Applied Statistics and Probability for Engineers

13-4 RANDOMIZED COMPLETE BLOCK DESIGN 503

MIND-EXPANDING EXERCISES

13-40. Show that in the fixed-effects model analysis

of variance E(MSE)^2. How would your develop-

ment change if the random-effects model had been

specified?

13-41. Consider testing the equality of the means of

two normal populations where the variances are

unknown but are assumed equal. The appropriate test

procedure is the two-sample t-test. Show that the two-

sample t-test is equivalent to the single-factor analysis of

variance F-test.

13-42. Consider the ANOVA with a2 treatments.

Show that the MSEin this analysis is equal to the

pooled variance estimate used in the two-sample

t-test.

13-43. Show that the variance of the linear combina-

tion

13-44. In a fixed-effects model, suppose that there are

nobservations for each of four treatments. Let Q^21 , Q^22 ,

and Q^23 be single-degree-of-freedom sums of squares for

the orthogonal contrasts. Prove that SSTreatmentsQ^21 

Q^22 Q^23.

13-45. Consider the single-factor completely ran-

domized design with atreatments and nreplicates.

Show that if the difference between any two treatment

means is as large as D, the minimum value that the OC

curve parameter ^2 can take on is

13-46. Consider the single-factor completely ran-

domized design. Show that a 100(1) percent confi-

dence interval for ^2 is

where Nis the total number of observations in the

experimental design.

13-47. Consider the random-effect model for the

single-factor completely randomized design. Show that

1 Na 2 MSE

^2 2, Na

^2 

1 Na 2 MSE

^21  2, Na

^2 

nD^2

2 a^2

a

a

i 1

ciYi. is ^2 a

a

i 1

nic^2 i.

a 100(1)% confidence interval on the ratio of vari-

ance components ^2 ^2 is given by

where

and

13-48. Consider a random-effects model for the

single-factor completely randomized design. Show that

a 100(1)% confidence interval on the ratio ^2 

(^2 ^2 ) is

where Land Uare as defined in Exercise 13-47.

13-49. Continuation of Exercise 13-48.Use the

results of Exercise 13-48 to find a 100(1)% confi-

dence interval for ^2 (^2 ^2 ).

13-50. Consider the fixed-effect model of the com-

pletely randomized single-factor design. The model

parameters are restricted by the constraint.

(Actually, other restrictions could be used, but this one is

simple and results in intuitively pleasing estimates for

the model parameters.) For the case of unequal sample

size n 1 , n 2 , p, na, the restriction is .Use

this to show that

Does this suggest that the null hypothesis in this model

is H 0 : n 1  1 n 2  2 pnaa0?

13-51. Sample Size Determination.In the single-

factor completely randomized design, the accuracy of a

E 1 MSTreatments 2 ^2 

a

a

i 1

ni^2 i

a 1

gai 1 nii 0

gai 1 i 0

L

1 L



^2 

^2 ^2 



U

1 U

U

1

n^ c

MSTreatments

MSE

a

1

f 1  2,a1,Na

b 1 d

L

1

n^ c

MSTreatments

MSE

a

1

f 2,a1,Na

b 1 d

L

^2 

^2

U

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