Fundamentals of Probability and Statistics for Engineers

following Equation (10.6), ra ndom variable D thus approaches a chi-squared
distribution with one degree of freedom, and the proof is complete for k 2.
The proof for an arbitrary k proceeds in a similar fashion.

By means of Theorem 10.1, a test of hypothesis H considered above can be
constructed based on the assignment of a probability of Type-I error. Suppose
that we wish to achieve a Type-I error probability of. The^2 test suggests
that hypothesis H is rejected whenever

and is accepted otherwise, where d is the sample value of D based on sample
values xi,i 1,...,n,and^2 ,takes the value such that (see Figure 10.1)

Since D has a Chi-squared distribution with (k 1) degrees of fr eedom for
large n, an approximate value for can be found from Table A.5 in
Appendix A for the^2 distribution when is specified.
The probability of a Type-I error is referred to as the significance level in this
context. As seen from Figure 10.1, it represents the area under fD(d) to the right
of. Letting 0 05, for example, the criterion given by Equation (10.7)
implies that we reject hypothesis H whenever deviation measure d as calculated
from a given set of sample values falls within the 5% region. In other words, we
expect to reject H about 5% of the time when in fact H is true. Which significance
level should be adopted in a given situation will, of course, depend on the

d

fD(d)

1–

Figure 10.1 Chi-squared distribution with (k 1) degrees of freedom

Model Verification 319

ˆ

dˆ

Xk

iˆ 1

n^2 i

npi

n>^2 k 1 ;; … 10 : 7 †

ˆ ^2 k1,

P…D>^2 k 1 ;†ˆ:



^2 k1,

^2 k1, ˆ

α

α

χ (^2) k–1,α


: