one are free to be any number at all, but the final one is fixed because the
sum of the data in the sample, divided by n, must equal the mean.
Here is an example. If you have a specified sample mean of 4.25 and n =2,
then the first value in the sample is free to be any value at all, but the second
must be one that gives a mean of 4.25, so it is a fixed number. Thus, the
number of degrees of freedom for a sample of n = 2 is 1. For n = 100 and a
specified mean (e.g. 4.25), 99 of the values are free to vary, but the final value
is also determined by the requirement for the mean to be 4.25, so the
number of degrees of freedom is 99.
The number of degrees of freedom determines the critical value of the t
statistic. For a single-sample t test, if your sample size is n, then you need to
use the t value that has n − 1 degrees of freedom. Therefore, for a sample size
of 10, the degrees of freedom are 9 and the critical value of the t statistic for
an α = 0.05 is 2.262 (Table 8.1). If your calculated value of t is less than
− 2.262 or more than +2.262, then the expected probability of that outcome
is < 0.05. From now on, the appropriate t value will have a subscript to show
the degrees of freedom (e.g. t
7
indicates 7 degrees of freedom).
8.4.2 One-tailed and two-tailed tests
All of the alternate hypotheses dealt with so far in this chapter do not specify
anything other than “The mean of the population from which the sample
has been drawn is different to an expected value” or “ The two samples are
from populations with different means.” Therefore, these are two-tailed
hypotheses because nothing is specified about the direction of the differ-
ence. The null hypothesis could be rejected by a difference in either a
positive or negative direction.
Sometimes, however, you may have an alternate hypothesis that specifies
a direction. For example, “The mean of the population from which the
sample has been taken is greater than an expected value” or “The mean of
the population from which sample A has been taken is less than the mean
of the population from which sample B has been taken.” These are called
one-tailed hypotheses.
If you have an alternate hypothesis that is directional, the null hypothesis
will not just be one of no difference. For example, if the alternate hypothesis
states that the mean of the population from which the sample has been
taken will be less than an expected value, then the null should state, “The
8.4 One sample mean to an expected value 91