will be rejected only if the sample evidence strongly suggests that
H
0
is false. In this
case
H
a
must be true.
•
Otherwise
H
0
will not be rejected.
So there are two possible conclusions:
•
reject
H
0
(accept
H
a
)
•
do not reject
H
0
7

Note that these decisions are not symmetric, there is no way you can say you
accept
H
0
.
Remark:
Hypotheses should be the logical complement of each other.
Common choices of hypotheses are
•
H
0
: population characteristic = specific value versus
H
a
: population characteristic
̸
= specific value
•
H
0
: population characteristic
≤
specific value versus
H
a
: population characteristic
>
specific value
•
H
0
: population characteristic
≥
specific value versus
H
a
: population characteristic
<
specific value
Example 3
•
H
0
:
p
= 0
.
25 versus
H
a
:
p
̸
= 0
.
25
•
H
0
:
μ
≥
100 versus
H
a
:
μ <
100
•
We can’t test
H
0
:
μ
≤
100
versus
H
a
:
μ >
150
Be careful when choosing hypotheses, because a statistical test can only support the alternative
hypothesis, by rejecting
H
0
. Is
H
0
not being rejected
does not
mean strong support for
H
0
,
but lack of strong evidence against
H
0
.
Example 4
A company is advertising that the average lifetime of their light bulbs is 1000 hours.
You
might question this, and want to show that in fact the lifetime is shorter.
Let
μ
= mean lifetime of the light bulbs.
You would test
H
0
:
μ
≥
1000 versus
H
a
:
μ <
1000.
Rejection of
H
0
would then support your claim, that the mean lifetime is less than 1000 hours.
However, non-rejection of
H
0
does not necessarily provide strong support for the advertised
claim, that the mean lifetime is at least 1000 hours.
How to make a decision (reject
H
0
, or do not reject
H
0
)
Since we do Statistics the decision to reject, or not to reject
H
0
is based on information
contained in a sample drawn from the population of interest. We use the sample to
1. Calculate a test statistic,

2. Evaluate the value of the test statistic based on its distribution under the assumption
that
H
0
is true.
8

3. Make a decision: Is the value of the test statistic highly unlikely to occur, under the
assumption that
H
0
is true, we will interpret this as a contradiction to the assumption
that
H
0
is true, and reject this hypothesis and decide that
H
a
must be true.

Example 5
Suppose
μ
is the mean in a given population which follows a normal distribution with standard
deviation
σ
= 0
.
2.
The investigator wants to test
H
0
:
μ
≥
0
.
5 versus
H
a
:
μ <
0
.
5.
A random sample of size 100 showed a sample mean of ¯
x
= 0
.
3.
which is less than 0
.
5, as claimed in the alternative hypothesis.