Week 6
Mean testing: T.Test
READING
Text reading
Dancey and Reidy, Chapter 6 in the 3rd edition, but Chapt 7 in 4th.
Let us start with some warnings: We are looking at different procedures each week. Virtually
one each week. You will get overwhelmed. This is a simple function of information overload,
rather than your personal fault. The important thing is to keep thinking “what is this
procedure for?”, or “when would this get used?”. It is NOT important to keep memorising the
information from the textbook. Assume that in the future you will need to use some of these
statistics. When this happens, you will go back to these books. Most of us learn from what
are called ‘worked examples’. This means we have a good look at how other people have
used a statistic. If you get really lucky, we can find something that is very similar to what we
plan to do.
So what does a course like this one really achieve? It provides an essential overview of the
major types of statistics that are commonly used across different disciplines. We feel a bit
self-conscious in that the text actually says “for Psychology” on it. But in fact all data driven
disciplines use the very same body of statistical procedures. There is no choice here.
Certainly, there will be some methods used more in certain areas. But there is a common core
of statistical procedures as shared by all empirical researchers. This is necessary since the
statistics we need are driven by the very problems we must solve, and not by the label of the
area we are apparently working within. By understanding this common core of statistics, you
become able to use more specialist statistics, ones which are perhaps more attuned to specific
problems. But to walk before running is still good advice. To repeat, it is wonderful to
analyse WHY certain procedures, such as t-tests (this chapter) are being used, what they do,
and where they might be found, etc. However, it is not important to be memorizing the
textbook information which will still be there when you need it.
And a final counsel before moving on: SPSS printouts, in common with all such products, are
not easy to read. Many procedures will generate paper garbage. Often there is no quick way
to stop this. Sometimes, the options allow us to shape the printouts, either leaving out or
including certain bits. But do not get alarmed when these programmes appear to generate
tables of relatively meaningless information. At times they dump out some of the
intermediate statistical data, which becomes highly meaningful to statisticians. Very often
such data is there to help us ‘double check’ important issues. Please keep in mind that SPSS
has developed over 40 year’s time to serve the needs of remarkably diverse groups of people.
CHAPTER OVERVIEW
In this chapter we will be looking at the differences between participants and within
participants. For example, you could compare scores on math achievement between males
and females. This is a between-participants, independent, or unrelated design, since two
different groups (i.e., males and females) are measured on one variable (i.e., math
achievement). On the other hand, one group of participants may be compared over two
variables (i.e., before and after treatment). This is a within-participants, repeated measures, or
related design, because the same people are measured on both variables (i.e., math and
science). T. test statistic is used to compare means between participants or within
participants. In this chapter we are going to show you how to analyse the data from such
designs.
To enable you to understand the tests presented in this chapter, you will need to have an
understanding of the following concepts:
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the mean, standard deviation, and standard error (Chapter 2)
z-scores and the normal distribution (Chapter 3)
assumptions underlying the use of parametric tests (Chapter 4)
probability distributions like the t-distribution (Chapter 4)
one- and two-tailed hypotheses (Chapter 4)
statistical significance (Chapter 4)
confidence intervals (Chapter 3).
KEY TERMS
1- Independent sample t-tests
2- Paired sample t-tests
3- Cohen effect size (d)
4- Leven tests of homogeneity of variance
KEY POINTS
1- Independent sample t-test is used to compare the means of two independently sampled
groups (between-participants design) over one variable/condition. For example, do
students who attend school regularly differ in their academic achievement from
students who attend school irregularly?
2- Dependent sample t-test is used to compare the means of one sample group over two
variables/conditions. For example, does students’ mathematics achievement differs
before and after a math anxiety remediation program?
Further readings
T-test
http://www.socialresearchmethods.net/kb/stat_t.htm
Independent T-test
http://www.ruf.rice.edu/~bioslabs/tools/stats/ttest.html
Dependent T-test
class.unl.edu/psycrs/handcomp/hcwgt.PDF
http://www-
3- Cohen’s effect size shows the magnitude of difference between two independent
means, expressed in standard deviations. Cohen’s d can be calculated from this
x1  x 2
formula d 
meanSD
Where x1 is the mean for the first group and x2 is the mean for the second group, and
meanSD is the mean standard deviation of the two groups.
Further readings
What is effect size?
http://davidmlane.com/hyperstat/effect_size.html
What is an 'Effect Size'? A Downloadable PDF guide for users
http://davidmlane.com/hyperstat/effect_size.html
Interactive tool to calculate effect size http://web.uccs.edu/lbecker/Psy590/escalc3.htm
Effect Size Calculator (Excel sheet downloadable version)
http://davidmlane.com/hyperstat/effect_size.html
4- Levene’s test of homogeneity of variance tests the assumption that the variance in
each group or category of the independent variable is the same. Therefore if Levene’s
test is significant at p < .05 then we can conclude that the null hypothesis is incorrect
and that the variances are significantly different, therefore, the assumption of
homogeneity of variance has been violated. If, however, Levene’s test is not
significant at p > .05, then we accept the null hypothesis that the assumption of
homogeneity of variance is tenable.
SPSS activities
1- Independent T-test
http://distdell4.ad.stat.tamu.edu/spss_1/TwoSampleTTest.html
2- Paired sample T-test http://distdell4.ad.stat.tamu.edu/spss_1/PairedTtest.html
ACTIVE LEARNING AND OPPORTUNITIES
Multiple Choice Questions
1- The DF for independent samples t-test analysis with 20 participants in each group is:
ABCD-
38
20
40
68
2- The most important assumption to meet when using a t-test is
ABCD-
The variation in scores should be minimal
Scores should be drawn from a normally distributed distribution
Conditions should have equal means
All of the above
3- In an analysis using independent t-test, you find the following result
Levene’s test for equality of variance: F = .15, p = .58
This result show the variances of the two groups are:
A- Dissimilar
B- Similar
C- Exactly the same
D- Indeterminate
Questions 4 to 8 relate to the tables of results
Group Statistics
math
achieve
Gender
0
1
N
254
192
Mean
19.30
16.68
Std. Deviation
4.123
5.443
Std. Error
Mean
.259
.393
Independent Samples Test
Levene's Tes t for
Equality of Variances
F
Math
achieve
Equal variances
ass umed
Equal variances
not as sumed
28.413
Sig.
.000
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
5.769
444
.000
2.61
.453
1.723
3.503
5.555
343.770
.000
2.61
.470
1.688
3.538
4- Are the variances of math achievement homogenous in males and females?
5- Are there any significant differences between males and females in math
Explain?
achievement?
6- Who achieved higher in math; males or females?
7- What does the confidence interval of the mean difference tell?
8- Calculate the effect size for the math achievement mean difference between males and
females.