Statistical Methods in Medical Research 4ed

100 Analysing means and proportions
x 2 58s= n
p :
In general, the 1 a confidence limits are
x zas= n
p ,
where za is the standardized normal deviate exceeded (in either direction) with
probability a. (The notation here is not universally standard: in some usages the
subscript a refers to either the one-tailed probability, which we write as 1
2 a,or
1
the distribution function, 1 2 a.)
Unknown s: the t distribution
Suppose now that we wish to test a null hypothesis which specifies the mean
value of a normal distribution (m ˆ m0) but does not specify the variance s2 , and
that we have no evidence about s2 besides that contained in our sample. The
procedure outlined above cannot be followed because the standard error of
the mean, s= n
p , cannot be calculated. It seems reasonable to replace s by the
estimated standard deviation in the sample, s, giving a standardized deviate
t ˆ x m0 p …4:6†
s= n
instead of the normal deviate z given by (4.3). The statistic t would be expected to
follow a sampling distribution close to that of z (i.e. close to a standard normal
distribution with mean 0 and variance 1) when n is large, because then s will be a
good approximation to s. When n is small, s may differ considerably from s,
purely by chance, and this will cause t to have substantially greater random
variability than z.
In fact, t follows what is known as the t distribution on n 1 degrees of
freedom (DF). The t distributions form a family, distinguished by an index, the
`degrees of freedom', which in the present application is one less than the sample
size. As the degrees of freedom increase, the t distribution tends towards the
standard normal distribution (Fig. 4.7). AppendixTable A3 shows the percentiles
of t, i.e. the values exceeded with specified probabilities, for different values
of the degrees of freedom, n. For n ˆ1, the tabulated values agree with those of
the standard normal distribution. The 5% point, which always exceeds the
normal value of 1 960, is nevertheless close to 2 0 for all except quite small values
of n. The t distribution was derived by W.S. Gosset (1876±1937) and published
under the pseudonym of `Student' in 1908; the distribution is frequently referred
to as Student's t distribution.
The t distribution is strictly valid only if the distribution of x is normal.
Nevertheless, it is reasonably robust in the sense that it is approximately valid for
quite marked departures from normality.