Tag: business economics and quantitative methods

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Questions Related to business economics and quantitative methods

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

If x, y are independent variable, then

  1. $Cov\left ( x, y \right )=1$
  2. $r _{xy}=0$
  3. $r _{xy}=1$
  4. $Cov\left ( x, y \right )=0$
Reveal answer Fill a bubble to check yourself
B,D Correct answer
Explanation

Fact. If the variables are uncorrelated or independent then covariance
and coefficient of correlation between the variable both are equal to 0
i.e. $r _{xy}=Cov\left ( x, y \right )=0$ 

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

If $n=10, \sum x=4,\sum y=3, \sum x^2=8,\sum y^2=9$ and $\sum xy=3,$ then the coefficient of $r _{x,y}$ is

  1. $\frac{3}{4}$
  2. $\frac{1}{5}$
  3. $\frac{1}{6}$
  4. $\frac{1}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Correlation coefficient 
${ r } _{ x,y }=\dfrac { n\sum { xy } -\sum { x } \sum { y }  }{ \sqrt { \left[ n\sum { { x }^{ 2 }-{ \left( \sum { x }  \right)  }^{ 2 } }  \right] \left[ n\sum { { y }^{ 2 }-{ \left( \sum { y }  \right)  }^{ 2 } }  \right]  }  } $

$=\displaystyle\frac { 30-12 }{ \sqrt { 64\times 81 }  } $
$\Rightarrow r _{x,y}=\dfrac{1}{4}$

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

FInd the rank correlation from the following data:

S. No. 1 2 3 4 5 6 7 8 9 10
Rank Differences -2 -4 -1 3 2 0 -2 3 3 -2

  1. 0.64

  2. 0.50

  3. 0.45

  4. 0.34
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Rank Difference $(d)$ | $d^2$ | | --- | --- | --- | | 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. | -2 -4 -1 3 2 0 -2 3 3 -2 | 4 16 1 9 4 0 4 9 9 4 |

 $\sum d^2=60,\quad n=10$

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

$r=1-\cfrac{6(60)}{10(10^2-1)}$

$r=1-\cfrac{360}{990}$

$r=0.6363....\approx 0.64$

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 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

calculate spearman's coefficient.

  1. $r=0.66$
  2. $r=0.32$
  3. $r=0.53$
  4. $r =0.28$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Descending order arranged data will be as follows:

Physics: $72,62,60,56,52,48,40,39,30$
MAthematics: $78,70,65,62,60,54,38,32,31$
Thus data will be

Mathematics $(M)$ | Rank $(P)$ | Rank $(P)$ | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 48 60 72 62 56 40 39 52 30 | 62 78 65 70 38 54 60 32 31 | 6 3 1 2 4 7 8 5 9 | 4 1 3 2 7 6 5 8 9 | 2 2 2 0 3 1 3 3 0 | 4 4 4 0 9 1 9 9 0 |

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

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

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

Find the spearman's rank coefficient of correlation from the following data:

X 48 33 40 9 16 16 65 25 16 57
Y 13 13 24 6 15 4 20 9 6 19
  1. $0.76$
  2. $0.52$
  3. $0.61$
  4. $0.85$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Rank | $Y$ | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 48 33 40 9 16 16 65 25 16 57 | 3 5 4 10 7 7 1 6 7 2 | 13 13 24 6 15 4 20 9 6 19 | 5 5 1 8 4 10 2 7 8 3 | 2 0 3 2 3 3 1 1 1 1 | 4 0 9 4 9 9 1 1 1 1 |

$n=10,\quad \sum d^2=39$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 39}{10(10^2-1)}=1-\cfrac{234}{990}=0.76$

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

The final position of twelve clubs in a football league and the average attendance at their home matches were as follows. Calculate a coefficient of correlation by ranks.

Club A B C D E F G H I J K L
Position 1 2 3 4 5 6 7 8 9 10 11 12
Attendance (thousands) 27 30 18 25 32 12 19 11 32 12 12 15

  1. 0.34

  2. 0.56

  3. 0.32

  4. 0.48
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Attendance | Rank | Position | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | A B C D E F G H I J K L | 27 30 18 25 32 12 19 11 32 12 12 15 | 4 3 7 5 1 9 6 12 1 9 9 8 | 1 2 3 4 5 6 7 8 9 10 11 12 | 3 1 4 1 4 3 1 4 8 1 2 4 | 9 1 16 1 16 9 1 16 64 1 4 16 |

$n=12,\quad \sum d^2=154$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 154}{12(12^2-1)}=1-\cfrac{924}{1716}=0.48$

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

Find the rank correlation coefficient between the heights of fathers and sons from the following data:

Heights of fathers in inches  65 66 67 67 68 69 70 72
Height of sons in inches 67 68 65 68 72 72 69 71
  1. $0.67$
  2. $0.58$
  3. $0.42$
  4. $0.92$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Rank | Height(Son) | Rank | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 65 66 67 67 68 69 70 72 | 8 7 5 5 4 3 2 1 | 67 68 65 68 72 72 69 71 | 7 5 8 5 1 1 4 3 | 1 2 3 0 3 2 2 2   | 1 4 9 0 9 4 4 4 |

$n=08,\quad \sum d^2=35$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 35}{8(8^2-1)}=1-\cfrac{210}{504}=0.58$

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

The marks in History and Mathematics of twelve students in a public examination are given below. Calculate a coefficient of correlation by ranks.

Student $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$ $K$ $L$
History $69$ $36$ $39$ $71$ $67$ $76$ $40$ $20$ $85$ $65$ $55$ $34$
Mathematics $33$ $52$ $71$ $25$ $79$ $22$ $83$ $81$ $24$ $35$ $46$ $64$

Interpret the result.

  1. A very good student of history is a very bad student in mathematics.

  2. A bad student of history is even bad student in mathematics.

  3. This is positive correlation

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

History $(H)$ | Mathematics$(M)$ | Rank $(H)$ | Rank$(M)$ | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | --- | | A B C D E F G H I J K L | 69 36 39 71 67 76 40 20 85 65 55 34 | 33 52 71 25 79 22 83 81 24 35 46 64 | 4 10 9 3 5 2 8 12 1 6 7 11 | 9 6 4 10 3 12 1 2 11 8 7 5 | 5 4 5 7 2 10 7 10 10 2 0 6 | 25 6 25 49 4 100 49 100 100 4 0 36   |

$n=12,\quad \sum d^2=508$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 508}{12(12^2-1)}=1-\cfrac{3048}{1716}=-0.77$

Since $r<0,$ we can say that a very good student of history is a very bad student in mathematics.

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

The marks in history and mathematics of twelve students in a public examination are given below. Calculate a coefficient of correlation by ranks.

Student A B C D E F G H I J K L
History 69 36 39 71 67 76 40 20 85 65 55 34
Mathematics 33 52 71 25 79 22 83 81 24 35 46 64


  1. -0.77

  2. -0.92

  3. 0.77

  4. 0.92
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

History $(H)$ | Mathematics$(M)$ | Rank $(H)$ | Rank$(M)$ | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | --- | | A B C D E F G H I J K L | 69 36 39 71 67 76 40 20 85 65 55 34   | 33 52 71 25 79 22 83 81 24 35 46 64 | 4 10 9 3 5 2 8 12 1 6 7 11 | 9 6 4 10 3 12 1 2 11 8 7 5 | 5 4 5 7 2 10 7 10 10 2 0 6   | 25 6 25 49 4 100 49 100 100 4 0 36   |

$n=12,\quad \sum d^2=508$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 508}{12(12^2-1)}=1-\cfrac{3048}{1716}=-0.77$

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

Following are the rank obtained by 10 students in two subjects , Statistics and Mathematics . To what extent the knowledge of the students in the two subjects is related?

Statistics 1 3 3 4 5 6 7 8 9 10
Mathematics 2 4 1 5 3 9 7 10 6 8

  1. 0.76

  2. 0.66

  3. 0.56

  4. 0.48
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mathematics $(Y)$ | $XY$ | $X^2$ | $Y^2$ | | --- | --- | --- | --- | --- | | 1 3 3 4 5 6 7 8 9 10 | 2 4 1 5 3 9 7 10 6 8 | 2 12 3 20 15 54 49 80 54 80 | 1 9 9 16 25 6 49 64 81 100 | 4 16 1 25 9 81 49 100 36 64 |

 $\sum X=56,\quad \sum Y=55,\quad \sum XY=369,\quad \sum X^2=390,\quad \sum Y^2=385$

$N=10$

Cov$(x,y)=\cfrac{\sum XY}{N}-\cfrac{\sum X}{N}.\cfrac{\sum Y}{N}=\cfrac{369}{10}-\cfrac{56}{10}.\cfrac{55}{10}=6.1$

$\sigma _x=\sqrt{\cfrac{\sum X^2}{N}-\left(\cfrac{\sum X^2}{N}\right)^2}=\sqrt{\cfrac{390}{10}-\left(\cfrac{56}{10}\right)^2}=2.76$

$\sigma _y=\sqrt{\cfrac{\sum Y^2}{N}-\left(\cfrac{\sum Y^2}{N}\right)^2}=\sqrt{\cfrac{385}{10}-\left(5.5\right)^2}=2.87$

$r=\cfrac{Cov(x,y)}{\sigma _x.\sigma _y}=\cfrac{6.1}{2.87\times 2.76}=0.77$