Basic Statistics

86 ESTIMATION OF POPULATION MEANS: CONFIDENCE INTERVALS

To find an estimate of the sample size needed for an interval of length L, we must
assume a value of 0 and proceed as if 0 were known. If 0 is unknown, an approximate
estimate can be obtained from the range of likely values for the variable in question
using the method given in Section 5.2.2. The difficulty with the use of approximate
estimates is that the size of n varies with the square of 0, so modest inaccuracies in
o result in larger inaccuracies in n. Sometimes all the researcher can reasonably do
is try several likely estimates of 0 to gain some insight on the needed sample size.

7.5 ESTIMATING THE DIFFERENCE BETWEEN TWO MEANS:

UNPAIRED DATA

In the example used in Section 7.1.1, we estimate the mean gain in weight of infants

given a supplemented diet for a 1-month period. Very often, however, our purpose is

not merely to estimate the gain in weight under the new diet, but also to find out what

difference the new diet supplement makes in weight gained. Perhaps it makes little

or no difference, or possibly the difference is really important.

To compare gains in weight under the new supplemented diet with gains in weight

under the standard, unsupplemented diet, we must plan the experiment differently.

Instead of just giving the entire group of infants the new supplemented diet, we

now randomly assign the infants into two groups and give one group (often called the

treatment or experimental group) the new supplemented diet and the other group (often

called the control group) the standard diet. These two groups are called independent

groups; here the first group has 16 infants in it and the second group has 9 (see

Table 7.1). Whenever possible it is recommended that a control group be included in

an experiment.

We compare separate populations: the population of gains in weight of infants

who might be given the supplemented diet, and the population of gains in weight of

infants who might be given the standard diet. A gain in weight in the first population

is denoted by XI, the mean of the first population is called 1-11, and the mean of the

first sample is 71. Similarly, a member of the second population is called X2, the

mean of the second population is called 1-12, and the mean of the second sample is

X2. The purpose of the experiment is to estimate p1 - p2, the difference between

the two population means. We calculate 7, using the first group of 16 infants and

XZ from the second group of 9 infants. If 7, = 311.9 g and x2 = 206.4 g, the

estimate of p1 - p2 is f?l - f?2 = 105.5 g (see Table 7.1). This is the point estimate

for p1 - p2 and indicates that the new diet may be superior to the old.

7.5.1

After calculating x1 - 72 = 105.5 g, we calculate a 95% confidence interval. for
p1- 1-12, the difference between the two population means. To calculate the confidence
interval, the distribution of - X2 is needed. This presents no problem since
XI - x2 is simply a statistic that can be computed from the experiment. If the
experiment were repeated over and over, the value of - 312 would vary from one
experiment to another, and thus it has a sampling distribution.

The Distribution of XI - X2