Statistical Methods in Medical Research 4ed

96 Analysing means and proportions
Frequency
300
200
100
0
0 . 5 1 . 3 2 . 1 2 . 9 3 . 7 4 . 5 5 . 3 6 . 1 6 . 9 7 . 7 8 . 5
Mean, x
Fig. 4.4 The distribution of means from 2000 samples of five random digits (Table 4.1), with the
approximating normal distribution.
as the sample size approaches the population size. Clearly, when n ˆ N, f ˆ 1
and SE…x† ˆ0, there is only one possible random sample, consisting of all the
members of the population, and for this sample x ˆ m.
The sampling error of the sample median has no simple general expression. In
random samples from a normal distribution, however, the standard error of the
median for large n is approximately 1 253s= n
p . The fact that this exceeds s= n
p
shows that the median is more variable than the sample mean (or, technically, it
is less efficient as an estimator of m). This comparison depends on the assumption
of normality for the distribution of x, however, and for certain other
distributional forms the median provides the more efficient estimator.
Inferences from the sample mean
We consider first the situation in which the population standard deviation, s, is
known; later we consider what to do when s is unknown.
Known s
Let us consider in some detail the problem of testing the null hypothesis (which
we shall denote by H0) that the parameters of a normal distribution are m ˆ m0 and s ˆ s0, using the mean, x, of a random sample of size n.
If H0 is true, we know that the probability is only 0 05 that x falls outside
the interval m0 1 96s0= n
p to m0 ‡ 1 96s0= n
p . For a value of x outside this
range, the standardized normal deviate
z ˆ x m0 p …4:3†
s0= n