Tag: business economics and quantitative methods

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Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

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
  1. $0.4$
  2. $0.56$
  3. $0.67$
  4. $0.89$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$1st$ Judge | Rank | $2nd$ Judge | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | --- | | A B C D E | 5.7 5.8 5.9 5.6 5.5 | 3 2 1 4 5 | 5.6 5.7 6.0 5.5 5.8 | 4 3 1 5 2 | 1 1 0 1 3 | 1 1 0 1 9 |

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

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

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

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.

  1. 0.2

  2. 0.3

  3. 0.7

  4. 0.9
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Marks in Maths$(Y)$ | $XY$ | $X^2$ | $Y^2$ | | --- | --- | --- | --- | --- | | 35 23 47 17 10 43 9 6 28 | 30 33 45 23 8 49 12 4 31 | 1050 759 2115 391 80 2107 108 24 868 | 1225 529 2209 289 100 1849 81 35 784 | 900 1089 2025 529 64 2401 144 16 961 |

 

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

$N=09$

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

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

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

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


Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

What is correction factor(C.F) in the rank correlation coefficient.

  1. C.F $=\sum (m^{2}-1)$
  2. C.F $=\sum (m^{2}+1)$
  3. C.F $=\sum m^{2}(m^{2}-1)$
  4. C.F $=\sum m(m^{2}-1)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The correction factor in the rank correlation coefficient is given by $\sum m(m^2-1)$


where $m$ is the number of times the data repeats.
Hence, C.F $=\sum m(m^2-1)$.

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

Based on the following data, find coefficient of rank correlation.

x 43 96 74 38 35 43 22 56 35 80
y 30 94 84 13 30 18 30 41 48 95
  1. $0.3456$
  2. $0.5621$
  3. $0.6303$
  4. $0.7326$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the ranks of students obtained in Physics be $x$ and the ranks of students obtained in Chemistry be $y$.

 $X$  $Y$  Rank $X$       $(x)$  Rank $Y$      $(y)$  $d=x-y$  $d^2$
 $43$ $30$   $5.5$  $7$ $-1.5$   $2.25$
 $96$  $94$  $1$ $2$   $-1$  $1$
 $74$  $84$  $3$  $3$  $0$  $0$
 $38$  $13$  $7$  $10$  $-3$  $9$
 $35$  $30$  $8.5$  $7$  $1.5$  $2.25$
 $43$  $18$  $5.5$  $9$ $-3.5$   $12.25$
 $22$  $30$  $10$ $7$   $3$  $9$
 $56$  $41$  $4$  $5$  $-1$  $1$
 $35$  $48$  $8.5$  $4$ $4.5$   $20.25$
 $80$  $95$  $2$  $1$  $1$  $1$
       $\sum$  $0$  $58$


In the $X$ series $43$ has repeated twice and given ranks $5.5$ instead of $5$ and $6$. 

For this the correction factor is $\dfrac{2(4-1)}{12}=\dfrac{1}{2}$.

Also $35$ has repeated twice and given ranks $8.5$ instead of $8$ and $9$. For this the correction factor is $\dfrac{2(4-1)}{12}=\dfrac{1}{2}$.

In the $Y$ series $30$ has repeated thrice and given ranks $7$ instead of $6,7,8$. 

For this the correction factor is $\dfrac{3(9-1)}{12}=2$.

So, the total correction factors $C.F=\dfrac{1}{2}+\dfrac{1}{2}+2=3$

The rank correlation coefficient is given by,

$r=1-\dfrac{6(\sum d^2-C.F)}{n(n^2-1)}$
$=1-\dfrac{6(58+3)}{10(100-1)}$
$=1-\dfrac{276}{10 \times 99}$
$=1-\dfrac{366}{990}$
$=1-0.3696$
$=0.6303$
Therefore the rank correlation coefficient is $0.6303$.

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

If the correlation coefficient between $x$ and $y$ is $0.6$, covariance is $27$ and variance of $y$ is $25$, then what is the variance of $x$?

  1. $9/5$
  2. $81/25$
  3. $9$
  4. $81$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$Correlation \:coefficient = \dfrac{cov (x,y)}{std\: deviation (x) \times std\: deviation (y)}$

Let std deviation of $x$ be $x$.

We have $correlation \:coefficient=0.6, cov(x,y)=27, std\:deviation(y)=\sqrt{25}=5$

Substituting respective values

$0.6=\dfrac{27}{5\times x}$

$\Rightarrow x=9$

So variance of $x$ is $9^2=81$

Multiple choice business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

For a small group of $9$ candidates who appeared for CPT examination the sum of squares of deviation in ranks for Paper $1$ and Paper $2$ marks was found to be $44$, the rank correlation coefficient will be____.

  1. $0.51$
  2. $0.63$
  3. $0.75$
  4. $0.9$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Spearman's rank correlation r = 1 - (6 * sum(D^2)) / (n * (n^2 - 1)). Given n = 9 and sum(D^2) = 44. r = 1 - (6 * 44) / (9 * 80) = 1 - 264 / 720 = 1 - 0.366 = 0.633.

Multiple choice business economics and quantitative methods public economics components of budget and budgetary procedure government budget and taxation government budget and economy

Which among the following is a type of government budget?

  1. Balanced budget

  2. Unbalanced budget

  3. Family budget

  4. Both A & B
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

There are two types of budget in general. 

1. Balanced Budget: In this type of budget, government expenditure is equal to government revenue. 

2. Unbalanced budget: In this type of budget, government expenditure is not equal to government revenue. 

 (i) Surplus Budget: In this type of budget, government revenue is greater than government expenditure due to which there is a surplus in the budget. 

 (ii) Deficit budget: In this type of budget, government expenditure is greater than government revenue due to which there is a deficit in the budget. 

Multiple choice business economics and quantitative methods public economics components of budget and budgetary procedure government budget and taxation government budget and economy

The budget where the government revenue is equal to the government expenditure is termed as ___________.

  1. balanced budget

  2. unbalanced budget

  3. constant budget

  4. none of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Budget is an statement of the estimates of the government receipts and government expenditure during the period of the financial year. Balanced Budget is the type of budget where government expenditure is equal to government revenue.

Multiple choice business economics and quantitative methods public economics components of budget and budgetary procedure government budget and taxation government budget and economy

The classical economists believed that the government can rescue the people in times of economic distortions through unbalanced budgets.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

False. 

The classical economist advocated and proposed the concept of balanced where according to them, a balanced budget refers to a budget where the government expenditure and government revenue are equal and there is no surplus budget or deficit budget in the economy. According to them, government could rescue the people in times of distortions through balanced budgets. 

Multiple choice business economics and quantitative methods public economics components of budget and budgetary procedure government budget and taxation government budget and economy

The budgetary process in India involves _____________ different operations.

  1. three

  2. four

  3. two

  4. five
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In India, the Union Budget is prepared by the Department of Economic Affairs of Ministry of Finance. The budget has four stages viz., 

(1) estimates of expenditures and revenues, 
(2) first estimate of deficit, 
(3) narrowing of deficit and 
(4) presentation and approval of budget. 
The process begins with various ministries providing initial estimates of plan and non-plan expenditures.