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Chapter 4 The Two-Sample Location Problem
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Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 1: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Chapter 4The Two-Sample Location Problem

Page 2: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 3: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 4: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

§4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model

Order the combined X-sample and Y-sample in an ascending order.

Let S1 = the rank of Y1

S2 = the rank of Y2

. ….

Sn = the rank of Yn

in the combined sample of m+n.

Page 5: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 6: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 7: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 8: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 9: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 10: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Large Sample Tests

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Page 11: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Table 4.1 Tritiated Water Diffusion Across Human Chorioamnion

Pd (10-4 cm/s)

At Term 12-26 weeks Gestational Age

0.80 1.15

0.83 0.88

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Page 12: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Example 4.1: Water Transfer in Placental Membrane. The data in Table 4.1are a portion of the data obtained by Lloyd et al. (1969). Among other things, these authors investigated whether there is a difference in the transfer of tritiated water (water containing tritium, a radioactive isotope of hydrogen) across the tissue layers in the term human chorioamnion (a placental membrane) and in the human chorioamnion between 3 to 6 months gestation age. The objective measure used was the permeability constant Pd of the human chorioamnion to water. The tissues used for the study were obtained within 5 min of delivery from the placentas of healthy, uncomplicated pregnancies in the following gestation age categories: (a) between 12 and 26 weeks following termination of pregnancy via abdominal hysterotomy (surgical incision of the uterus) for psychiatric indications; and (b) term, uncomplicated vaginal deliveries. Tissues from ten term pregnancies and five terminal pregnancies were used in the experiment. Table 4.1 gives the average permeability constant (in units of 10-4 cm/s) for six measurements on each of the 15 tissues in the study.

Page 13: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

In this example, the alternative of interest is greater permeability of the human chorioamnion for the term pregnancy. Thus, if we let X correspond to the Pd values of tissues from term pregnancies and Y to the Pd values of tissues from terminated pregnancies, we perform test (4.5), which is designed to detect the alternative Δ < 0.

For purpose of illustration we choose α to be 0.082. From Table A.6 we find w0.082 = 52. Now we list the combined sample in increasing order to facilitate the joint ranking. The ranks are given in parentheses

Page 14: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

X Y X X Y Y X Y

0.73 0.74 0.80 0.83 0.88 0.90 1.04 1.15

(1) (2) (3) (4) (5) (6) (7) (8)

Y X X X X X X

1.21 1.38 1.45 1.46 1.64 1.89 1.91

(9) (10) (11) (12) (13) (14) (15)

We see that the Y-ranks are 2,5,6,8,9 and thus

W = 2+5+6+8+9 = 30.

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Page 15: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 16: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Control SST

1042 (13) 874 (9)

1617 (23) 389 (2)

1180 (18) 612 (4)

973 (12) 798 (7)

1552 (22) 1152 (17)

1251 (19) 893 (10)

1151 (16) 541 (3)

1511 (21) 741 (6)

728 (5) 1064 (14)

1079 (15) 862 (8)

951 (11) 213 (1)

1319 (20)

Table 4.2 Alcohol Intake for 1 Year (cl of Pure Alcohol)

Page 17: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Example 4.2 : Alcohol Intakes. Eriksen, Bjornstad, and Gotestam (1986) studied a social skills training program for alcoholics. Twenty-four “alcohol-dependent” male inpatients at an alcohol treatment center were randomly assigned to two groups. The control group patients were given a traditional treatment program. The treatment group patients were given the traditional treatment program plus a class in social skills training (SST). After being discharged from the program, each patient reported – in 2-week intervals – the quantity of alcohol consumed, the number of days prior to his first drink, the number of sober days, the days worked, the times admitted to an institution, and the nights slept at home. Reports were verified by other sources (wives or family members). (Such data can be unreliable!) One patient in the SST group, discovered to be an opiate addict, disappeared after discharge and submitted no reports. The remaining 23 patients reported faithfully for a year. The results for alcohol intake are given in Table 4.2. The ranks in the joint ranking of the 23 observations are given in parentheses in Table 4.2.

Page 18: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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To test H0 vs the alternative that the SST group tends to have lower alcohol intakes, we need to test H0 : Δ = 0 vs H2 : Δ < 0. Suppose, for example, we choose α = 0.05. Then z0.05 = 1.645 and the normal approximation given by the display (4.11) is Reject H0 if W* -1.645; otherwise do not reject.

From Table 4.2, we find the sum of the SST ranks is W = 9+2+4+7+17+10+3+6+14+8+1=81

Then from equation (4.9) we obtain

Page 19: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 20: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 21: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 22: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 23: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 24: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 25: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 26: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 27: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 28: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 29: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 30: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 31: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 32: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 33: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 34: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 35: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 36: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 37: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 38: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 39: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 40: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 41: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 42: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

)jPr(J wherejJ if

),()(:H vs)()(:HReject

:procedureTest

n.m assume ,generality of lossWithout

10 xGxFxGxF

Page 43: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

Example 5.4: Effect of Feedback on Salivation Rate

The effect of enabling a subject to hear himself salivate while trying to increase or decrease his salivary rate has been studied by Delse and Feather (1968). Two groups of subjects were told to attempt to increase their salivary rates upon observing a light to the left, and decrease their salivary rates upon observing a light to the right. The apparatus for collecting and recording the amounts of saliva was described by Delse and Feather (1968) and also Feather and Wells (1966). Members of the feedback group received a 0.2-s, 1000-cps tone for each drop collected, whereas members of the no-feedback group did not receive any indication of their salivary rates. Table 5.7 gives differences of the form mean number of drops over 13 increase signals- mean number of drops over 13 decrease signals for the feedback group and the no-feedback group, each group consisting of 10 subjects.

Since the sample sizes are both equal to 10, we arbitrarily choose to label the feedback group data as the X-sample and the no-feedback group data as the Y-sample. Thus we have m = n = 10, N = (10+10)=20,

Page 44: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

and d = 10. We simultaneously illustrate the calculation of the values of the empirical distribution functions F10(t) and G10(t) at the ordered combined sample values Z(1) ≤…≤ Z(20) from Table 5.7, as well as the absolute differences | F10(Z(i) ) - G10(Z(i) )|, in the following display.

Page 45: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

0 2.071 20

8.60 19

0 8.38 18

7.74 17

5.00 16

4.29 15

3.71 14

3.25 13

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1.73 9

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0.15- 5

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Page 46: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

rate. salivationon effect an have

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Page 47: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

atoccur can functionson distributi empirical Y and X in the jumps

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Page 48: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 49: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

6 31.9,1.9,6. 3.6 ,.36

4 31.9,1.9,6. 3.6 ,.23

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Page 50: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

3 )1,1,,( (0,0,0,1)

2 )1,,,( (0,0,1,1)

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Page 51: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

:obtain weJ ofn computatio

for 5.4 Example ofapproach tabular theFollowing 5.d and 15,N

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Page 52: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

0 2.071 20

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Page 53: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 54: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 55: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 56: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 57: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 58: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 59: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

0.l)j,N(k, and l)-kk,N(j,l)-kN(j,

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Page 60: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 61: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

(2). implieswhich

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Page 62: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 63: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 64: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 65: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 66: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 67: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 68: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 69: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 70: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 71: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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Page 72: Chapter 4 The Two-Sample Location Problem. §4.1 The Wilcoxon-Mann-Whitney Rank Sum Test for the Location-Shift Model Order the combined X-sample and.

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