Applied Statistics and Probability for Engineers

576 CHAPTER 15 NONPARAMETRIC STATISTICS

The Normal Approximation

When p0.5, the binomial distribution is well approximated by a normal distribution when

nis at least 10. Thus, since the mean of the binomial is npand the variance is np(1p), the

distribution of Ris approximately normal with mean 0.5nand variance 0.25nwhenever nis

moderately large. Therefore, in these cases the null hypothesis can be tested using

the statistic

H 0 :  0

Z 0  (15-5)

R0.5n

0.5 1 n

The two-sided alternative would be rejected if the observed value of the test statistic

, and the critical regions of the one-sided alternative would be chosen to reflect the

sense of the alternative. (If the alternative is , reject H 0 if z 0

 z, for example.)

EXAMPLE 15-2 We will illustrate the normal approximation procedure by applying it to the problem in

Example 15-1. Recall that the data for this example are in Table 15-1. The eight-step proce-

dure follows:

1. The parameter of interest is the median of the distribution of propellant shear strength.


2. 
   


3. 
   


4. 0.05


5. The test statistic is


6. Since 0.05, we will reject H 0 in favor of H 1 if


7. Computations: Since r14, the test statistic is


8. Conclusions: Since z 0 1.789 is not greater than z0.0251.96, we cannot reject the
   null hypothesis. Thus, our conclusions are identical to those in Example 15-1.

15-2.2 Sign Test for Paired Samples

The sign test can also be applied to paired observations drawn from continuous populations.

Let (X 1 j, X 2 j),j1, 2,... , nbe a collection of paired observations from two continuous pop-

ulations, and let

DjX 1 jX 2 j j1, 2,... , n

z 0 

14 0.5 1202

0.5 220

1.789

0 z 00

 z0.0251.96.

z 0 

r0.5n

0.5 1 n

H 1 :  2000 psi

H 0 : 2000 psi

H 1 : 

 0

0 z 00

 z 2

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