Basic Statistics

90 ESTIMATION OF POPULATION MEANS: CONFIDENCE INTERVALS

Table 7.2 Weight Loss (lb) Between Time 1 and Time 2 for Eight Adults

Adult Number Time 1, XI Time 2, XZ Difference d

278

183

175

199

210

167

245

225

27 1

181

175

190

204

164

239

22 1

• n = 8x1 = 210.25 xz = 205.625 d = 4.625
  ~1 = 37.8 S2 = 36.2 sd = 2.92

is commonly used in casekontrol studies where patients are matched by gender and
age. It is often used in surgical experiments where two surgical treatments are used on
the opposite sides of animals such as rats to compare the outcomes. Another common
use of paired comparisons is to contrast data gathered in two time periods.
For example, consider data on eight adults before and after going on a diet given in
Table 7.2. We will let X1 denote the weight before dieting (time 1) and X2 the weight
after dieting for 2 months (time 2). The difference in the before and after weights can
be computed for each adult as d = X1 - X2 and there are eight differences. We will
assume that we have a simple random sample of differences and that the differences
are normally distributed.
We now analyze these differences instead of the 16 original weights. We reduced
the problem from one of two sets of observations on weights to a single set of dif-
ferences. We now treat the d’s as eight observations and first compute the mean and
standard deviation using a statistical package. We obtain

= 2.9246

d=- ‘ = 4.625 and Sd =

n

Then, the 95% confidence interval can be computed using the formula given earlier

for a single mean where the X’s are replaced by d’s. There are 8 differences and

n - 1 = 7 d.f. The 95% interval is

or

2.9246

4.625 h (2.365)-

Js

or

4.625 i 2.445 or 2.18-7.07 lb

The interval contains only positive numbers and we conclude that pd is positive and

that, on average, the diet decreases weight for the adults.