Database Reference
In-Depth Information
Consider the hypothetical case where we are comparing the weights of people
before and after an “intervention.” The “intervention” has the result that the “before”
state is the person's normal weight in pounds and the “after” state is his/her weight
a moment later, after picking up a handful of 1-pound weights and putting them in
his/her pocket. The “intervention” in this example is a bit silly, but its purpose is to
help you understand the issue. We have paired data and, say, n = 6 (i.e., 6 people in
the study).
Let's assume that the “handful” is between ive and seven 1-pound weights,
depending on how big one's hand is (we told you it would sound a bit silly, but trust
us—the point of the example will be made clear!). Suppose that the data values are
as follows:
Before
After
d = Weight Gain
135
140
5
155
161
6
203
210
7
178
184
6
182
188
6
143
148
5
Notice that some “handfuls” were ive 1-pound weights, while some were six
1-pound weights, and one person, apparently with larger hands, had a handful con-
sisting of seven 1-pound weights. It is clear from the description of the problem that
the true weight gain is not 0, and averages about 6.
So, when we test using the paired-data test, we write
H0:μ1=μ2
H1:μ1≠μ2
which, as we noted earlier, can be “converted” to
H0:D=0
H1:D≠0
Looking above at the “ d ” (weight gain) column, we ind that the mean of
d = 5.833, and there is overwhelming evidence that the true average difference (i.e.,
D) is near 6 and H0 should be, and will be, rejected. Indeed, the paired t-test output
for this example is shown in Figure 3.8 .
The result we get is what is expected. The two-sided p -value is extremely small,
being 7.5 E−06 (“E” stands for “exponent of 10,” meaning 7.5 × 10 −6 , or a p -value of
0.0000075), indicating beyond a reasonable doubt that the “before” and “after” true
means differ. Again, there's absolutely no surprise here; the sample means differ by
5.833, which is logical, given each person picks up between 5 and 7 pounds of extra
weight. There is very little variability in the “ d ” column, and it's virtually impossible
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