Page 343 - Applied Statistics And Probability For Engineers

c09.qxd 5/15/02 8:02 PM Page 291 RK UL 9 RK UL 9:Desktop Folder:

9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 291

We may solve this problem by following the eight-step procedure outlined in Section 9-1.4.
This results in
1. The parameter of interest is , the mean burning rate.
2. H 0 : 50 centimeters per second
3. H 1 : 50 centimeters per second
4. 0.05
5. The test statistic is
x
0
z
0

1n

6. Reject H 0 if z 0 1.96 or if z 0
1.96. Note that this results from step 4, where we
specified 0.05 , and so the boundaries of the critical region are at z 0.025 1.96
and
z 0.025
1.96.
7. Computations: Since x 51.3 and 2,

51.3
50
z 3.25
0
2
225
8. Conclusion: Since z 0 3.25 1.96, we reject H 0 : 50 at the 0.05 level of
significance. Stated more completely, we conclude that the mean burning rate dif-
fers from 50 centimeters per second, based on a sample of 25 measurements. In
fact, there is strong evidence that the mean burning rate exceeds 50 centimeters
per second.

We may also develop procedures for testing hypotheses on the mean where the alter-
native hypothesis is one-sided. Suppose that we specify the hypotheses as

H : 0
0
H : 0 (9-11)
1
In defining the critical region for this test, we observe that a negative value of the test statistic
Z 0 would never lead us to conclude that H 0 : 0 is false. Therefore, we would place the
critical region in the upper tail of the standard normal distribution and reject H 0 if the com-
puted value of z 0 is too large. That is, we would reject H 0 if

z z (9-12)
0
as shown in Figure 9-6(b). Similarly, to test

H : 0
0
(9-13)
H 1 : 0
we would calculate the test statistic Z 0 and reject H 0 if the value of z 0 is too small. That is, the
critical region is in the lower tail of the standard normal distribution as shown in Figure
9-6(c), and we reject H 0 if

z (9-14)
z 0