an F with 3 and 96 df that equals 2.272 has a two tailed
probability of 0.085
When “F” ratio is close to “1”, the estimates will be said to be similar (no indication
of detectable differences between subgroup means). When F-ratio is large, the
estimates are dissimilar (indication of differences between subgroup means).
SPSS calculated that an F with 3 and 96 df that equals 2.272 has a two tailed
probability of 0.085
Edition

One Way ANOVA – Post Hoc TestsOne Way ANOVA – Post Hoc Tests
Ref: SPSS Software

Anova and T-Test

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