The t-test:

1
The t-test
2
Answers the question is the difference between
the two conditions in my experiment "real" or due
to chance? Two versions (a) Dependent-means
t-test (Matched-pairs" or "one-sample"
t-test). Same subjects do both experimental
conditions e.g., two conditions A and B half
subjects do A then B rest do B then A. (Randomly
allocated to one order or the other).
3
(b) Independent-means t-test (Two-sample"
t-test). Different subjects do each
experimental condition. e.g., two conditions A
and B half subjects do A rest do B. (Randomly
allocated to A or B).
4
Both types of t-test have one independent
variable, with two levels (the two different
conditions of our experiment). There is one
dependent variable (the thing we actually
measure). Effects of alcohol on reaction-time
performance. I.V. is "alcohol consumption". Two
levels - drunk and sober. D.V. is RT. Use a
repeated-measures t-test measure each subject's
RT twice, once while drunk and once while sober.
Effects of personality type on a memory test.
I.V. is "personality type". Two levels -
introversion and extraversion. D.V. is memory
test score. Use an independent-measures t-test
measure each subject's memory score once, then
compare introverts and extraverts.
5
Rationale behind the t-test Experiment on the
effects of alcohol on RT. Measure RT for subjects
when drunk, and when sober. Null hypothesis
alcohol has no effect on RT variation between
the drunk sample mean and the sober sample mean
is due to sampling variation. The drunk and
sober scores are samples from the same population
(sober RTs).
6
If the difference between the sober and drunk
means is large, we might prefer to believe that
alcohol has affected RT the difference is not
due to sampling variation, but arose because the
drunk and sober scores are samples from two
different populations - the population of sober
RTs and the population of drunk RTs. How large
is large? The t-test enables us to decide.
7
Both types of t-test are similar in principle to
the z-score.
Observed difference between sample means
Predicted difference between sample means (that
there will be no difference at all)
measure of the extent to which pairs of sample
means might differ
8
t-distribution becomes progressively more like
the normal distribution as sample size (n)
increases
9
1. We have two sample means, which differ. 2.
Null hypothesis is that the two samples come from
the same population if so, ideally the sample
means should be identical.
RT for drunk sample 800 ms RT for sober sample
300 ms Difference 500 ms
Sober sample RT really reflects sober
population RT say, 600 ms Drunk sample RT also
really reflects sober population RT 600
ms Difference 0 The difference between 500 ms
and 0 ms is due to chance (sampling variation).
10
3. Alternative hypothesis is that our
experimental manipulation has affected our
subjects. The two samples (drunk and sober) are
samples from different populations with different
means. If so, the samples might well have
different means. (e.g. the sober sample mean of
300 ms might reflect a sober population mean of
300 ms the drunk sample mean of 800 ms might
reflect a drunk population mean of 800 ms).
11
A big difference between our two sample means
therefore suggests that either (a) the two
sample means are poor reflections of the mean of
the single population that they are supposed to
represent (i.e., our samples are atypical ones).
OR (b) The two sample means are actually from
two different parent populations, and our initial
assumption that the samples both come from the
same population is wrong. The bigger the
difference between our two sample means, the less
plausible (a) becomes, and the more likely that
(b) is true.
12
Repeated Measures t-test, step-by-step
Does Prozac affect driving ability? Ten
subjects have their driving performance tested
twice on a sheep farm test A after they have
taken Prozac ( experimental condition) test B
while they are drug-free ( control condition).
Each subject thus provides two scores (one for
each condition). Five do A then B, five do B then
A.
13
Number of sheep hit during a 30-minute driving
test Subject Test A Score Test B Score
Difference, D 1 28 25 3 2 26 27 -1 3
33 28 5 4 30 31 -1 5 32 29
3 6 30 30 0 7 31 32 -1 8 18 21 -3
9 22 25 -3 10 24 20 4 27.4 26.8
?D 6
14
the average difference between scores in our two
samples (should be close to zero if there is no
difference between the two conditions)
the predicted average difference between scores
in our two samples (usually zero, since we assume
the two samples dont differ )
estimated standard error of the mean difference
(a measure of how much the mean difference might
vary from one occasion to the next).
15
1. Add up the differences ?D 6 2. Find the
mean difference
3. Estimate of the population standard deviation
(the standard deviation of the differences)
16
4. Estimate of the population standard error
(the standard error of the differences between
two sample means)
17
5. Hypothesised difference between the sample
means. Our null hypothesis is usually that there
is no difference between the two sample means.
(In statistical terms, that they have come from
two identical populations) ? D (hypothesised)
0 6. Work out t
7. "Degrees of freedom" (d.f.) are the number of
subjects minus one d.f. n - 1 10 - 1
9
18
critical values of t (two-tailed)
8. Find the critical value of t from a table (at
the back of many statistics books also on my
website). (a) Two-tailed test if we are
predicting a difference between tests A and B
find the critical value of t for a "two-tailed"
test. With 9 d.f., critical value 2.26. (b)
One-tailed test if we are predicting that A is
bigger than B, or A is smaller than B, find the
critical value of t for a "one-tailed" test. For
9 d.f., critical value 1.83.
19
If obtained t is bigger or equal to the critical
t-value, "reject the null hypothesis" - the
difference between our sample means is probably
too large to have arisen by chance. Here,
obtained t 0.65. This is less than 2.262. There
was no significant difference between performance
on the two tests the observed difference is so
small, it probably arose by chance. Conclusion
Prozac does not significantly affect driving
ability.
20
Results of analysis using SPSS (Analyze
gtCompare Means gt Paired samples t-test)