Basic Statistics

88 ESTIMATION OF POPULATION MEANS: CONFIDENCE INTERVALS

the statistic XI - XZ has a normal distribution with mean p1 - pz and variance
o2(l/n1 + l/nz). In Section 7.1.2, when we had normally distributed with mean
p and with variance a2/n, and g2 was known, the 95% confidence interval for p was
X & 1.960/&. Analogous to this, if o2 is known, a 95% confidence interval for

PI - PZ is

(Xi - Xz) f 1.960

For XI = 3 1 1.9 g and XZ = 206.4 g, if it is known that o = 120 g, the interval is

(311.9 - 206.4) i 1.96(120)dm

105.5 i 1.96(120)a

105.5 i 1.96(120)(.417)

105.5 & 98.1

or

or

or

or

7.4-203.6 g

Both confidence limits are positive, so we conclude that the supplemented diet does
increase weight gains on the average in the population (i.e., we conclude that p1 - pz
is positive and p1 > pz). It is also obvious from the length of the confidence interval
(203.6 - 7.4 = 196.2 g) that the size of the difference covers a wide range.
Here, we are assuming that we have simple random samples from two indepen-
dent populations that both have a variance o2 and that the computed z is normally
distributed. To estimate the sample size needed for a confidence interval of length L
when n1 = n2 or simply n, we replace o with d'% in the formula for sample size
for a single group.

7.5.3

Usually, the population variance o2 is unknown and must be approximated by some

estimate that can be calculated from the samples. An estimate s: can be calculated

from the first sample using s: = C(X1 - F~)~/(nl - l), and similarly, an estimate

s; can be computed from the second sample. These two estimates are then pooled to

obtain the pooled estimate of the variance, s:. The pooled estimate of the variance is

a weighted mean of s: and sz and its formula is

Confidence Intervals for p1 - pa: Unknown Variance

s2 = (n1 - 1,s: + (nz - 1)s;

P 121 + 722 - 2

When

l/nz), the quantity

• XZ are normally distributed with mean p1 - p2 and variance o'( l/nl+

(Xl - X2) - (Pl - pz)

aJl/n1+ l/nz

z=