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Question-1

Draw a scatter diagram and indicate the nature of correlation.

X

10

20

30

40

50

60

70

80

Y

5

10

15

20

25

30

35

40


Solution:

Comment : The diagram indicates that there is perfect negative correlation between the values of the two variables X and Y.

Question-2


Draw a scatter diagram and interpret whether the correlation is positive or negative.

X

4

5

6

7

8

9

10

11

12

13

14

15

Y

78

72

66

60

54

48

42

36

30

24

18

12


Solution:




Comment : The diagram indicates that there is perfect negative correlation between the values of the two variables X and Y.

Interpretation : The diagram indicates that there is high degree of positive correlation because the plotted points are near to each other and the trend of the points is upward

Question-3


Compute karl pearson's coefficient of correlation and interpret the result :

Marks in Mathematics

15

18

21

24

27

Marks in Accountancy

25

25

27

31

32


Solution:
Let X and Y denote marks in mathematics and accountancy

X

(xx)2

Y

(y – y)2

(xx) (y y)

15

18

21

24

27

-6

-3

0

+3

+6

36

9

0

9

36

25

25

27

31

32

-3

-3

-1

+3

+4

9

9

1

9

16

18

9

0

9

24

It indicates that there is high degree of positive correlation between marks in Mathematics and Accountancy.

Question-4

Compute coefficient of correlation from the following data :

X-series

Y-series

Mean

15

28

Sum of squares deviations from mean

144

225

Sum of products of deviations of X and Y series from their respective mean is 20. Number of pairs of observations is 10.


Solution:
Given :

Question-5


Calculate coefficient of correlation of the following data by the product Moment Method :

X

8

6

4

3

4

Y

9

7

4

4

6


Solution:

X

X2

Y

Y2

XY

8

6

4

3

4

64

36

16

9

16

9

7

4

4

6

81

49

16

16

39

72

42

16

12

24

Using product Moment Method

Question-6

Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y) :

X

65

66

67

68

69

70

71

Y

67

68

66

69

72

72

69


Solution:

 

X

Y

65

66

67

68

69

70

71

-3

-2

-1

0

+1

+2

+3

9

4

1

0

1

4

9

67

68

66

69

72

72

69

-2

-1

-3

0

+3

+3

0

4

1

9

0

9

9

0

6

2

3

0

3

6

0



Question-7


Calculate the correlation coefficient between X and Y and comment on the relationship :

X

-3

-2

-1

1

2

3

Y

9

4

1

1

4

9


Solution:
In this question the mean of X and Y comes zero or in fractions. It will create a problem in computing deviations. So here product Moment Method will be used.

X

X2

Y

Y2

XY

-3

-2

-1

1

2

3

9

4

1

1

4

9

9

4

1

1

4

9

81

16

1

1

16

81

-27

-8

-1

1

8

27

Comment : r=0 shows that there is absence of correlation between the variables X and Y. But, we observe that there remains a non-linear correlation between the two variables, ie., Y=X2. So in this question, the correlation coefficient fails to indicate the correct correlation between these two variables.

Question-8


Calculate Karl pearson's coefficient of correlation between ages of husband and wife from the following data :

Age of husband (in yrs.)

21

22

23

24

25

26

27

Age of wife (in yrs.)

16

15

17

18

19

20

21


Solution:
Let X and Y denote ages of husband and wife

X

X2

Y

y2

Xy

21

22

23

24

25

26

27

-3

-2

-1

0

1

2

3

9

4

1

0

1

4

9

16

15

17

18

19

20

21

-2

-3

-1

0

+1

+2

+3

4

9

1

0

1

4

9

6

6

1

0

1

4

9

Question-9

Calculate Karl Pearson's correlation coefficient from the following data :

X

14

15

18

20

25

30

Y

40

45

65

28

30

40

Take 20 and 40 as assumed mean for X and Y series.


Solution:

X

Y

14

15

18

20 A

25

30

-6

-5

-2

0

+5

+10

36

25

4

0

25

100

40 A

45

65

28

30

40

0

5

25

-12

-10

0

0

25

625

144

100

0

0

-25

-50

0

-50

0

Since actual means are not in whole numbers, we take 20 as assumed mean for X and 40 as assumed mean for Y.

It shows that there is weak negative correlation between X and Y.

Question-10


Calculate coefficient of correlation between age group and rate of mortality from the following data :

Age group

0-20

20-40

40-60

60-80

80-100

Rate of Mortality

350

280

540

760

900


Solution:
Since class intervals are given for age, their and values should be used for the calculation of r.

Age group

M.V.

Rate of Mor. (Y)

0-20

20-40

40-60

60-80

80-100

10

30

50 A

70

90

-40

-20

0

+20

+40

-2

-1

0

+1

+2

4

1

0

1

4

350

280

540 A

760

900

-190

-260

0

+220

+360

-19

-26

0

+22

+36

361

676

0

484

1296

38

26

0

22

72

N = 5

Question-11


From the data given below, calculate Karl pearson's coefficient of correlation between density of population and death rate :

Region

Area in Sq.Km.

Population

Deaths

A

B

C

D

200

150

120

80

40,000

75,000

72,000

20,000

480

1200

1080

280


Solution:
First of all, we shall compute density of population, i.e., population per sq.km and death rate per 1000.

Density of population =

 

Region

Density (X)

dx

Death Rate (Y)

dy

dy'

dy'2

dx'.dy'

A

B

C

D

200

500 A

600

250

-300

0

+100

-250

-6

0

+2

-5

36

0

4

25

12

16 A

15

14

-4

0

-1

-2

-4

0

-1

-2

16

0

1

4

+24

0

--2

+10

Question-12

The ranking of ten students in two subjects A and B are as below :

A

3

5

8

4

7

10

2

1

6

9

B

6

4

9

8

1

2

3

10

5

7

What is the coefficient of rank correlation ?


Solution:

R1

R2

3

5

8

4

7

10

2

1

6

9

6

4

9

8

1

2

3

10

5

7

-3

+1

-1

-4

+6

+8

-1

-9

+1

+2

9

1

1

16

36

64

1

81

1

4

N = 10

Question-13

Five competitors in a beauty contest are ranked by three judges in the following order :

Rank by Judge A

1

2

3

4

5

Rank by Judge B

2

4

1

5

3

Rank by Judge C

1

3

5

2

4

Using rank correlation coefficient, determine which pair of judges has the nearest approach to common tastes in beauty.


Solution:

R1

R2

R3

1

2

3

4

5

2

4

1

5

3

1

3

5

2

4

-1

-2

+2

-1

+2

+1

+1

-4

+3

-1

0

-1

-2

+2

+1

1

4

4

1

4

1

1

16

9

1

0

1

4

4

1

N= 5

Applying the formulae,

Since the coefficient of rank correlation is positive and highest in the judgement of the judeges a and c, we conclude that they have the similar tastes in beauty. Judges B and C have very different tastes.

Question-14

Two judges in a baby competition rank the 12 entries as follows :

Entries

A

B

C

D

E

F

G

H

I

J

K

L

X-Judge

1

2

3

4

5

6

7

8

9

10

11

12

Y-Judge

12

9

6

10

3

5

4

7

8

2

11

1

What degree of agreement is there between the judges ?


Solution:

Entry

Rank by X (R1-)

Rank by Y (R2)

D=R1-R2

D2

A

B

C

D

E

F

G

H

I

J

K

L

1

2

3

4

5

6

7

8

9

10

11

12

12

9

6

10

3

5

4

7

8

2

11

1

-11

-7

-3

6

+2

+1

+3

+1

+1

+8

0

+11

121

49

9

36

4

1

9

1

1

64

0

121

N = 12

It indicates that the judges X and Y have fairly strong divergent likes and dislikes so far as ranking of the babies is concerned.

Question-15

Given below is the percentage of marks secured by 5 students in Economics and Statistics :

Student

A

B

C

D

E

Marks in Economis

60

48

49

50

55

Marks in statistics

85

60

55

65

75

Calculate the coefficient of rank correlation.


Solution:

Marks in Eco. (X)

R1

Marks in Stat. (Y)

R2

D=R1-R2

D2

60

48

49

50

55

1

5

4

3

2

85

60

55

65

75

1

4

5

3

2

0

+1

-1

0

0

0

1

1

0

0

N = 5

It indicates that there is high degree of relationship between the marks in Economics and statistics.

Question-16

In a poem recitation Competition, ten participants were recorded following marks by two different judges Xand Y :

X

15

17

14

13

11

12

16

18

10

9

Y

15

12

4

6

7

9

3

10

2

5

Calculate the coefficient of rank correlation. [KVS 2004]


Solution:

X

R1

Y

R2

D=R1=R2

D2

15

17

14

13

11

12

16

18

10

9

4

2

5

6

8

7

3

1

9

10

15

12

4

6

7

9

3

10

2

5

1

2

8

6

5

4

9

3

10

7

+3

0

-3

0

+3

+3

-6

-2

-1

+3

9

0

9

0

9

9

36

4

1

9

N = 10

Question-17

What is correlation?

Solution:
Correlation studies and measures the direction and intensity of relationship among variables.

Question-18

Complete the following : . [NCT 2006]

Solution:

Question-19

Give the meaning of positive correlation. [NCT 2003, 2006]

Solution:
Correlation is said to be positive when the variables move together in the same direction, i.e., when X rises. Y also rises and when X falls. Y also falls.

Question-20

What is the nature of correlation between two variables when they move in the same direction?

Solution:
Positive correlation

Question-21

What is the maximum and minimum value of coefficient of correlation?

Solution:
The maximum value of coefficient of correlation (r) = + 1 and

The minimum value of coefficient of correlation (r) = - 1

Question-22

When are the two variables said to be in perfect correlation?

Solution:
When the values of both the variables under study change at a constant ratio irrespective of its direction. It is a case of perfect correlation.

Question-23

What is the nature of relationship if both the variables change in 'constant proportion'?

Solution:
Perfect correlation.

Question-24

Under what circumstances is rank correlation preferred?

Solution:
Rank correlation is preferred to Pearsonian coefficient of correlation when extreme values are present.

Question-25

What do you mean by multiple correlation ?

Solution:
When relationship among three or more than three variables is studied simultaneously, then such correlation is called multiple correlation.

Question-26

What is simple correlation?

Solution:
When the relationship between two variables is studied, then such correlation is called simple correlation.

Question-27

What is the nature of correlation when value of r is + 1 ?

Solution:
When the values of r = + 1, there is perfect positive correlation.

Question-28

State the Spearman's formula of rank correlation.

Solution:

Question-29

Can coefficient of correlation be 1.98 ? Why ?

Solution:
Coefficient of correlation (r) cannot be 1.98 as its value is either equal to 1 or less than.

Question-30

Define Karl Pearson's coefficient of correlation.

Solution:
Karl Pearson's coefficient of correlation measures the degree of relationship between the two variables X and Y. It is denoted by 'r'.

Question-31

What is scatter diagram ? What is the limitation of scatter diagram as a method of estimating correlation?

Solution:
A scatter diagram is graphic method of measuring correlation between the two variables. In scatter diagram, we plot the values of two variables as a set of points on a graph paper. The cluster of points is called scatter diagram. Scatter diagram does not give us the degree of correlation between two variables. It simply indicates the direction of correlation.

Question-32

State the formula for calculating Karl Pearson's coefficient of correlation if the deviations are taken from actual mean.

Solution:

Question-33

Mention any two properties of Karl Pearson's coefficient of correlation.

Solution:
Two properties of correlation coefficient are :

(i) Correlation coefficient always remains between -1 and +1. Symbolically

(ii) The values of coefficient of correlation (r) is unaffected by change of origin and scale.

Question-34

What is Spearman's rank correlation?

Solution:
Under Spearman's rank correlation method, correlation is measured on the basis of ranks rather than the original values of the variables.

Question-35

Name various methods of studying correlation.

Solution:
The various methods of studying correlation are :

(i) Scatter Diagram,

(ii) Karl Pearson's coefficient of correlation and

(iii) Spearman's rank correlation.

Question-36

Does correlation imply causation ?

Solution:
No, correlation does not imply causation. It implies covariation. It should never be interpreted as implying cause and effect.

Question-37

Why r is preferred to covariance as a measure of association ?

Solution:
r is preferred to covariance as measure of association because it studies and measures the direction and intensity of relationship among variables.

Question-38

What are the limits of the coefficient of correlation? Or Can r lie outside - 1 and + 1 ?

Solution:
The values of coefficient of correlation always lies between -1 and +1. Symbolically

Question-39

Give formula for calculating Karl's Pearson's coefficient of correlation if the deviation if the deviations are taken from assumed mean.

Solution:

Question-40

Give the formula to calculate rank correlation coefficient with (i) non-repeated ranks and (ii) repeated ranks.

Solution:
(i) (where ranks are not repeated)

(ii) (where ranks are repeated).

Question-41

Define covariance.

Solution:
Covariance is defined and given by:

Cov

Question-42

If the correlation between X and Y is 0.58, then what would be the correlation coefficient between

Solution:
Correlation coefficient between U and V is the same as correlation coefficient betwen X and Y, i.e., since the coefficient of correlation is unaffected by change of origin and change scale.

Question-43

What does the value of r = 0 imply ?

Solution:
r = 0 imply that there is no linear relationship between X and Y.

Question-44

If r = + 1 or r = - 1, what kind of relationship exists between X and Y ?

Solution:
r = +1 imply that there is perfect positive correlation and r = -1 imply that there is perfect negative correlation.

Question-45

When is rank correlation more precise than simple correlation coefficient?

Solution:
When the variables cannot be measured meaningfully, then in that case rank correlation is more precise than simple correlation coefficient. Ranking may be a better alternative to quantitative fraction of quantities.

Question-46

Does zero correlation mean independence ?

Solution:
No. but there is possibility of dependence among the variables. A zero value of r simply indicates that there is no linear relationship between the variables. The variables may have quadratic relationship among the variables. I.e., Y=X2.

Question-47

Can simple correlation coefficient measure any type of relationship ?

Solution:
No, it simply indicates that there is no linear relationship between the variables.

Question-48

List some variables where accurate measurement is difficult.

Solution:
Honesty, beauty, judgement, fragment secularism etc. are some examples of variables where accurate measurement is difficult.

Question-49

Why does rank correlation coefficient differ from pearsonian correlation coefficient ?

Solution:
Because rank correlation coefficient provides a measure of linear association between the ranks assigned to the values of the variables and not their values.




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