Applied Statistics and Probability for Engineers

10-1

10-3.2 More About the Equal Variance Assumption (CD Only)

In practice, one often has to choose between case 1 and case 2 of the two-sample t-test. In case

1, we assume that  and use the pooled t-test. On the surface this test would seem to have

some advantages. It is a likelihood ratio test, whereas the case 2 test with is not.

Furthermore, it is an exact test (if the assumptions of normality, independence, and equal vari-

ances are correct), whereas the case 2 test is an approximate procedure. However, the pooled

t-test can be very sensitive to the assumption of equal variances, especially when the sample sizes

are not equal. To help see this, consider the denominator of the test statistic for the pooled t-test:

Because the variances are divided (approximately) by the wrong sample sizes, use of the

pooled t-test when the variances are unequal and when can lead to very frequent er-

roneous conclusions. This is why using n 1 n 2 is a good idea in general, and especially when

we are in doubt about the validity of the equal variance assumption.

It would, of course, be possible to perform a test of H 0 :  12 ^22 versus and

then use the pooled t-test if the null hypothesis is not rejected. This test is discussed in Section 10-

5. However, the test on variances is much more sensitive to the normality assumption than are
   t-tests. A conservative approach would be to always use the case 2 procedure. Alternatively, one
   can use the normal probability plot both as a check of the normality assumption and as a check for
   equality of variance. If there is a noticable difference in the slopes of the two straight lines on the
   normal probability plot, the case 2 procedure would be preferred, especially when

10-5.2 Development of the FDistribution (CD Only)

We now give a formal development of the Fdistribution. The development makes use of the

material in Section 5-8 (CD Only).

n 1 n 2.

H 1 : ^21 ^22

n 1 n 2

Sp

B

1

n 1 

1

n 2 B

1 n 1  12 S^21  1 n 2  12 S^22

n 1 n 2  2

n 1 n 2

n 1 n 2 B

S^21

n 2 

S^22

n 1

^21 ^22

^21 ^22

Let U 1 and U 2 be independent chi-square random variables with v 1 and v 2 degrees of

freedom, respectively. Then the ratio

has the probability density function

This is the F-distribution with v 1 degrees of freedom in the numerator and v 2 degrees

of freedom in the denominator.

f 1 x 2 

 a

 1  2

2

b a

 1

 2 b

 1 2

x^1    2 ^1

 a

 1

2

b  a

 2

2

b ca

 1

 2 b^ x^1 d

1  1  22 2 ,^0 x^

F

U 1   1

U 2   2

Theorem: The

F-Distribution

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