Page 142 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

122 4 Parametric Tests of Hypotheses

H 0: µ ≤ µ , H 1: µ > µ or
0
0
H 0: µ ≥ µ , H 1: µ < µ , for a one-sided test.
0
0

We assume that the random variable being tested has a normal distribution. We
then recall from section 3.2 that when the null hypothesis is verified, the following
random variable:

X − µ
T = 0 , 4.2
s / n

has a Student’s t distribution with n − 1 degrees of freedom. We then use as the test
statistic, t n(x), the following quantity:

x µ− 2
*
t = 0 .
s / n

*
When a statistic as t is standardised using the estimated standard deviation
instead of the true standard deviation, it is called a studentised statistic.
For large samples, say n > 25, one could use the normal distribution instead,
since it will yield a good approximation of the Student’s t distribution. Even with
small samples, we can use the normal distribution if we know the true value of the
standard deviation. That’s precisely what we have done in the preceding sections.
However, in normal practice, the true value of the standard deviation is unknown
and the test relies then on the Student’s t distribution.
Assume a two-sided t test. In order to determine the critical region for a level of
significance α, we compute the 1–α/2 percentile of the Student s t distribution with
’
df = n–1 degrees of freedom:

−
T df (t ) = 1 α 2 / ⇒ t df 1 , α 2 / , 4.3
−

and use this percentile in order to establish the non-critical region C of the test:

C = [ df 1 , α 2 / , + t df 1 , α 2 / ] −t . 4.4
−
−

Thus, the two-sided probability of C is 2(α /2) = α. The non-critical region can
also be expressed in terms of X , instead of T (formula 4.2):

C = [ µ − t df 1 α 2 / s / n, µ 0 t + df 1 α 2 / s / ] n . 4.4a
−
−
,
0
,

Notice how the test of a mean is similar to establishing a confidence interval for
a mean.

2
We use an asterisk to denote a test statistic.