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value of t to see if you either accepted the null hypothesis, or rejected the null
hypothesis and accepted the research hypothesis.
The two-group t-test requires a different formula because you have two groups,
each with a mean score on some variable. You are trying to determine whether to
accept the null hypothesis that the population means of the two groups are equal
(in other words, there is no difference statistically between these two means), or
whether the difference between the means of the two groups is ‘‘sufﬁciently large’’
that you would accept that there is a signiﬁcant difference in the mean scores of
the two groups.
The numerator of the two-group t-test asks you to ﬁnd the difference of the
means of the two groups:
X 1 X 2
ð 5 : 4 Þ
The next step in the formula for the two-group t-test is to divide the answer you
get when you subtract the two means by the standard error of the difference of the
two means, and this is a different standard error of the mean that you found for the
one-group t-test because there are two means in the two-group t-test.
The standard error of the mean when you have two groups is called the
‘‘standard error of the difference of the means’’ between the means of the two
groups. This formula looks less scary when you break it down into four steps:
1. Square the standard deviation of Group 1, and divide this result by the sample
size for Group 1 (n 1 ).
2. Square the standard deviation of Group 2, and divide this result by the sample
size for Group 2 (n 2 ).
3. Add the results of the above two steps to get a total score.
4. Take the square root of this total score to ﬁnd the standard error of the
difference of the means between the two groups, S X 1 X 2 ¼
q
S 1
n 1 þ S 2
:
n 2
This last step is the one that gives students the most difﬁculty when they are
ﬁnding this standard error using their calculator, because they are in such a hurry
to get to the answer that they forget to carry the square root sign down to the last
step, and thus get a larger number than they should for the standard error.
5.2.1 An Example of Formula #1 for the Two-Group t-Test
Now, let’s use Formula #1 in a situation in which both groups have a sample size
greater than 30.
Suppose that a large university offered several sections of Introductory Biology
101 to undergraduates last semester and that it wanted to compare the results of the
student evaluation form at the end of the course to see if there were gender
differences between males and females. Suppose, further, that Item #12 of the
student evaluation form is the item given in Fig. 5.7 .
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