Applied Statistics and Probability for Engineers

318 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

6. Reject H 0 if


7. Computations:


8. Conclusions: Since ^20 2.94 ^2 0.05,13.84, we are unable to reject the null hypothesis
   that the distribution of defects in printed circuit boards is Poisson. The P-value for the
   test is P0.0864. (This value was computed using an HP-48 calculator.)

EXAMPLE 9-13 A Continuous Distribution

A manufacturing engineer is testing a power supply used in a notebook computer and, using

    0.05, wishes to determine whether output voltage is adequately described by a normal dis-

tribution. Sample estimates of the mean and standard deviation of V and s 0.08 V

are obtained from a random sample of n100 units.

A common practice in constructing the class intervals for the frequency distribution used

in the chi-square goodness-of-fit test is to choose the cell boundaries so that the expected fre-

quencies Einpiare equal for all cells. To use this method, we want to choose the cell bound-

ariesa 0 , a 1 ,p , akfor the kcells so that all the probabilities

are equal. Suppose we decide to use k8 cells. For the standard normal distribution, the inter-

vals that divide the scale into eight equally likely segments are [0, 0.32), [0.32, 0.675) [0.675,

1.15), [1.15, ) and their four “mirror image” intervals on the other side of zero. For each inter-

val pi 1  8 0.125, so the expected cell frequencies are Ei npi100(0.125)12.5. The

complete table of observed and expected frequencies is as follows:

piP 1 ai

1 Xai 2  

ai

ai 

1

f 1 x 2 dx

x5.04

^20 

132

 28.32 22

28.32

115

 21.24 22

21.24

113

 10.44 22

10.44

2.94

^20 ^2 0.05,13.84.

Class Observed Expected

Interval Frequency oi Frequency Ei

x4.948 12 12.5

4.948x4.986 14 12.5

4.986 x5.014 12 12.5

5.014  x5.040 13 12.5

5.040 x5.066 12 12.5

5.066 x5.094 11 12.5

5.094 x5.132 12 12.5

5.132 x 14 12.5

Totals 100 100

The boundary of the first class interval is. The second class interval is

and so forth. We may apply the eight-step hypothesis-testing proce-

dure to this problem.

1. The variable of interest is the form of the distribution of power supply voltage.


2. H 0 : The form of the distribution is normal.

3 x 1.15s, x 0.675s 2

x 1.15s4.948

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