Question

The table below gives the scores of Students on a English Examinations

under KCSE

Student

1

2

3

4

5

6

7

8

9

Oral test

B-

B+

B

A-

A

B

C+

B

C-

Composition

10

21

22

19

17

14

13

16

12

Literature

34

76

74

60

68

44

45

53

43

(Composition test is out of 30 and literature test is out of 100 marks)

(a) Calculate the value of the most appropriate measure of correlation between the results

in the oral and composition tests, justifying your choice of measure. Interpret the value you

obtain.

(b) Calculate the value of the most appropriate measure of correlation between the results in

the composition and literature tests, justifying your choice of measure. Interpret the value

you obtain.

Answer 1

(a) The ranks of oral test and composition are:

Spearman's correlation coefficient:

$\bar{x}=5\\\bar{y}=5\\\sum{\left( x_i-\bar{x} \right) ^2}=58\\\sum{\left( y_i-\bar{y} \right) ^2}=60\\\sum{\left( x_i-\bar{x} \right) \left( y_i-\bar{y} \right)}=42\\\rho =\frac{\sum{\left( x_i-\bar{x} \right) \left( y_i-\bar{y} \right)}}{\sqrt{\sum{\left( x_i-\bar{x} \right) ^2}\sum{\left( y_i-\bar{y} \right) ^2}}}=\frac{42}{\sqrt{58\cdot 60}}=0.711967$

There is a strong positive correlation between the scores.

(b)

Pearson's correlation coefficient:

$\bar{x}=16\\\bar{y}=55.22222\\\sum{\left( x_i-\bar{x} \right) ^2}=136\\\sum{\left( y_i-\bar{y} \right) ^2}=1805.556\\\sum{\left( x_i-\bar{x} \right) \left( y_i-\bar{y} \right)}=473\\\rho =\frac{\sum{\left( x_i-\bar{x} \right) \left( y_i-\bar{y} \right)}}{\sqrt{\sum{\left( x_i-\bar{x} \right) ^2}\sum{\left( y_i-\bar{y} \right) ^2}}}=\frac{473}{\sqrt{136\cdot 1805.556}}=0.954523$

There is a strong positive correlation between the scores.