Question

The city of Tucson, Arizona, employs people to assess the value of homes for the purpose of calculating real estate tax. The city manager sends each assessor to the same five homes and then compares the results. The information is given below, in thousands of dollars. Can we conclude that there is a difference in the assessors, at α=0.05

Assessor
Home Zawodny Norman Cingle Holiday
A $54 $52 $48 $45
B 58 54 52 58
C 43 53 44 54
D 75 66 63 64
E 82 85 96 86

Required:
a. Is there a difference in the treatment means?
b. Is there a difference in the block means?

Answer 1

1. There is not a difference between treatment means.
2. There is a difference between block means.

Explanation:

1) We set up our null and alternative hypothesis as

H0 : there is no difference between treatment means

H0 : there is no difference between block means

against

Ha: treatment means are not equal.

Ha: block means are not equal.

2) We chose the significance level alpha= 0.01

3) The test statistic to use are

F1= estimated variance from Between hospitals SS/ Estimated Variance from Error SS

F2= estimated variance from Between days SS/ Estimated Variance from Error SS

Which have F distributions with v1=4 and v2= 4 d.f when null hypothesis is true.

Computations:

The necessary computations are shown below.

Assessor Home Zawodny Norman Cingle Holiday Ti Ti2 Sum X2ij

A $54 $52 $48 $45 199 39601 9949

(2916) (2704) (2304) (2025)

B 58 54 5 2 58 222 49284 12348

(3364) (2916) (2704) (3364)

C 43 53 44 54 194 37636 9510

(1849) (2809) (1936) (2916)

D 75 66 63 64 268 71824 18711

(5625) (4356) (3969) (4761)

E 82 85 96 86 349 121801 30561

(6724) (7225) (9216) (7396)

T.j 312 310 303 305 1232 320146 81079

T2.j 97344 96100 91809 93025 378278

Sum X 2 ij 20478 20010 20129 20462 81079

Now Total SS= 81079- (1232)2/ 20

= 81079- 75891.2= 5187.8

Between Treatments SS= 378278/ 5 -75891.2= -235.6

Between Blocks SS= 320146 /4- 75891.2= = 4145.3

Error SS= 5187.8- (-235.6+4145.3)= 1278.1

Source SS df MS F

Treatments -235.6 3 78.533 F1= 0.737

Blocks 4145.3 4 1036.325 F2= 9.73

Error 1278.1 12 106.508

Total 19

For Treatment:

Reject H0 if F> F (0.01) (3,12)=5.95

For blocks:

Reject H0 if F> F (0.01) (4,12)=5.41

Since the computed value F1= 0.737 does not fall in the critical region F> F (0.01) (3,12)= 5.95 and the calculated value F2= 9.73 falls in the region F> F (0.01) (4,12)=5.41 so we conclude:

1. There is not a difference between treatment means.
2. There is a difference between block means.