Chapter 9: Introduction to Hypothesis Testing

412 CHAPTER 9 • INTRODUCTION TO HYPOTHESIS TESTING
error. To find beta for this test procedure under these conditions, the engineers can use the
following steps:
Step 1 Specify the null and alternative hypotheses.
The null and alternative hypotheses are
H 0
: 2.25 (status quo)
H A
: 2.25
Step 2 Specify the significance level.
The two-tailed hypothesis test will be conducted using α 0.05.
Step 3 Determine the critical values, z (/2)L
and z (/2)U
, from the standard
normal distribution.
The critical value from the standard normal is z (α/2)L
and
z (α/2)U
z 0.025
1.96.
Step 4 Calculate the critical values.
x
.
,
z0. 025
225 . 196 .
0 005 → xL2. 2478;
x U
2.
2522
n
20
LU
Thus, the null hypothesis will be rejected if x 2.2478 or x 2.2522
Step 5 Specify the stipulated value of .
The stipulated value of is 2.255.
Step 6 Compute the z-values based on the stipulated population mean.
The z-values based on the stipulated population mean is
xL
z 2. 24782.
255
xU
644
. and z 2. 25222.
255
250
.
0.
005
0.
005
n 20
n 20
Step 7 Determine beta and reach a conclusion.
Beta is the probability from the standard normal distribution between
z 6.44 and z 2.50. From the standard normal table, we get
(0.5000 0.5000) (0.5000 0.4938) 0.0062
Thus, beta 0.0062. There is a very small chance (only 0.0062) that
this hypothesis test will fail to detect that the mean diameter has shifted
to 2.255 inches from desired mean of 2.25 inches. This low beta will
give the engineers confidence that their test can detect problems when
they occur.
As shown in Section 9.2, many business applications will involve hypotheses tests
about population proportions rather than population means. Example 9-12 illustrates the
steps needed to compute the beta for a hypothesis test involving proportions.