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Procedure for testing hypotheses
The procedure for hypothesis testing involves the following six steps:
1. Declare a null hypothesis, H
0
. This is the hypothesis to be tested. For
example, H
0
:
µ
1
-
µ
2
= 0, i.e., we hypothesize that the mean value of
population 1 and the mean value of population 2 are the same. If H
0
is
true, any observed difference in means is attributed to errors in random
sampling.
2. Declare an alternate hypothesis, H
1
. For the example under consideration,
it could be H
1
:
µ
1
-
µ
2
≠
0 [Note: this is what we really want to test.]
3. Determine or specify a test statistic, T. In the example under consideration,
T will be based on the difference of observed means,
X
1
-
X
2
.
4. Use the known (or assumed) distribution of the test statistic, T.
5. Define a rejection region (the critical region, R) for the test statistic based
on a pre-assigned significance level
α
.
6. Use observed data to determine whether the computed value of the test
statistic is within or outside the critical region. If the test statistic is within
the critical region, then we say that the quantity we are testing is
significant at the 100
α
percent level.
Notes:
1. For the example under consideration, the alternate hypothesis H
1
:
µ
1
-
µ
2
≠
0 produces what is called a two-tailed test. If the alternate hypothesis is
H
1
:
µ
1
-
µ
2
> 0 or H
1
:
µ
1
-
µ
2
< 0, then we have a one-tailed test.
2. The probability of rejecting the null hypothesis is equal to the level of
significance, i.e., Pr[T
∈
R|H
0
]=
α
. The notation Pr[A|B] represents the
conditional probability of event A given that event B occurs.
Errors in hypothesis testing
In hypothesis testing we use the terms errors of Type I and Type II to define the
cases in which a true hypothesis is rejected or a false hypothesis is accepted
(not rejected), respectively. Let T = value of test statistic, R = rejection region,
A = acceptance region, thus, R
∩
A =
∅
, and R
∪
A =
Ω
, where
Ω
= the
parameter space for T, and
∅
= the empty set. The probabilities of making
an error of Type I or of Type II are as follows:
