Tag: statistics

Quizzes (51) Questions (221)

Questions Related to statistics

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

State the following statement is True or False
The value of correlation coefficient is always $2$.

  1. True

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

Correlation coefficient is the measure of the degree of linear relationship between two variables, usually labelled X and Y. While in regression the emphasis is on predicting one variable from the other, in correlation the emphasis is on the degree to which a linear model may describe the relationship between two variables. In regression the interest is directional, one variable is predicted and the other is the predictor; in correlation the interest is non-directional, the relationship is the critical aspect.

$-1\le r\le 1$

Hence statement is false

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

Consider the following statements :
1. Two independent variables are always uncorrelated.
2. The coefficient of correlation between two variables X and Y is positive. When X decreases then Y decreases.
Which of the above statements is/are correct ?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Two independent variables have coefficient of correlation $0$, So they are always uncorrelated.

Positive coefficient means they both move in same direction, so if $X$ decreases then $Y$ also decreases.
So, both the statements are correct.
Hence, C is correct.

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

State the following statement is true or false

The value of coefficient of correlation is always $2$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Value of coefficient of Correlation is always between $-1 $ and $+1$, depending on the strength and direction of a linear relationship between the variables.
Value of correlation coefficient lies between $-1$ and $+1$.
Therefore, the given statement is FALSE. 
Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

Calculate the correlation coefficient between the corresponding values of X and Y in the following table:

X 2 4 5 6 8 11
Y 18 12 10 8 7 5
  1. $-0.65$
  2. $-0.82$
  3. $-0.92$
  4. $-0.48$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$X\\ 2\\ 4\\ 5\\ 6\\ 8\\ 11\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { x } =36$             $Y\\ 18\\ 12\\ 10\\ 08\\ 07\\ 05\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { y } =60$           $x=x-\overline { x } \\ -4\\ -2\\ -1\\ \quad 0\\ \quad 2\\ \quad 5\\ \ _ \ _ \ _ \ _ \ _ \ _ \\ \quad 0$              $Y=y-\overline { y } \\ \quad 8\\ \quad 2\\ \quad 0\\ -2\\ -3\\ -5\\ \ _ \ _ \ _ \ _ \ _ \\ \quad 0$               $XY\\ -32\\ -4\\ \quad 0\\ \quad 0\\ -6\\ -25\\ \ _ \ _ \ _ \ _ \ _ \ _ \\ -67$          ${ X }^{ 2 }\\ 16\\ 4\\ 1\\ 0\\ 4\\ 25\\ \ _ \ _ \ _ \ _ \ _ \\ \quad 50$         ${ Y }^{ 2 }\\ 64\\ 4\\ 0\\ 4\\ 9\\ 25\\ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \ _ \\ \sum { { Y }^{ 2 }=106 } $

Therefore, $\overline { a } =\cfrac { 36 }{ 6 } \\ \quad =6$
$\overline { y } =\cfrac { 60 }{ 6 } \\ \quad =10$

Therefore, $=\quad \cfrac { \sum { XY }  }{ \sqrt { \sum { { X }^{ 2 } } \sum { { Y }^{ 2 } }  }  } \\ =\cfrac { -67 }{ \sqrt { 50*106 }  } \\ =-0.92$
Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

State the following statement is True or False
If two regression coefficients are $0.8$ and $0.2$ respectively, then correlation coefficient is  equal to zero.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let $r _x=0.8, r _{y}=0.2$
Coeff. of correlation = Geometric Mean of Regression Coeff. 
$=\sqrt { { r } _{ x }.{ r } _{ y } } $

$=\sqrt { 0.8\times 0.2 } $ 

$=\sqrt { 0.16 } $

$\therefore $   CorrelationCoeff. $= 0.4\neq 0$

$\therefore $ The given statement is false.
Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

Two variates, x and y, are uncorrelated and have standard deviations $\sigma _x$ and $\sigma _y$ respectively. What is the correlation coefficient between x + y and x - y?

  1. $\dfrac{\sigma _x \sigma _y}{\sigma _x^2 + \sigma _y^2}$
  2. $\dfrac{\sigma _x + \sigma _y}{2\sigma _x \sigma _y}$
  3. $\dfrac{\sigma _x^2 - \sigma _y^2}{\sigma _x^2 + \sigma _y^2}$
  4. $\dfrac{\sigma _y - \sigma _x}{\sigma _x \sigma _y}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let $u = (x + y); v = (x - y)$
$\overline{u} = (\overline{x} + \overline{y}); \overline{v} = (\overline{x} - \overline{y})$

$cov(u,v) = E{(u - \overline{u})(v - \overline{u})}=E\{(x - \overline{x}) + ( y - \overline{y})\}\times \{(x - \overline{x}) - ( y - \overline{y})\}$

= $E\{(x - \overline{x} )^2 - ( y - \overline{y})^2\} = \sigma^2 _x - \sigma^2 _y$
Also, $var(u) = E\{( u - \overline{u}\} = E( x - \overline{x}) + (y - \overline{y})\}^2 = \sigma^2 _x + \sigma _y^2$

Also, $var(u) = E\{( v - \overline{v}\} = E( x - \overline{x}) + (y - \overline{y})\}^2 = \sigma^2 _x + \sigma _y^2$

Thus, $\rho=\dfrac{cov(u,v)}{\sigma _u\times\sigma _v}=\dfrac{\sigma^2 _x - \sigma^2 _y}{\sigma^2 _x + \sigma^2 _y}$
Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

The regression coefficients of a bivariate distribution are -0.64 and -0.36. Then the correlation coefficient of the distribution is

  1. 0.48

  2. -0.48

  3. 0.50

  4. -0.50
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solution:

We have,
$b _{xy}=-0.64$ and $b _{yx}=-0.36$
$\therefore$ Correlation coefficient $=\sqrt{b _{xy}\times b _{yx}}$
$=\pm\sqrt{(-0.64)(-0.36)}=\pm0.48$
$\Longrightarrow \sigma=-0.48$
[$\because b _{xy}$ and $b _{yx}$ both are negative.]
Hence, B is the correct option.

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

For two variables x and y. the two regression coefficients are $b _{x}= -\dfrac{3}{2}$ and $b _{y}=-\dfrac{1}{6}$.
The correlation coefficient between x and y is : 

  1. $-\dfrac{1}{4}$
  2. $\dfrac{1}{4}$
  3. $-\dfrac{1}{2}$
  4. $\dfrac{1}{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We know that, for two variables $x$ and $y$

$r=\pm \sqrt { { b } _{ x }{ b } _{ y } } $
Where ${b} _{x} $ and ${b} _{y}$ are the regression coefficients 
and $r$ is the correlation coefficient between $x$ and $y$
Also, the sign of $r$ is same as the sign of regression coefficients
$\therefore r=-\sqrt{\cfrac{-3}{2}\times\cfrac{-1}{6}}$
$\therefore r=\cfrac{-1}{2}$

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

If the covariance between x and y is $30$, variance of x is $25$ and variance of y is $144$, then what is the correlation coefficient?

  1. $0.4$
  2. $0.5$
  3. $0.6$
  4. $0.7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given:-
$Cov(x,y)=30$
$V(x)=25$
$V(y)=144$
As we know formula of Corelation coefficient is$:-$
Let $r$ be Corelation coefficient of $x,y$ 
Then,
$r=\dfrac{Covariance(x,y)}{\sqrt{V(x)\times V(y)}}$
on solving$:-$
$\Rightarrow r=\dfrac{30}{\sqrt{25\times 144}}$
$\Rightarrow r=0.5$
Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

For two variables $x$ and $y$ regression equations are given as $7x-3y-18=0$ and $4x-y-11=0$ then the correlation coefficient between $x$ and $y$ is 

  1. $0.7048$
  2. $0.7500$
  3. $0.7638$

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

Let $7x-3y-18=0$ represents the regression line of $y$ on $x$
$\Rightarrow y=-6+\dfrac { 7 }{ 3 } x\ \Rightarrow { b } _{ yx }=\dfrac { 7 }{ 3 } $
Then $4x-y-11=0$ is the regression line of $x$ on $y$.
$\Rightarrow x=\frac { 11 }{ 4 } +\dfrac { y }{ 4 } \ \Rightarrow { b } _{ xy }=\dfrac { 1 }{ 4 } $
Both ${ b } _{ yx }$ and ${ b } _{ yx }$ are positive. 
$\Rightarrow r=\sqrt { { b } _{ yx }\times { b } _{ xy } } \ \Rightarrow r=\sqrt { \dfrac { 7 }{ 3 } \times \dfrac { 1 }{ 4 }  } =0.7638$
So, option C is correct.