Statistical Methods in Medical Research 4ed

would be less than 1 96 or greater than 1 96. Such a value of x could be
regarded as sufficiently far from m 0 to cast doubt on the null hypothesis.
Certainly, H0 might be true, but if so an unusually large deviation would have
arisenÐone of a class that would arise by chance only once in 20 times. On the
other hand such a value of x would be quite likely to occur if m had some value
other than m 0, closer, in fact, to the observed x. The particular critical values
adopted here for z, 1 96, correspond to the quite arbitrary probability level of
0 05. If z is numerically greater than 1 96 the difference between m 0 and x is said
to be significant at the 5% level. Similarly, an even more extreme
difference yielding a value of z numerically greater than 2 58 is significant at
the 1% level. Rather than using arbitrary levels, such as 5% or 1%, we might
enquire how far into the tails of the expected sampling distribution the
observed value of x falls. A convenient way of measuring this tendency is to
measure the probability, P, of obtaining, if the null hypothesis were true, a value
of x as extreme as, or more extreme than, the value observed. If x is just
significant at the 5% level, z ˆ 1 96 and P ˆ 0 05 (the probability
being that in both tails of the distribution). If x is beyond the 5% significance
level, z > 1 96 or < 1 96 and P < 0 05. If x is not significant at the 5%
level, P > 0 05 (Fig. 4.5). If the observed value of z were, say 2 20, one
could either give the exact value of P as 0 028 (from Table A1), or, by comparison
with the percentage points of the normal distribution, write
0 02 < P < 0 05.
Just significant at
5% level
P = 0 . 05
Not significant at
5% level
P > 0 . 05
Significant at
5% level
P < 0 . 05
–1
Standardized deviate, z Standardized deviate, z
. 96 1 . 0
96
–1 . 96 1 . 0
96
–1
Standardized deviate, z
. 96 1 . 0
96
4.2 Inferences from means 97
Fig. 4.5 Significance tests at the 5% level based on a standardized normal deviate. The observed
deviate is marked by an arrow.