Biostatistics

148 CHAPTER 5 SOME IMPORTANT SAMPLING DISTRIBUTIONS
difference between population means, the probability of obtaining a difference between
sample means as large as or larger than 13 is .0375.
Sampling from Normal Populations The procedure we have just followed
is valid even when the sample sizes, n 1 and n 2 , are different and when the population
variances, s 2 1 and s2 2 have different values. The theoretical results on which this procedure
is based may be summarized as follows.
Given two normally distributed populations with means m 1 and m 2 and variances s 2 1
and s 2 2 , respectively, the sampling distribution of the difference, x 1 x 2 , between the
means of independent samples of size n 1 and n 2 drawn from these populations is
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normally distributed with mean m 1 m 2 and variance s 2 1 =n
1 þ s
2
2 =n 2 .
Sampling from Nonnormal Populations Many times a researcher is
faced with one or the other of the following problems: the necessity of (1) sampling from
nonnormally distributed populations, or (2) sampling from populations whose functional
forms are not known. A solution to these problems is to take large samples, since when the
sample sizes are large the central limit theorem applies and the distribution of the
difference between two sample means is at least approximately normally distributed
with a mean equal to m 1 m 2 and a variance of s 2 1 =n
1
þ s
2
2 =n 2 . To find probabilities
associated with specific values of the statistic, then, our procedure would be the same as
that given when sampling is from normally distributed populations.
EXAMPLE 5.4.2
Suppose it has been established that for a certain type of client the average length of a home
visit by a public health nurse is 45 minutes with a standard deviation of 15 minutes, and that
for a second type of client the average home visit is 30 minutes long with a standard
deviation of 20 minutes. If a nurse randomly visits 35 clients from the first and 40 from the
second population, what is the probability that the average length of home visit will differ
between the two groups by 20 or more minutes?
Solution:
No mention is made of the functional form of the two populations, so let us
assume that this characteristic is unknown, or that the populations are not
normally distributed. Since the sample sizes are large (greater than 30) in
both cases, we draw on the results of the central limit theorem to answer the
question posed. We know that the difference between sample means is at
least approximately normally distributed with the following mean and
variance:
m x1 x 2
¼ m 1 m 2 ¼ 45 30 ¼ 15
sx 2 1 x 2
¼ s2 1
þ s2 2
¼ ð15Þ2
n 1 n 2 35 þ ð20Þ2
40 ¼ 16:4286