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Eleven neighborhoods have been rated by an urban sociologist on a “quality-of-life” scale (which includes measures of affluence, availability of medical care, and recreational facilities) and a social cohesion scale. The results are presented here in scores. Higher scores indicate higher quality of life and greater social cohesion. Are the two variables associated? What is the strength and direction of the association? Summarize the relationship in a sentence or two. (HINT: Don forget to square the Value of Spearman’s rho for a PRE interpretation.) Neighborhood Quality of Life Social Cohesion Beaconsdale 17 8.8 Bretwood 40 3.9 Chesapeake Shores 47 4.0 College Park 90 3.1 Denbigh Plantation 35 7.5 Kingswood 52 3.5 North End 23 6.3 Phoebus 67 1.7 Riverview 65 9.2 Queens Lake 63 3.0 Windsor Forest 100 5.3

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Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836

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Chapter
Section
BuyFindarrow_forward

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
Chapter 11, Problem 11.10P
Textbook Problem
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Eleven neighborhoods have been rated by an urban sociologist on a “quality-of-life” scale (which includes measures of affluence, availability of medical care, and recreational facilities) and a social cohesion scale. The results are presented here in scores. Higher scores indicate higher quality of life and greater social cohesion. Are the two variables associated? What is the strength and direction of the association? Summarize the relationship in a sentence or two. (HINT: Don forget to square the Value of Spearman’s rho for a PRE interpretation.)

Neighborhood Quality of Life Social Cohesion
Beaconsdale 17 8.8
Bretwood 40 3.9
Chesapeake Shores 47 4.0
College Park 90 3.1
Denbigh Plantation 35 7.5
Kingswood 52 3.5
North End 23 6.3
Phoebus 67 1.7
Riverview 65 9.2
Queens Lake 63 3.0
Windsor Forest 100 5.3

To determine

To find:

The spearman’s rho correlation.

Explanation of Solution

Given:

The table of eleven neighborhoods rated on urban sociologist on a quality of life scale and social cohesion is given below,

Neighborhood Quality of Life Social Cohesion
Beaconsdale 17 8.8
Bretwood 40 3.9
Chesapeake Shores 47 4.0
College Park 90 3.1
Denbigh Plantation 35 7.5
Kingswood 52 3.5
North End 23 6.3
Phoebus 67 1.7
Riverview 65 9.2
Queens Lake 63 3.0
Windsor Forest 100 5.3

Formula used:

The formula to calculate the Spearman’s rho is given by,

rs=16D2N(N21)

Where, D2 is the sum of the squares of the differences in ranks, and N is the number of observations.

The formula to calculate the difference in ranks is calculated as,

D1=Rank(X1)Rank(Y1)

Calculation:

Let the Quality of life be X and social cohesion be Y.

Consider the following table of differences in ranks,

Quality of Life (X) Social Cohesion (Y) Rank (X) Rank (Y) D D2
17 8.8 1 10 9 81
40 3.9 4 5 1 1
47 4.0 5 6 1 1
90 3.1 10 3 7 49
35 7.5 3 9 6 36
52 3.5 6 4 2 4
23 6.3 2 8 6 36
67 1.7 9 1 8 64
65 9.2 8 11 3 9
63 3.0 7 2 5 25
100 5.3 11 7 4 16
D=0 D2=322

Rank the highest score on each variable with 1, and so on.

The formula to calculate the difference in ranks is calculated as,

D1=Rank(X1)Rank(Y1)

Substitute 1 for X1 and 10 for Y1 in the above mentioned formula,

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