The twosample ttest

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The two-sample t-test

• Expanding t to two groups

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t-tests used for population mean diffs

• With 1-sample t, we have a single sample and a
  population value in mind
• With 2-sample t, we have two groups.
• Experimental vs. Control
• Drug vs. placebo
• Psychotherapy, wait list
• New training, status quo
• Brand Preference
• Pepsi, Coke
• Coors, Red Stripe
• Male, Female
• Husband, wife Brother, sister Sorority,
  fraternity
• Use samples to decide about populations
  treatment effectiveness, differences between
  brands, groups

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Varieties of t

• 1-Sample (only one kind)
• 2-Sample (two kinds)
• Independent samples groups are unrelated
• Experimental vs. control groups (at random)
• Male vs. female participants (unrelated)
• Dependent samples groups are related
• Same person in both groups
• Left hand vs. right hand
• Pain-free vs. placebo
• Groups are related somehow
• Wife, husband
• Cancer patient, friend

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Common Structure of t test
Many statistical tests have this
form Statistic/(Standard error of
Statistic). All t-tests have this form.
Example, 1-sample t
It may not look like it, but it is basically the
difference in means divided by the standard error
of the difference in means.
The t-test and many other stats can be thought of
as a fancy z score.
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Independent Samples t
The formula for the two-sample t-test for
independent samples looks like this
This says we find the value of t by taking the
difference in the two sample means and dividing
by the standard error of the difference in means.

The top part is just the means for the two
groups. The bottom part is a function of the
group standard deviations. Let us take a look.
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Example of the Standard Error of the Difference
in Means
Suppose that at USF the mean height is 68 inches
and the standard deviation of height is 6 inches.
Suppose we sampled people 100 at a time into two
groups. We would expect that the average mean
difference would be zero. What would the
standard deviation of the distribution of
differences be?
The standard error for each group mean is .6, for
the difference in means, it is .85.
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Estimating the Standard Error of Mean Differences
The USF scenario we just worked was based on
population information. That is
We generally dont have population values. We
usually estimate population values with sample
data, thus
All this says is that we replace the population
variance of error with the appropriate sample
estimators.
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Pooled Standard Error
We can use this formula when the sample sizes for
the two groups are equal.
When the sample sizes are not equal across
groups, we find the pooled standard error. The
pooled standard error is a weighted average,
where the weights are the groups degrees of
freedom.
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Example of the two-sample t, Empathy by College
Major
Suppose we have a professionally developed test
of empathy. The test has people view film clips
and guess what people in the clips are feeling.
Scores come from comparing what people guess to
what the people in the films said they felt at
the time. We want to know whether Psychology
majors have higher scores on average to this test
than do Physics majors. No direction, we just
want to know if there is a difference. So we
find some (N15) of each major and give each the
test. Results look like this
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Empathy Scores
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Empathy
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Six Steps
1. Set alpha. Set at .05.

• State hypotheses

Null hypothesis
Alternative (substantive) hypothesis
3. Calculate the statistic.
4. Determine the critical value. We are looking
for a value of t in the table. N 30, df
30-2 28, alpha .05, 2 tails. The critical
value is 2.05.
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Six Steps
5. State the decision rule. If the absolute
value of the sample t is greater than or equal to
critical t, then reject H0. If not, then fail to
reject H0. In this case .62 lt 2.05, so we
cannot reject H0.
6. State the conclusion. In our sample, t is
not significant. Based on the results of this
study, there is no evidence that Psychology
majors and Physics majors differ on a test of
empathy.