Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.7. Chi-Square Tests 283

4.6.8.Letpequal the proportion of drivers who use a seat belt in a country that

does not have a mandatory seat belt law. It was claimed thatp =0.14. An

advertising campaign was conducted to increase this proportion. Two months after

the campaign,y= 104 out of a random sample ofn= 590 drivers were wearing

their seat belts. Was the campaign successful?

(a)Define the null and alternative hypotheses.

(b)Define a critical region with anα=0.01 significance level.

(c)Determine the approximatep-value and state your conclusion.

4.6.9.In Exercise 4.2.18 we found a confidence interval for the varianceσ^2 using
the varianceS^2 of a random sample of sizenarising fromN(μ, σ^2 ), where the mean
μis unknown. In testingH 0 :σ^2 =σ 02 againstH 1 :σ^2 >σ 02 , use the critical region
defined by (n−1)S^2 /σ^20 ≥c. That is, rejectH 0 and acceptH 1 ifS^2 ≥cσ^20 /(n−1).
Ifn= 13 and the significance levelα=0.025, determinec.

4.6.10. In Exercise 4.2.27, in finding a confidence interval for the ratio of the
variances of two normal distributions, we used a statisticS^21 /S 22 , which has anF-
distribution when those two variances are equal. If we denote that statistic byF,
we can testH 0 :σ^21 =σ^22 againstH 1 :σ^21 >σ^22 using the critical regionF≥c.If
n=13,m= 11, andα=0.05, findc.

4.7 Chi-SquareTests

In this section we introduce tests of statistical hypotheses calledchi-square tests.
A test of this sort was originally proposed by Karl Pearson in 1900, and it provided
one of the earlier methods of statistical inference.
Let the random variableXibeN(μi,σi^2 ),i=1, 2 ,...,n,andletX 1 ,X 2 ,...,Xn
be mutually independent. Thus the joint pdf of these variables is

1

σ 1 σ 2 ···σn(2π)n/^2

exp

[

−

1

2

∑n

1

(

xi−μi

σi

) 2 ]

, −∞<xi<∞.

The random variable that is defined by the exponent (apart from the coefficient

−^12 )is

∑n

1 [(Xi−μi)/σi]

(^2) , and this random variable has aχ (^2) (n) distribution. In
Section 3.5 we generalized this joint normal distribution of probability tonrandom
variables that aredependentand we called the distribution amultivariate normal
distribution. Theorem 3.5.1 shows a similar result holds for the exponent in the
multivariate normal case, also.
Let us now discuss some random variables that have approximate chi-square
distributions. LetX 1 beb(n, p 1 ). Consider the random variable
Y=
X 1 −np 1
√
np 1 (1−p 1 )
,