Applied Statistics and Probability for Engineers

578 CHAPTER 15 NONPARAMETRIC STATISTICS

15-2.3 Type II Error for the Sign Test

The sign test will control the probability of type I error at an advertised level for testing the null

hypothesis for any continuous distribution. As with any hypothesis-testing procedure,

it is important to investigate the probability of a type II error,. The test should be able to effec-

tively detect departures from the null hypothesis, and a good measure of this effectiveness is the

value of for departures that are important. A small value of implies an effective test procedure.

In determining , it is important to realize not only that a particular value of , say ,

must be used but also that the formof the underlying distribution will affect the calculations. To

illustrate, suppose that the underlying distribution is normal with 1 and we are testing the

hypothesis versus. (Since in the normal distribution, this is equiv-

alent to testing that the mean equals 2.) Suppose that it is important to detect a departure from

to. The situation is illustrated graphically in Fig. 15-1(a). When the alternative

hypothesis is true (H 1 : ), the probability that the random variable Xis less than or equal to

the value 2 is

Suppose we have taken a random sample of size 12. At the 0.05 level, Appendix Table VII

indicates that we would reject if rr*0.052. Therefore, is the probability that

we do not reject when in fact , or

 1  a

2

x 0

a

12

x

b 1 0.1587 2 x 1 0.8413 212 x0.2944

H 0 :  2  3

H 0 :  2

pP 1 X 22 P 1 Z 12  1  12 0.1587

 3

 2  3

H 0 :  2 H 1 :  

2 

  0 

H 0 : 

–1 0 1 2 3 4 5 6 x

σ= 1

0.1587

–1 0 1 2 3 4 5 x

σ= 1

Under H 0 : μ∼= 2 Under H 1 : ∼μ= 3

(a)

μ∼= 2 μ= 2.89

Under H 0 : μ∼= 2

2 μ= 4.33

Under H 1 : μ∼= 3

(b)

0.3699

x x

Figure 15-1 Calcula-

tion of for the sign

test. (a) Normal

distributions. (b)

Exponential

distributions.

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