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

Answer:

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

Step-by-step 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.