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Correlation coefficients - class-XI

Description: correlation coefficients
Number of Questions: 17
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Tags: mathematics and statistics linear correlation skewness correlation analysis correlation and regression analysis business maths introduction to statistical method and econometrics statistics correlation economics skewness of frequency distribution
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 Given  $\sum xy=120, \sigma _x=8  ,\sum x^2=90,\sigma _ y=7 ,n=10$ ; where x and y are deviations from mean, the  correlation coefficient is 

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. $0.21$

  2. $0.9$

  3. $0.36$

  4. $0.6$

Show answer
Correct Option: A

Karl Pearson's coefficient of skewness of a distribution is 0.32.Its s.d.is 6.5 and mean is 29.6.The mode and median of the distribution are

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. 27.52,28.91

  2. 26.92,27.23

  3. 25.67,26.34

  4. none of these

Show answer
Correct Option: A
Explanation:

Karl Pearson's coefficient of skewness $\displaystyle =\frac{Mean-Mode}{S.D.}$ $\displaystyle \therefore 0.32=\frac{29.6-Mode}{6.5}\Rightarrow Mode=27.52$ Also Karl Pearson's coeff.of skewness $\displaystyle =\frac{3\left ( Mean-Median \right )}{S.D}$ $\displaystyle \because 0.32=\frac{3\left ( 29.6-Median \right )}{6.5}$ $\displaystyle \Rightarrow Median=28.91$

The sum of the deviations of the variates 6,8,10,16,20,24 

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. -1

  2. 1

  3. 0

  4. 4

Show answer
Correct Option: C
Explanation:

$\Rightarrow$   $Mean = \dfrac{6+8+10+16+20+24}{6}=14$

$\Rightarrow$   Sum of the deviations = $(6-14)+(8-14)+(10-14)+(16-14)+(20-14)+(24-14)$
$\Rightarrow$   Sum of the deviation = $-8-6-4+2+6+10$
$\therefore$    Sum of the deviation = $-18+18$
$\therefore$    Sum of the deviation = $0$

Choose the statement which consists of two correlated variables.

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. Increase in the intensity of cold results in greater sale of woollen clothes

  2. Increase in temperature of delhi has led to congestion

  3. Increase in weight of children s accompanied by increase in weight of their mother

  4. None of the above

Show answer
Correct Option: A
Explanation:

Since increase in intensity of cold result in greater scale of woolen clothes,

Calculate Pearson's coefficient of correlation between the values of $X$ and $Y$.

X 1 2 3 4 5
Y 7 6 5 4 3
statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. $-0.3$

  2. $0.3$

  3. $1$

  4. $-1$

Show answer
Correct Option: D
Explanation:
$X\\ 1\\ 2\\ 3\\ 4\\ 5\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \overline { x } \cfrac { 15 }{ 5 } =3$                $Y\\ 7\\ 6\\ 5\\ 4\\ 3\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \overline { y } =\cfrac { 25 }{ 5 } =5$                  $Y=y-\overline { y } \\ \quad 02\\ \quad 01\\ \quad 00\\ -1\\ -1\\ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 0$                    $XY\\ -4\\ -1\\ \quad 0\\ -1\\ -4\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { XY=-10 } $                 ${ X }^{ 2 }\\ 4\\ 1\\ 0\\ 1\\ 4\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { { X }^{ 2 }=10 } $        

${ Y }^{ 2 }\\ 4\\ 1\\ 0\\ 1\\ 4\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { { Y }^{ 2 }=10 } $

Therefore, $r=\cfrac { \sum { XY }  }{ \sqrt { \sum { { X }^{ 2 }\sum { { Y }^{ 2 } }  }  }  } \\ =\cfrac { -10 }{ \sqrt { 10*10 }  } \\ =\cfrac { -10 }{ 10 } \\ =-1.$

For $ n=25,\sum x=125,\sum x^2=650,\sum y=100,\sum y^2=460,\sum xy=508$, correlation coefficient is 

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. $0.99$

  2. $0.207$

  3. $0.66$

  4. $0.89$

Show answer
Correct Option: B
Explanation:

$n=25$

$\sum x=125,\ \sum{{x}^{2}}=650,\ \sum{xy}=334$  $\sum y=100,\ \sum{{y}^{2}}=508$
$r=\cfrac { n\sum { xy } -(\sum { x } \times \sum { y } ) }{ \sqrt { (n\sum { { x }^{ 2 } } -\sum { { x }^{ 2 } } )(n\sum { { y }^{ 2 } } -\sum { { y }^{ 2 } } ) }  } =\cfrac { 25\times 508 -100\times 125 }{ \sqrt { (25\times 650 -{ 125 }^{ 2 }  )(25\times 460 -{100}^{2}) }  }=0.207 $

$n=25,\sum x=125,\sum x^2=650, \sum y=100,\sum y^2=460,\sum xy=508$. It was observed that two pair of values of $(x,y)$ were copied as $ (6,14)$ and $(8,6) $ instead of $(8,12),(6,8).$ The correct correlation coefficient is 

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. $0.667$

  2. $0.87$

  3. $-0.25$

  4. $0.356$

Show answer
Correct Option: A
Explanation:

Corrected $\sum x=125-6-8+8+6=125$

Corrected $\sum y=100-14-6+12+8=100$
Corrected $\sum x^2=650-(6)^2-(8)^2+(6)^2+(8)^2=650$
Corrected $\sum y^2=460-(14)^2-(6)^2+(12)^2+(8)^2=436$
Corrected $\sum xy=508-6\times 18-8\times 6+8\times 12+6\times 8=520$
The formula for Pearson product moment correlation is
$r=\dfrac{n\sum xy-(\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}$

    $=\dfrac{25\times 520-125\times 100}{\sqrt{[25\times 650-(125)^2][25\times 436-(100)^2]}}$

   $=\dfrac{13000-12500}{\sqrt{[16250-15625][10900-10000]}}$

   $=\dfrac{500}{\sqrt{[625][900]}}$

   $=\dfrac{500}{(25)(30)}$

   $=\dfrac{2}{3}$

   $=0.667$

The coefficient of correlation when coefficients of regression are $0.2$ and $1.8$ is

statistics skewness of frequency distribution karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. $0.36$

  2. $0.2$

  3. $0.6$

  4. $0.9$

Show answer
Correct Option: C
Explanation:

$Given\quad { b } _{ xy }=0.2\\ { b } _{ yx }=0.8\\ r=\sqrt { { b } _{ xy }\times { b } _{ yx } } \\ r=\sqrt { 0.36 } \\ r=0.6$

Where r is coefficient of correlation.

Standard error of regression analysis is classified as

business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. average of coefficient

  2. variance of residual

  3. mean of residual

  4. average of residual

Show answer
Correct Option: B
Explanation:

Standard error of regression analysis is classified as variance of residuals.

Variance of residuals also known as error variance.

The independent variable is also called: 

business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. Regressor

  2. Regressand

  3. Predictand

  4. None of these

Show answer
Correct Option: A
Explanation:

A linear regression line has an equation of the form $Y=a+bX$


where $Y$ is called dependent variable or response or regressand
$X$ is called independent variable or predictors or explanatory variable or regressor
$a$ is the y-intercept and
$b$ is the slope of the line.

To determine the height of a person when his weight is given is:

business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. Correlation problem

  2. Regression problem

  3. Association problem

  4. None of these

Show answer
Correct Option: B
Explanation:

A regression analysis gives the relationship between the variables. If the weight of the person is given then to determine the height of a person we formulate it as a regression problem. 

A linear regression line has an equation of the form $Y=a+bX$

where $Y$ is called dependent variable or response or regressand
$X$ is called independent variable or predictors or explanatory variable or regressor
$a$ is the y-intercept and
$b$ is the slope of the line.

A process by which we estimate the value of dependent variable on the basis of one or more independent variables is called: 

business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. Correlation

  2. Regression

  3. Residual

  4. Slope

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  A process by which we estimate the value of dependent variable on the basis of one or more independent variables is called: $Regression$.

$\Rightarrow$  More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed.
$\Rightarrow$  Regression takes a group of random variables, thought to be predicting $Y$, and tries to find a mathematical relationship between them. 

If all conditions or assumptions of regression analysis then simple regression can give

business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

  1. dependent estimation

  2. reliable estimates

  3. independent estimation

  4. unreliable estimates

Show answer
Correct Option: B
Explanation:

The simple regression gives the reliable estimates, if all the conditions or assumptions of the regression analysis are satisfied.

In regression analysis, testing of assumptions if these are true or not is classified as

    business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

    1. weighted analysis

    2. average analysis

    3. significance analysis

    4. specification analysis

    Show answer
    Correct Option: D
    Explanation:

    In regression analysis, testing of assumptions if these are true or not is classified as specification analysis. It refers to the determination of which independent variables to be included or excluded from a regression equation.

    The dependent variable is also called: 

    business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

    1. Regression

    2. Continuous variable

    3. Regressand

    4. None of these

    Show answer
    Correct Option: C
    Explanation:

    A linear regression line has an equation of the form $Y=a+bX$


    where $Y$ is called dependent variable or response or regressand
    $X$ is called independent variable or predictors or explanatory variable or regressor
    $a$ is the y-intercept and
    $b$ is the slope of the line.

    For the variables $x$ and $y$, the two regression lines are $6x+y=30$ and $3x+2y=25$. What are the value of $\bar{x},\bar{y}$ and $r$ respectively ?

    business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

    1. $\dfrac{20}{3},\dfrac{35}{9},-0.5$

    2. $\dfrac{20}{3},\dfrac{35}{9},0.5$

    3. $\dfrac{35}{9},\dfrac{20}{3},-0.5$

    4. $\dfrac{35}{9},\dfrac{20}{3},0.5$

    Show answer
    Correct Option: D
    Explanation:

    $6x+y=30$                    ----- ( 1 )

    $3x+2y=25$                  ----- ( 2 )
    Multiplying  equation ( 1 ) by $2$ we get,
    $\Rightarrow$  $12x+2y=60$         ---- ( 3 )
    Subtracting equation ( 2 ) from equation ( 3 ) we get,
    $\Rightarrow$  $9x=35$
    $\therefore$  $x=\dfrac{35}{9}$
    Substituting value of $x$ in equation ( 1 ) we get
    $\Rightarrow$  $6\times\dfrac{35}{9}+y=30$
    $\Rightarrow$  $\dfrac{70}{3}+y=30$
    $\Rightarrow$  $y=30-\dfrac{70}{3}$
    $\Rightarrow$  $y=\dfrac{90-70}{3}$
    $\therefore$  $y=\dfrac{20}{3}$

    $\therefore$  $\overline{x}=\dfrac{35}{9}$ and $\overline{y}=\dfrac{20}{3}$
    Now, we are going to find $r:$
    $6x+y=30$                            [ Given ]
    $6x=30-y$
    $x=\dfrac{30}{6}-\dfrac{y}{6}$
    $\Rightarrow$  $b _{xy}=\dfrac{-1}{6}$
    $3x+2y=25$              [ Given ]
    $2y=25-3x$
    $y=\dfrac{25}{2}-\dfrac{3}{2}x$
    $\Rightarrow$  $b _{yx}=\dfrac{-3}{2}$
    $\Rightarrow$  Coefficient of correlation $(r)=\sqrt{b _{xy}\times b _{yx}}$
                                                            $=\sqrt{\dfrac{-1}{6}\times \dfrac{-3}{2}}$

                                                            $=\sqrt{\dfrac{1}{4}}$

                                                            $=\dfrac{1}{2}$

                                                            $=0.5$
    $\therefore$  $r=0.5$

    Who has coined the term Regression?

    business maths correlation and regression analysis karl pearson coefficient of correlation karl pearson's product moment method karl's pearson method

    1. Francis Galton

    2. R.A. Fisher

    3. Daniel

    4. Karl Pearson

    Show answer
    Correct Option: A
    Explanation:

    Regression is a technique for determining the statistical relationship between two or more variables where a change in a dependable variable is associated with and depends on a change in one or more independant variables. The term regression has been coined by Francis Galton.

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