Tag: business economics and quantitative methods

Questions Related to business economics and quantitative methods

Height Long Jump High Jump
A 158 324 175
B 165 365 185
C 162 380 180
D 170 400 184
E 175 350 199
F 163 350 172
G 178 425 189

Find rank correlation between height & long jump and height & high jump.

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

  1. $0.9 \ and \ 0.25$

  2. $0.86 \ and\ 0.52$

  3. $0.52 \ and \ 0.86$

  4. $0.89 and \ 0.63$

Show answer
Correct Option: C
Explanation:
$Height$  $Rank(Height)$  $Long\,Jump$  $Rank(Long\,jump)$  $d$  $d^2$ 
$158$  $7$  $324$  $6$  $1$  $1$ 
$165$  $4$  $365$  $4$  $0$  $0$ 
$162$  $6$  $380$  $3$  $3$  $9$ 
$170$  $3$  $400$  $2$  $1$  $1$ 
$175$  $2$  $350$  $6$  $4$  $16$ 
$163$  $5$  $350$  $5$  $0$  $0$ 
$178$ $1$ $425$  $1$  $0$  $0$ 
          $\sum d^2=26$ 

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


    $=1-\dfrac{6\times 26}{7((7)^2-1)}$

    $=0.52$

$Height$  $Rank(Height)$  $High\,jump$  $Rank(High\,jump)$  $d$  $d^2$ 
$158$  $7$  $175$  $6$  $1$  $1$ 
$165$  $4$  $185$  $3$  $1$  $1$ 
$162$  $6$  $180$  $5$  $1$  $1$ 
$170$  $3$  $184$  $4$  $1$  $1$ 
$175$  $2$  $199$  $1$  $1$  $1$ 
$163$  $5$  $172$  $7$  $2$  $4$ 
$178$  $1$  $189$  $2$  $1$  $1$ 
          $\sum d^2=10$ 

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


    $=1-\dfrac{6\times 10}{7((7)^2-1)}$

    $=0.86$

The formula for speraman's rank coefficient is 

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

  1. $\dfrac{1}{n}\sum(x-\overline x)(y-\overline y)$

  2. $\dfrac{\dfrac{1}{n}\sum(x-\overline x)(y-\overline y)}{\sigma _x\sigma _y}$

  3. $1-\dfrac{6\sum D^2}{n(n^2-1)}$

  4. $1-\dfrac{6[\sum D^2+\frac{1}{12}(m _1^3-m _1)+frac{1}{12}(m-2^3-m _2)+....].}{n(n^2-1)}$

Show answer
Correct Option: C
Explanation:

The formula for speraman's rank coefficient is

$1-\dfrac{6\sum D^2}{n(n^2-1)}$
Where,
$n$ is the number of data points of the two variables 
$D=$ difference between ranks and $D^2=$ difference squared.

Following are the marks of $10$ students obtained in Physics and Chemistry in an examination. Find the rank-correlation coefficient.

x 43 96 74 38 35 43 22 56 35 80
y 30 94 84 13 30 18 30 41 48 95
business economics and quantitative methods linear correlation spearman's coefficient of correlation spearman's rank correlation method correlation coefficients

  1. $0.3697$

  2. $0.4673$

  3. $0.6303$

  4. $0.7834$

Show answer
Correct Option: C
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 $0.6303$

In a dance competition, the marks given by two judges to $10$ participants are given below. 

Participant A B C D E F G H I J
1st Judge 1 5 4 8 9 6 10 7 3 2
2nd Judge 4 8 7 6 5 9 10 3 2 1

Find the rank correlation coefficient.

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

  1. $0.5515$

  2. $0.4485$

  3. $0.3995$

  4. $0.2348$

Show answer
Correct Option: A
Explanation:

Rank Correlation Coefficient,$r _s=1-\cfrac{6\sum{\rho _i^2}}{n(n^2-1)}$

where, $\rho _i=$Difference between two ranks of each observation.
Here, $n=10$
$r _s=1-\cfrac{6\sum{\rho _i^2}}{n(n^2-1)}$
$=1-\cfrac{6(9+9+9+4+16+9+0+16+1+1)}{10(100-1)}$
$=1-\cfrac{6\times 74}{10\times 99}$
$=1-0.4484848$
$=0.5515$
Hence, A is correct option. 

The ranks in the statistics table are called tied ranks if 

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

  1. all ranks are unique

  2. more than one observation are equal

  3. more than one ranks are equal

  4. none of the above

Show answer
Correct Option: B,C
Explanation:

The term tie is used in connection with rank order statistics. 


Tied observations are observations having the same value, which prohibits the assignment of unique rank numbers.

Hence options $(B)$ and $(C)$ are correct.

The defects of rank correlation is/are_______________.

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

  1. the original values are taken

  2. the original values are not taken

  3. it becomes tedious to calculate if number exceeds 30

  4. both (B) and (C)

Show answer
Correct Option: D
Explanation:

Rank correlation is the technique in which ranks are provided to the data after sorting it so sometimes, it becomes very difficult to assign ranks if the variables are large in numbers and also in this, ranks are taken inspite of original values.

Rank correlation co-efficient was developed by___________.

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

  1. Karl Pearson

  2. C. Spearman

  3. Francis Cotton

  4. Carly

Show answer
Correct Option: B
Explanation:

Rank correlation coefficient is developed by Charles Spearman, a renowned statistician that measures the strength between the ranked variables.

Rank correlation is useful where____________.

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

  1. we place things in an order of merit

  2. the number of variables is more than 30

  3. there is a need to calculate the co-efficient of frequency distribution

  4. none of the above

Show answer
Correct Option: A
Explanation:

Rank correlation technique measures the strength and direction between two variables by providing suitable ranks to the concerned variables, e.g. marks of students in a class can be easily ranked.

The coefficient of rank correlation of marks obtained by 10 students in English and Economics was to be fount 0.5. It was later discovered that the difference in ranks in the two subjects obtained by one of the students was wrongly taken as 3 instead of 7. Find correct coefficient of rank correlation.

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

  1. 122.5

  2. 132.7

  3. 142.3

  4. 145.6

Show answer
Correct Option: A

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.

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

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

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

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

  4. None of the above

Show answer
Correct Option: A
Explanation:

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

Mathematics $(M)$ | Rank $(P)$ | Rank $(P)$ | $|d|$ | $d^2$ | | --- | --- | --- | --- | --- | --- | | 98 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 |

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

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

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