MN1025 – Business Statistics
19
Lecture 5—Friday 8/2/2008
TESTING HYPOTHESES:
TESTING THE MEAN
Reference: Lind
et al.
, Chapters 9, 10.
5.1
Example: less than 99 defects?
Last week we used the
T
statistic to derive con
fidence intervals for the unknown mean of a nor
mally distributed population. In this lecture we will
use the same formalism to answer yes/no questions
about the population mean. We are again concerned
with a sample drawn from an (approximately) nor
mally distributed population. We start with an ex
ample.
A manufacturer has a production line which has had
99 defects per day, on average.
A new system is
introduced and the manufacturer wants to know if
there has been a
significant decrease
in the average
number of defects.
Hypothesis testing proceeds in five steps.
5.2
Step 1: Setting up the hypotheses
In our example, the previous system had an average
fault rate of 99 defects per day.
The average fault
rate of the new system, denoted by
μ
, is unknown.
Our test will consider two hypotheses:
The
Null
Hypothesis
,
H
0
,
is
that
nothing
has
changed, i.e. that
μ
= 99 as previously.
The
Alternative Hypothesis
,
H
1
, is that there has
been a decrease, i.e. that
μ <
99 or, in words, the
new population mean is less than the previous one.
In some books, the alternative hypothesis is written
as
H
A
or
H
a
.
We have to choose between
H
0
and
H
1
.
We will
choose, or
accept
,
H
0
(no change) unless there is
sig
nificant
evidence in favour of
H
1
, in which case we
say that we
reject
H
0
. We will see below what ex
actly is meant by this.
Notice that we never test an hypothesis in isolation;
we always need to know the alternative.
5.3
Step 2: Setting a level of significance
The level of significance is the probability of reject
ing the null hypothesis when it is actually true (this
is called a
type I error
).
Commonly used levels of
significance are 10%, 5% and 1%. The level of sig
nificance must be set before the sample is taken.
To
choose,
e.g.,
a
5%
level
of
significance
in
Minitab’s 1sample t test, click on Options and en
ter a 95% confidence level.
5.4
Step 3: Choosing a test statistic
In this chapter, we use
T
as the test statistic (see lec
ture 4). We will encounter other useful test statistics
later in the course.
5.5
Step 4: Setting a decision rule
Assume we take a sample of size
n
from the produc
tion line and compute the sample mean, ¯
x
, and the
sample standard deviation,
s
. If the average number
of defects for the new system,
μ
, is less than 99, we
expect the sample mean to be also less than 99, i.e.,
¯
x <
99. But of course, the sample mean could turn
out to be less than 99 even if
μ
is equal to 99, i.e. if
the population mean
μ
has not decreased.
We should therefore reject the null hypothesis
μ
=
99 only if ¯
x
is
significantly
smaller than 99. To make
this idea precise, we assume that
μ
= 99
and look
at the
T
statistic, which is in this case given by
T
=
¯
x
−
μ
s/
√
n
=
¯
x
−
99
s/
√
n
.
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 Spring '08
 SCHACK
 Statistics, Normal Distribution, Null hypothesis, Statistical hypothesis testing

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