Exercise 9.5

Johnson* conducted a study to determine whether, in collegiate schools of nursing,

relationships bertween certain variables could be identified. Two variables of interest for

which indixes were constucted were “extent of agreement (between the dean and the faculty)

on the responsibilities for decision making” and “faculty satisfaction.” The ranks on the two

variables of the 12 institutions that participated in the study are shown in Table 9.11. The

author computed a value of rs =-0.336 from the data, which she declared not significant.

Compute from the data and test significance against the alternative that < 0. what is the p-

value?

*Johnson, Betty M.,”Decision Making, Faculty Satisfaction, and The Place of the School of Nursing in the

University,” Nursing Res.,22(1973),100-107.

Table 9.11

School Rank on faculty satisfaction Rank on decision-making agreement

A 1 12

B 7 11

C 6 10

D 2 9

E 8 8

F 4 7

G 10 6

H 12 5

I 11 4

J 5 3

K 9 2

L 3 1

Solution :

1) Hypothesis :

H 0 : X∧Y areindependent

H 1: τ<0 (claim)

x y (x,y) ranking y pairs in natural order y pairs in reverse natural order1 12 (1,12) 0 112 9 (2,9) 2 83 1 (3,1) 9 04 7 (4,7) 3 55 3 (5,3) 6 16 10 (6,10) 1 57 11 (7,11) 0 58 8 (8,8) 0 49 2 (9,2) 3 010 6 (10,6) 0 211 4 (11,4) 1 012 5 (12,5) 0 0

P = 25 Q = 41

2) Test statistic :

S=25−41=−16

¿ −1612(11)

2

=−0.2424

τ∗¿−0.303

−0.2424>−0.303

3) Decision :

Do not reject H 0

4) Conclusion :

T h ereis no enoughevidence¿ support t h eclaimt hat τ<0

Exercise 9.7

Pierce* points out that is most investigations of lightning discharges to earth, the estimated

quantity of electricity passing from the cloud to the ground is around 20 to 30 coulombs.

However, Pierce cites the data of Meese and Evans*, who reported much larger values. Their

data as reported by Pierce are shown in table 9.13, along with the distance of the observing

site of the discharge. Pierce computes a Pearson product-moment correlation coefficient of

r=0.877 and a P value of 0.01. Compute and the corresponding P value for Hi : > 0.

*Pierce, E.T.,”The Charge Transferred to Earth by a lightning Flash,” J.Franklin Inst.,286 (1968), 353-354

*Meese,A.D., and W.H.Evans.”Charge Transfer in the Lightning Stroke as Determined by the

Magnetograph,”J.Franklin Inst.,273(1963),375-382.

Table 9.13

Distance, kilometres Charge, coulombs 6 23 6 46 6 46 6 47 6 94 7 80 9 133 10 81 10 114 10 274 11 260 12 378 15 197 15 234 18 1035 23 1065

Solution :

1) Hypothesis :

H 0 : Distance of lig h ting flas h∧c h arge transfered¿ eart h are independent

H 1: Distance of lig hting flas h∧ch argetransfered ¿eart h are direcly related

τ>0(claim)

x y y pairs in natural order y pairs in reverse natural order6 23 11 06 46 11 06 46 11 06 47 11 06 94 9 27 80 10 09 133 7 210 81 6 010 114 6 010 274 3 311 260 3 212 378 2 215 197 2 015 234 2 018 1035 1 023 1065 0 0

P = 95 Q=11

2) Test statistic :S=95−11=84

T x=12

[5 (4 )+3 (2 )+2(1)]=14

T y=12

[2 (1 ) ]=1

¿ 84

√ 12

16 (15 )−14√ 12

16 (15 )−1

=0.7479

τ∗¿0.317

0.7479>0.317

3) Decision :

Reject H0

4) Conclusion :

There is enough evidence to support the claim that the distance of lighting flash &

charged transferred to earth are directly related.

Exercise (large sample)

Cravens and Woodruff * conducted a study to design and test a methodology for analytically

determining standards of sales performance. They reported the data on benchmark

achievement and management rating for 41 sales territories shown in the table 9.14. They

computed benchmark achievement as being sales volume divided by benchmark sales, and

based management ratings on salesperson motivation and effort.

We wish to compute for these data to see whether there is sufficient evidence to conclude

that benchmark achievement and management rating are directly related. Although the data

are reported as ranks, we follow the same procedure in computing as we would if the data

were reported in absolute quantities.

*Cravens, David W., and Robert B. Woodruff, “An Approach for Determining Criteria of Sales Perfomance,”J.

Appl. Psychol., 57 (1973), 242-247.

Table 9.14

Territory Benchmark achievement

(X)

Management rating (Y)

Territory Benchmark achievement

(X)

Management rating (Y)

1 2 4 22 19 16 2 9 2 23 24 23 3 7 20 24 6 22 4 23 17 25 12 12 5 5 5 26 28 34 6 17 7 27 30 41 7 16 6 28 26 38 8 25 24 29 29 36 9 4 3 30 27 32 10 10 21 31 33 29 11 20 18 32 35 31 12 15 9 33 31 26 13 8 8 34 34 28 14 11 10 35 32 30 15 1 1 36 39 33 16 21 14 37 37 35 17 14 15 38 36 37 18 3 11 39 41 39 19 13 13 40 38 40 20 18 19 41 40 27 21 22 25

Solution :

1) Hypotheses :

H 0 : Benchmark achievement and management rating are independent

H 1: Benchmark achievement and management rating are directly related (τ>0) (claim)

(x,y) in ranking y pairs in natural order y pairs in reverse natural order(1,1) 40 0(2,4) 38 2(3,11) 30 8(4,3) 36 1(5,5) 35 1(6,22) 19 16(7,20) 20 14(8,8) 30 4(9,2) 32 0

(10,21) 19 12(11,10) 27 3(12,12) 26 3(13,13) 25 3(14,15) 23 4(15,9) 24 2(16,6) 25 0(17,7) 24 0(18,19) 19 4(19,16) 21 1(20,18) 19 2(21,14) 20 0(22,25) 16 3(23,17) 18 0(24,23) 17 0(25,24) 16 0(26,38) 3 12(27,32) 8 6(28,34) 6 7(29,36) 4 8(30,41) 0 11(31,26) 10 0(32,30) 6 3(33,29) 6 2(34,28) 6 1(35,31) 5 1(36,37) 2 3(37,35) 2 2(38,40) 0 0

(39,33) 1 1(40,27) 1 0(41,39) 0 0

P = 679 Q = 140

2) Test statistic :

S=679−140=539

¿ 53941(40)

2

=0.6573

z=3 (0.6573 ) √41 (40 )

√2 (2 (41 )+5 )=6.0539

P ( z<6.0539 )=1>0.05

3) Decision :

Do not reject H0

4)Conclusion :

There is no enough evidence to support the claim that benchmark achievement and

management rating are directly related.