Applied Statistics and Probability for Engineers

If the null hypothesis H 0 :  0 is true, , and it follows that the distribution of Z 0

is the standard normal distribution [denoted N(0, 1)]. Consequently, if H 0 :  0 is true, the

probability is 1 that the test statistic Z 0 falls between and , where is the

percentage point of the standard normal distribution. The regions associated with

and are illustrated in Fig. 9-6(a). Note that the probability is that the test statistic Z 0

will fall in the region or when H 0 :  0 is true. Clearly, a sample

producing a value of the test statistic that falls in the tails of the distribution of Z 0 would be

unusual if H 0 :  0 is true; therefore, it is an indication that H 0 is false. Thus, we should

reject H 0 if the observed value of the test statistic z 0 is either

(9-9)

and we should fail to reject H 0 if

(9-10)

The inequalities in Equation 9-10 defines the acceptance regionfor H 0 , and the two inequali-

ties in Equation 9-9 define the critical regionor rejection region.The type I error probability

for this test procedure is.

It is easier to understand the critical region and the test procedure, in general, when the

test statistic is Z 0 rather than. However, the same critical region can always be written in

terms of the computed value of the sample mean. A procedure identical to the above is as

follows:

where

EXAMPLE 9-2 Aircrew escape systems are powered by a solid propellant. The burning rate of this pro-

pellant is an important product characteristic. Specifications require that the mean burning

rate must be 50 centimeters per second. We know that the standard deviation of burning

rate is 2 centimeters per second. The experimenter decides to specify a type I error

probability or significance level of    0.05 and selects a random sample of n25 and

obtains a sample average burning rate of centimeters per second. What conclu-

sions should be drawn?

x51.3

a 0 z

2
1 n and b 0
z

2
1 n

Reject H 0 :  0 if either xa or xb

x

X

z   

2 z 0 z   

2

z 0 z

2 or z 0 
z

2

Z 0 z  

2 Z 0 

z   

2

z   

2 z     

2

100
2

z   

2 z     

2 z     

2

E 1 X 2  0

290 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

Figure 9-6 The distribution of Z 0 when H 0 :  0 is true, with critical region for (a) the two-sided alternative H 1 :  0 ,

(b) the one-sided alternative H 1 :  0 , and (c) the one-sided alternative H 1 :  0.

(a)

0

N(0,1)

–z /α 2 z /α 2 Z 0

/2α Acceptance /2α

region

Critical region

(c)

0

N(0,1)

–z α Z 0

α Acceptance

region

(b)

0

N(0,1)

z α Z 0

α

Critical region

Acceptance

region

Critical region

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