TESTS OF HYPOTHESES FOR A SINGLE MEAN 99acyanotic heart disease children may learn to walk at the same average age as other
children. It should be emphasized that this conclusion may be incorrect; the mean
may actually be something different from 12.0. But in any case, a sample mean of
11.6 could easily have arisen from a population whose mean was 12.0, so if we still
wish to establish that the age of walking is different for acyanotic children from that
for normal children, we must gather more data.
In these examples, it was fairly easily decided to reject the null hypothesis when
P was .0008, and similarly it was easily decided to accept the null hypothesis when
P was equal to .33. It is not so apparent which decision to make if we obtain a P
somewhere in between, say P = .1, for example. Where should the dividing line be
between the values of P that would lead us to reject the null hypothesis and those that
would lead us to accept the null hypothesis? In many medical or public health studies,
the dividing line is chosen in advance at .05. If P is < .05 and if the population mean
is 12.0, the chance is < 1 in 20 that we will have a sample mean as unusual as that
13.38 months occur; we therefore conclude that the mean is not 12.0.
In statistical literature, the number picked for the dividing line is called a (alpha);
it is seen as our chance of making the mistake of deciding that the null hypothesis is
false when it is actually true. Alpha is sometimes called the level ofsign$cance. It
is advisable to choose the level of a before obtaining the results of the test so that the
choice is not dependent on the numerical results. For example, when a is chosen to be
.05 in advance, we only have to see if the absolute value of the computed z value was
z[.975] = 1.96 for HO : p = po. When the null hypothesis is rejected, the results
are said to be statistically sign$cant. If the computed z is > 1.96 or < -1.96, we
say that the value of z lies in the rejection region and the null hypothesis is rejected.
If the value of z lies between -1.96 and f1.96, it lies in the acceptance region, and
the null hypothesis is not rejected. If the null hypothesis is not rejected, the results
are often stated to be nonsign$cant.
8.1.2 One-sided Tests When 0 Is Known
In the previous example concerning age of walking for acyanotic children, we wished
to find out whether or not the mean is 12.0; the test that was made is called a two-sided
test because computing P included the chance of getting a sample mean above the
one actually obtained (13.38) and also the chance of getting a sample mean less than
10.625.
Sometimes it is appropriate, instead, to make a one-sided test. We may consider
it highly unlikely that acyanotic children learn to walk earlier on the average than
normal children and may not be interested in rejecting the null hypothesis if this
occurs. Then the question to be answered by the experiment is no longer whether the
mean is 12.0 or something else. Instead, the question is whether or not the mean is
12.0. We do not want to reject the null hypothesis for small values of the mean.
Then, to calculate P we find the proportion of sample means that lie above 7. If
from the sample, fT is calculated to be 13.38, P from Table A.2 is found to be .0004.
The null hypothesis is stated HO : p 5 12 or, in general, HO : p 5 po. If the
null hypothesis is rejected, we conclude that the population mean is > 12.0; that is,