Applied Statistics and Probability for Engineers

262 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

0 510152025 x

k = 10

k = 5

k = 2

f(x)

Figure 8-8 Proba-

bility density functions

of several ^2 distribu-

tions.

The probability density function of a^2 random variable is

(8-20)

where kis the number of degrees of freedom. The mean and variance of the ^2 distribution are

kand 2k, respectively. Several chi-square distributions are shown in Fig. 8-8. Note that the

chi-square random variable is nonnegative and that the probability distribution is skewed to

the right. However, as kincreases, the distribution becomes more symmetric. As the

limiting form of the chi-square distribution is the normal distribution.

The percentage pointsof the ^2 distribution are given in Table III of the Appendix.

Define as the percentage point or value of the chi-square random variable with kdegrees

of freedom such that the probability that X^2 exceeds this value is . That is,

This probability is shown as the shaded area in Fig. 8-9(a). To illustrate the use of Table III,

note that the areas are the column headings and the degrees of freedom kare given in the left

column. Therefore, the value with 10 degrees of freedom having an area (probability) of 0.05

to the right is ^2 0.05,1018.31.This value is often called an upper 5% point of chi-square with

P 1 X^2

^2 ,k 2  

^2 ,k

f 1 u 2 du

^2 ,k

kS ,

f 1 x 2 

1

2 k^2  1 k     22

x^1 k^22 ^1 ex^2 x

0

Let X 1 , X 2 , p,Xnbe a random sample from a normal distribution with mean and

variance ^2 , and let S^2 be the sample variance. Then the random variable

(8-19)

has a chi-square (^2 ) distribution with n1 degrees of freedom.

X^2 

1 n 12 S^2

^2

Definition

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