Applied Statistics and Probability for Engineers

That is, the upper 5 percentage point of F5,10is f0.05, 5,103.33.

Table V contains only upper-tail percentage points (for selected values of f ,u,vfor 

0.25) of the Fdistribution. The lower-tail percentage points f 1 ,u,vcan be found as follows.

10-5 INFERENCES ON THE VARIANCES OF TWO NORMAL POPULATIONS 357

0 246810 x

u = 5, v = 15

f(x)

u = 5, v = 5

Figure 10-4 Probability density functions of

two Fdistributions.

Figure 10-5 Upper and lower percentage

points of the Fdistribution.

x

α α

f1 –α,,uv fα,,uv

f(x)

f 1 ,u,v (10-28)

1

f ,v,u

For example, to find the lower-tail percentage point f0.95, 5,10, note that

10-5.2 Development of the FDistribution (CD Only)

10-5.3 Hypothesis Tests on the Ratio of Two Variances

A hypothesis-testing procedure for the equality of two variances is based on the following result.

f0.95, 5,10

1

f0.05,10, 5



1

4.74

0.211

Let X 11 , X 12 , p , X 1 n 1 be a random sample from a normal population with mean  1 and

variance ^21 , and let X 21 , X 22 , p , X 2 n 2 be a random sample from a second normal pop-

ulation with mean  2 and variance ^22. Assume that both normal populations are

independent. Let and be the sample variances. Then the ratio

has an Fdistribution with n 1 1 numerator degrees of freedom and n 2 1 denom-

inator degrees of freedom.

F

S 12

 ^21

S^22 ^22

S 12 S^22

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