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Psyc 291: Experimental Statistics
Chapter 8: Two-Sample t-Tests
Paired Sample t-Tests
1. Paired t-tests are used to ask the question, is the mean of this interval/ratio) measure different
than the mean of another measure taken by the same or a related sample?

Question always compares two means when scores on one measure are linked to scores
on the other measure in some way; often (but not always) within-subjects.
o Does score improve or worsen for the same subjects from time one to time two?
o Does the same group score higher on this measure or that measure?
o Do husbands score higher than wives?

The variable is parametric (interval/ratio).

Both variables are continuous or have multiple possible responses

Hypothesis should include direction of relationship.
o The score on this measure is above the score on this other measure.
o The score on this measure is below the score on this other measure.

Null hypothesis: The sample means are the same for both measures; no difference.
2. Before we hypothesis test with paired samples, you need a different kind of mean.

We use the mean of the differences of the scores rather than of the scores themselves.

We also need the standard deviation of these differences:
Psych GPA
Overall GPA
Difference
Diff.
Deviation
Diff Dev Sq.
Subject 1
3.0
3.4
-.4
-.5
.25
Subject 2
2.7
2.1
.6
.5
.25
Mean of
Differences:
.1
Sum of
Squares:
.5

Variance of difference will be sum of squares divided by n-1; SD of difference will be the
square root of variance, as usual. The S.E.M is the SD divided by square root of n.

Ultimately it’s just like a regular mean and SD, except it’s of the difference between two
scores, not of one set of scores.
3. The t-statistic is set up the same way as a one-sample test, except using the difference score as
the “observed mean” and 0 as the “predicted mean”.
Psyc 291: Experimental Statistics
Chapter 8: Two-Sample t-Tests

Hypothesis: The difference will be positive/negative, different than 0.

Null Hypothesis: The mean of difference will be 0; there paired scores are identical.

The t-statistic is your observed difference minus your predicted difference, divided by the
standard error of the difference.
o Since predicted difference is always 0, it’s just observed differenced divided by
standard error.
4. To test your hypothesis….

Get the critical t value from a look-up table, using df (n-1) and appropriate level of
significance.
o n-1, because each member of the pair is only counted once (and we’re using one
mean, the mean of differences).

Compare the calculated value from #2 above to the critical value; if the calculated value
is greater, you reject the null hypothesis.
5. To write the results of a t-test, use real-world terms, provide the sample mean and standard
deviation for each group, then the stats (t(df) = calculated value, p < or > alpha).

Psychology students had a higher Psych GPA (M = 3.7 GPA units, SD = 1.1) than overall
GPA (M = 3.1 GPA units, SD = 0.7) (t(39) = 6.8, p < .05).

High schoolers’ consumption of caffeine was not lower after the intervention (M = 54
oz., SD = 14.1) than before the intervention (M = 57 oz, SD = 13.8) (t(39) = 6.8, p > .15).

With very small t values, you can omit the rest of the information, because they will
never be statistically significant: any time t < 1, you can just write t < 1 for the inferential
statistics.
Psyc 291: Experimental Statistics
Chapter 8: Two-Sample t-Tests
Independent Sample t-Tests
1. Independent t-tests are used to ask the question, is the mean of this (interval/ratio measure)
sample different than the mean of another, unrelated sample?

Question always compares two means when scores are independent; always betweensubjects design with different participants in each group.
o Do juniors have a lower GPA than sophomores?
o Do smokers have higher health costs than non-smokers?

The variable is parametric (interval/ratio).

Both variables are continuous or have multiple possible responses

Hypothesis should include direction of relationship.
o This group scores higher than this group.
o The group scores lower than this group.

Null hypothesis: The means are the same for both groups; no difference.
2. Before we hypothesis test with paired samples, you need a different kind of variance and
standard error.

We are still starting with the difference between the two group means.

When considering variability, we need a measure of pooled variance – we need to take
into account the variability of each group, weighted for the number in each group.
o To get pooled variance, add the two sums of squares together, then divide by the
combined degrees of freedom.
o pooled variance = (SS1 + SS2) / (df1 + df 2)

From the pooled variance, we can calculate the standard error of the difference of means.
o To get S.E. DM, divide the pooled variance by the first n. Then start with pooled
variance again and divide by the second n. Add those together, and take the
square root.
o S.E. DM = √(pooled var / n1 + pooled var / n2).

To calculate the t-statistic, divide the difference of means by the S. E. DM.
Psyc 291: Experimental Statistics
Chapter 8: Two-Sample t-Tests
4. To test your hypothesis….

Get the critical t value from a look-up table, using df (n-1) and appropriate level of
significance.
o n-1, because each member of the pair is only counted once (and we’re using one
mean, the mean of differences).

Compare the calculated value from #2 above to the critical value; if the calculated value
is greater, you reject the null hypothesis.
5. To write the results of a t-test, use real-world terms, provide the sample mean and standard
deviation for each group, then the stats (t(df) = calculated value, p < or > alpha).

Juniors had a higher GPA (M = 3.7 GPA units, SD = 1.1) than seniors did (M = 3.1 GPA
units, SD = 0.7) (t(39) = 6.8, p < .05).

Business majors consumed the same amount of caffeine caffeine (M = 54 oz., SD = 14.1)
than philosophy majors (M = 57 oz, SD = 13.8) (t(39) = 6.8, p > .15).

With very small t values, you can omit the rest of the information, because they will
never be statistically significant: any time t < 1, you can just write t < 1 for the inferential
statistics.