Applied Statistics and Probability for Engineers

13-11

13-3.3 Determining Sample Size in the Random Model (CD Only)

The power of the test for the random-effects model is

It can be shown that if H 1 is true (^2 0) the power can be computed using the central F

distribution, with a 1 and a(n 1) degrees of freedom. In fact, the ratio

has the F-distribution with a 1 and a(n 1) degrees of freedom. Then,

(S13-3)

This probability statement may be easily evaluated using certain hand-held calculators, or it

may be evaluated using tables of the F-distribution.

EXAMPLE S13-2 Consider a completely randomized design with five treatments selected at random and six

observations per treatment. If 0.05, what is the power of the test if ^2 ^2?

From Equation S13-3, we have the power as

since if ^2 ^2 the ratio ^2 ^2 1. Now f0.05,4,252.76, so

This probability was evaluated using a calculator that provided F-distribution probabilities.

Since the power of the test is 0.81, this implies that the null hypothesis H 0 : ^2 0 will be

rejected with probability 0.81 in this experimental situation.

It is also possible to evaluate the power of the test using the operating characteristic

curves on page 13-12 through 13-15. These curves plot the probability of the type II error 

against , where

 (S13-4)

B

1 

n^2 

^2

P 5 F4,25 0.39 6 0.81

1   F eF4,25

2.76

31  61124

fP eF4,25

2.76

7

f

1   P eF4,25

f0.05,4,25

31  61124

f

P eFa 1,a 1 n  12

f,a 1,a 1 n    12

11 n^2 ^22

f

P e

MSTreatments

MSE 11 n^2 ^22

f,a 1,a 1 n    12

11 n^2 ^22

f

1   P e

MSTreatments

MSE

f,a 1,a 1 n    120 ^2  

06

MSTreatments 1 ^2 n^2  2

MSE^2

P 5 F 0

 f,a 1,a 1 n   120 ^2  

06

1   P 5 Reject H 00 H 0 is false 6

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