Correlation Test

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Correlation Test - 18

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Weekly Quiz Competition

Question 1

1 / -0

The marks obtained by the students in physics and in mathematics are as follows.

Marks in Physics	35	23	47	17	10	43	9	6	28
Marks in Mathematics	30	33	45	23	8	49	12	4	31

Compute of correlation of ranks.

0.2

0.3

0.7

0.9 Solution

Marks of Physics $$(X)$$

Marks in Maths$$(Y)$$

$$XY$$

$$X^2$$

$$Y^2$$

1050

759

2115

391

2107

108

868

1225

529

2209

289

100

1849

784

900

1089

2025

529

2401

144

961

$$\sum X=218,\quad \sum Y=235,\quad \sum XY=7502,\quad \sum X^2=7102,\quad \sum Y^2=8129$$

$$N=09$$

Cov$$(x,y)=\cfrac{\sum XY}{N}-\cfrac{\sum X}{N}.\cfrac{\sum Y}{N}=\cfrac{7502}{09}-\cfrac{218}{9}.\cfrac{235}{9}=201.08$$

$$\sigma_x=\sqrt{\cfrac{\sum X^2}{N}-\left(\cfrac{\sum X^2}{N}\right)^2}=\sqrt{\cfrac{7102}{09}-\left(\cfrac{218}{09}\right)^2}=14.22$$

$$\sigma_y=\sqrt{\cfrac{\sum Y^2}{N}-\left(\cfrac{\sum Y^2}{N}\right)^2}=\sqrt{\cfrac{8129}{09}-\left(\cfrac{235}{9}\right)^2}=14.88$$

$$r=\cfrac{Cov(x,y)}{\sigma_x.\sigma_y}=\cfrac{201.08}{14.22\times 14.88}=0..95$$

Question 2
1 / -0

Calculate the coefficient of correlation between $$x$$ and $$y$$ for the data
x 1 2 3 4 5 6 7 8 9 10
y 3 10 5 1 2 9 4 8 7 6

A
$$0.12$$

B
$$0.19$$

C
$$0.22$$

D
$$0.62$$

Solution

$$x\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10\\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \sum { x=55 } $$ $$y\\ 3\\ 10\\ 5\\ 1\\ 2\\ 9\\ 4\\ 8\\ 7\\ 6\\ \_ \_ \_ \_ \_ \_ \_ \\ \sum { y=55 } $$ $$X=x-\overline { x } \\ -4.5\\ -3.5\\ -2.5\\ -1.5\\ -0.5\\ \quad 0.5\\ \quad 1.5\\ \quad 2.5\\ \quad 3.5\\ \quad 4.5\\ \_ \_ \_ \_ \_ \_ \_ \\ \quad 0$$ $$XY\\ \quad 11.25\\ -15.75\\ \quad 1.25\\ \quad 6.75\\ \quad 1.75\\ \quad 1.75\\ -1.25\\ \quad 6.25\\ \quad 1.75\\ \quad 2.25\\ \_ \_ \_ \_ \_ \_ \_ \\ \quad 16\\ $$ $${ \quad X }^{ 2 }\\ 20.25\\ 12.25\\ 06.25\\ 02.25\\ 00.25\\ 00.25\\ 02.25\\ 06.25\\ 12.25\\ 20.25\\ \_ \_ \_ \_ \_ \_ \_ \\ 82.50$$

$${ \quad Y }^{ 2 }\\ 06.25\\ 20.25\\ 00.25\\ 12.25\\ 12.25\\ 02.25\\ 06.25\\ 02.25\\ 00.25\\ \_ \_ \_ \_ \_ \_ \_ \\ \quad 82.50$$

Therefore, $$\overline { x } =\cfrac { 55 }{ 10 } \\ \quad =5.5$$
$$\cfrac { \sum { y } }{ 10 } =5.5$$

Therefore, $$r=\cfrac { \sum { XY } }{ \sqrt { \sum { { X }^{ 2 }\sum { { Y }^{ 2 } } } } } \\ =\cfrac { 16 }{ 82.5 } \\ =0.19$$

Question 3

1 / -0

The marks obtained by nine students in Physics and Mathematics are given below:

Physics	$$48$$	$$60$$	$$72$$	$$62$$	$$56$$	$$40$$	$$39$$	$$52$$	$$30$$
Mathematics	$$62$$	$$78$$	$$65$$	$$70$$	$$38$$	$$54$$	$$60$$	$$32$$	$$31$$

Interpret the result.

This indicates a moderate positive relationship between marks in Physics and Mathematics.

This indicates a high positive relationship between marks in Physics and Mathematics.

This indicates a negative relationship between marks in Physics and Mathematics.

None of the above

Solution

Descending order arranged data will be as follows:

Physics: $$72,62,60,56,52,40,39,30$$

Mathematics: $$78,70,65,62,60,54,38,32,31$$

Thus data will be

Physics$$(P)$$

Mathematics $$(M)$$

Rank $$(P)$$

$$|d|$$

$$d^2$$

$$n=9,\quad \sum d^2=40$$

$$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{40\times 6}{9(9^2-1)}=1-\cfrac{240}{720}=0.66$$

Since $$r>0$$ we can say that this indicate a moderate positive relationship between marks of physics and mathematics.

Question 4
1 / -0

The given data shows test scores and sales by nine salesman during last year in a firm. Find regression of Y on X.
Test Score 14 19 24 21 26 22 15 20 19
Sales (in 000'Rs ) 31 36 48 37 50 45 33 41 39

A
$$0.66x-y=96$$

B
$$1.61x-y+7.83=$$

C
$$.25x+2y=5$$

D
$$x-0.5y=45$$

Solution

Question 5

1 / -0

In a skating competition the judges gave the five competitors the following marks. Calculate a coefficient of rank correlation.

Competitors	A	B	C	D	E
1st judge	5.7	5.8	5.9	5.6	5.5
2nd judge	5.6	5.7	6.0	5.5	5.8

$$0.4$$

$$0.56$$

$$0.67$$

$$0.89$$

Solution

Competition

$$1st$$ Judge

Rank

$$2nd$$ Judge

Rank

$$|d|$$

$$d^2$$

5.7

5.8

5.9

5.6

5.5

5.6

5.7

6.0

5.5

5.8

$$n=5,\quad \sum d^2=12$$

$$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 12}{5(5^2-1)}=1-\cfrac{72}{240}=0.44$$

Question 6
1 / -0

If the dependent variable increases as the independent variable increases in an estimating equation, the coefficient of correlation will be in the range of _________.

A
0 to (-) 1

B
0 to (-) 0

C
0 to (-) 0.05

D
0 to 1
Question 7
1 / -0

Match the following items in List - I with most suitable options in List - II:
List-I List-II
(a) Fisher 1. Inverse probability
(b) Karl Pearson 2. Normal Distribution
(c) Thomas Baye's 3. Correlation Coefficient
(d) Karl Gauss 4. Index Numbers

A
$$(a) - 4, (b) - 3, (c) - 2, (d) - 1$$

B
$$(a) - 4, (b) - 3, (c) - 1, (d) - 2$$

C
$$(a) - 4, (b) - 2, (c) - 3, (d) - 1$$

D
$$(a) - 4, (b) - 2, (c) - 1, (d) - 3$$
Question 8
1 / -0

Karl Pearson's co-efficient of correlation between two variables is _____________.

A
the product of their standard deviations

B
the square root of the product of their regression co-efficient

C
the co-variance between the variables

D
none of the above

Question 9

1 / -0

Following are the ranks of $$10$$ students in Maths and Biology projects given by $$2$$ professors. Find the rank correlation coefficient.

Maths	4	8	5	3	1	2	10	7	6	9
Biology	1	4	5	6	2	3	8	9	7	10

$$0.2787$$

$$0.7213$$

$$0.8232$$

$$0.9031$$

Solution

Let the ranks of students obtained in Maths be $$x$$ and the ranks of students obtained in Biology be $$y$$.

$$x$$	$$4$$	$$8$$	$$5$$	$$3$$	$$1$$	$$2$$	$$10$$	$$7$$	$$6$$	$$9$$
$$y$$	$$1$$	$$4$$	$$5$$	$$6$$	$$2$$	$$3$$	$$8$$	$$9$$	$$7$$	$$10$$
$$d=x-y$$	$$3$$	$$4$$	$$0$$	$$-3$$	$$-1$$	$$-1$$	$$2$$	$$-2$$	$$-1$$	$$-1$$
$$d^2$$	$$9$$	$$16$$	$$0$$	$$9$$	$$1$$	$$1$$	$$4$$	$$4$$	$$1$$	$$1$$
					$$\sum d^2$$ $$=46$$

The rank correlation coefficient is given by

$$r=1-\dfrac{6\sum d^2}{n(n^2-1)}$$

$$=1-\dfrac{6 \times 46}{10(100-1)}$$

$$=1-\dfrac{276}{10 \times 99}$$

$$=1-\dfrac{276}{990}$$

$$=1-0.2787$$

$$=0.7213$$

Therefore, the rank correlation coefficient is $$0.7213$$.

Question 10
1 / -0

If bxy$$=0.25$$ and byx$$=0.64$$, correlation coefficient is _________.

A
$$0.16$$

B
$$0.40$$

C
$$0.89$$

D
$$0.30$$

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Re-Attempt Weekly Quiz Competition

Test Score	14	19	24	21	26	22	15	20	19
Sales (in 000'Rs )	31	36	48	37	50	45	33	41	39

List-I	List-II
(a) Fisher	1. Inverse probability
(b) Karl Pearson	2. Normal Distribution
(c) Thomas Baye's	3. Correlation Coefficient
(d) Karl Gauss	4. Index Numbers

Correlation Test - 18

Result

Correlation Test - 18

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Question 1

0.2

0.3

0.7

0.9

Solution

Question 2

$$0.12$$

$$0.19$$

$$0.22$$

$$0.62$$

Solution

Question 3

This indicates a moderate positive relationship between marks in Physics and Mathematics.

This indicates a high positive relationship between marks in Physics and Mathematics.

This indicates a negative relationship between marks in Physics and Mathematics.

None of the above

Solution

Question 4

$$0.66x-y=96$$

$$1.61x-y+7.83=$$

$$.25x+2y=5$$

$$x-0.5y=45$$

Solution

Question 5

$$0.4$$

$$0.56$$

$$0.67$$

$$0.89$$

Solution

Question 6

0 to (-) 1

0 to (-) 0

0 to (-) 0.05

0 to 1

Question 7

$$(a) - 4, (b) - 3, (c) - 2, (d) - 1$$

$$(a) - 4, (b) - 3, (c) - 1, (d) - 2$$

$$(a) - 4, (b) - 2, (c) - 3, (d) - 1$$

$$(a) - 4, (b) - 2, (c) - 1, (d) - 3$$

Question 8

the product of their standard deviations

the square root of the product of their regression co-efficient

the co-variance between the variables

none of the above

Question 9

$$0.2787$$

$$0.7213$$

$$0.8232$$

$$0.9031$$

Solution

Question 10

$$0.16$$

$$0.40$$

$$0.89$$

$$0.30$$

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