Correlation and Regression MCQ Quiz in தமிழ் - Objective Question with Answer for Correlation and Regression - இலவச PDF ஐப் பதிவிறக்கவும்
Last updated on Mar 14, 2025
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Correlation and Regression Question 1:
If two regression coefficients are -0.8 and -0.2, then the value of coefficient of correlation is
Answer (Detailed Solution Below)
Correlation and Regression Question 1 Detailed Solution
The geometric mean between the two regression coefficients is equal to the correlation coefficient.
R = √(byx*bxy) = √(-0.8 * -0.2) = √0.16 = - 0.40
The '+' or '-' sign is given to the correlation coefficient based on the signs of the two regression coefficients.
Therefore, If the two regression coefficients are -0.8 and -0.2, then the value of the coefficient of correlation is - 0.40.
Correlation and Regression Question 2:
Given two lines of regression x + 3y = 11 and 2x + y = 7. Find the coefficient of correlation between x and y.
Answer (Detailed Solution Below)
Correlation and Regression Question 2 Detailed Solution
Concept:
\({\rm{Coefficient\;of\;correlation}} = {\rm{\;}}\sqrt {{b_{yx}} \times {b_{xy}}} \)
Here, byx and bxy are regression coefficients.
Or the slopes of the equation y on x and x on y are denoted as byx and bxy
Calculation:
Given two lines of regression,
x + 3y = 11
⇒ y = 11/3 − x/3
So, byx = − 1/3
Again,
2x + y = 7
⇒ x = 7/2 - y/2
So, bxy = −1/2
\({\rm{Coefficient\;of\;correlation}} = {\rm{\;}}\sqrt {{b_{yx}} \times {b_{xy}}} = \;\sqrt {\frac{{ - 1}}{3} \times \frac{{ - 1}}{2}} = \;\sqrt {\frac{1}{6}} = \;0.408\)
Since bxy and byx are negative.
r = −0.408
Note: If bxy and byx are negative then coefficient of correlation would be negative and if If bxy and byx are positive then coefficient of correlation would be positive.Correlation and Regression Question 3:
The two lines of regression are 8x - 10y = 66 and 40x - 18y = 214 and variance of x series is 9. What is the standard deviation of y series?
Answer (Detailed Solution Below)
Correlation and Regression Question 3 Detailed Solution
Concept:
1) Coefficient of correlation = \(r=√{{b_{xy}\times b_{yx}}}\)
Where byx and bxy are regression coefficients or the slope of the equation 'y on x' and 'x on y' are denoted as byx and bxy
2) Standard deviation = \(√{Varience}\)
3) \(\frac{σ_y}{σ_x}=\frac{b_{yx}}{r},\) σy and σx are the standard deviations of y and x series respectively.
Calculation:
Here, The two lines of regression are 8x - 10y = 66 and 40x - 18y = 214
For line 10y = 8x - 66
⇒ byx = 8/10 = 4/5
And, for line 40x - 18y = 214
⇒ 40x = 18y + 214
⇒ bxy = 18/40 = 9/20
Now, Coefficient of correlation = \(r=√{b_{xy}\times b_{yx}}\)
r = ± \(√({\frac{4}{5}\times\frac{9}{20})}=\pm{√{(\frac{9}{25}})}=\pm\frac{3}{5}\)
Since, bxy > 0 and byx > 0.
So, r = 3/5
Here, variance of x series is 9
Standard deviation of x series is σx = √9 = 3
We know, \(\frac{σ_y}{σ_x}=\frac{b_{yx}}{r}\)
So, \(\sigma_y =\frac{b_{yx}}{r}\times σ_x\)
⇒ \(\sigma_y =\frac{\frac{4}{5}}{\frac{3}{5}}\times 3 =4\)
∴ The standard deviation of the y series = 4.
Correlation and Regression Question 4:
Find regression line equations if means \(\bar x\) = 20 and \(\bar y = 10\), \({\sigma _x} = 10\),\({\sigma _y} = 5\), r = 0.5.
Answer (Detailed Solution Below)
Correlation and Regression Question 4 Detailed Solution
CONCEPT:
\({b_{yx}} = r.\frac{{{\sigma _y}}}{{{\sigma _x}}}\)
Where \({\sigma _x}\) = standard deviation of x; \({\sigma _y}\) = standard deviation of y
Regression Line y on x is given as \(x - \bar x = {b_{xy}}\left( {y - \bar y} \right)\)
CALCULATION:
\({b_{yx}} = r.\frac{{{\sigma _y}}}{{{\sigma _x}}} = 0.5 \times \frac{5}{{10}} = 0.25\)
Regression Line y on x is given as \(x - \bar x = {b_{xy}}\left( {y - \bar y} \right)\)
y – 10 = 0.25(x - 20)
y – 10 = 0.25x - 5
y = 0.25x – 5 + 10
y = 0.25x + 5
Correlation and Regression Question 5:
Find Correlation Coefficient from two Regression line equations X + Y = 5, X + 2Y = 3.
Answer (Detailed Solution Below)
Correlation and Regression Question 5 Detailed Solution
CONCEPT:
If there are two variables x and y we can have the following two types of regression lines,
- Regression equation of y on x (y dependent, x independent)
- Regression equation of x on y (x dependent, y independent)
- Regression equation of y on x will be \({\bf{Y}} - {\bf{\bar Y}} = {{\bf{b}}_{{\bf{yx}}}}\left( {{\bf{X}} - {\bf{\bar X}}} \right)\)
Where byx stands for regression coefficient of y on x .Here y depends on x and y is a dependent and x is an independent variable. This equation will be of the form y = ax + b and it is used to estimate the value of y given the value of x. The slope of this equation is byx.
- Regression equation of x on y will be \({\bf{X}} - {\bf{\bar X}} = {{\bf{b}}_{{\bf{xy}}}}\left( {{\bf{Y}} - {\bf{\bar Y}}} \right)\)
Where bxy stands for regression coefficient of ‘x’ on ‘y’. Here x depends on y and x is a dependent and y is an independent variable
This equation will be of the form x = by + a and it is used to estimate the value of x given the value of y
The slope of this equation is bxy.
- Correlation coefficient is the geometric mean between regression coefficients i.e., \({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)
CALCULATION
Given equations are X + Y = 5, X + 2Y = 3.
On subtracting we get ⇒ - y = 2
Putting y = -2, in equation (1)
we have x ± 2 = 5
⇒ x = 5 + 2 ⇒ x = 7
∴ x = 7and y = -2
∴ \({\rm{\bar x}} = 7{\rm{\;and\;\bar y}} = - 2\)
Suppose x + y – 5 = 0 is regression equation of y on x
⇒ 1x + 1y – 5 = 0
⇒ 1y = -1x + 5
\( \Rightarrow {\rm{y}} = \frac{{ - 1}}{1}{\rm{x}} + \frac{5}{1}\)
⇒ y = -1x + 5
∴ byx = -1
Suppose X + 2Y -3 = 0 is regression equation of x on y
⇒ 1x + 2y – 3 = 0
⇒ 1x = -2y + 3
\( \Rightarrow {\rm{x}} = \frac{{ - 2}}{1}{\rm{y}} + \frac{3}{1}\)
⇒ x = -2y + 3
∴ bxy = -2
byx⋅bxy = 2 which is > 1
So, our guessing is wrong so we interchange our guessing
Now, Suppose X + 2Y - 3 = 0 is regression equation of y on x
⇒ 1x + 2y-3 = 0.
⇒ 2y = -1x + 3
\( \Rightarrow {\rm{y}} = \frac{{ - 1}}{2}{\rm{x}} + \frac{3}{2}\)
⇒ y = -0.5x + 1.5
∴ byx=-0.5
And suppose X + Y – 5 = 0 is regression equation of x on y
⇒ 1x + 1y – 5 = 0
⇒ 1x = -1y + 5 \( \Rightarrow {\rm{x}} = \frac{{ - 1}}{1}{\rm{y}} + \frac{5}{1}\)
⇒ x = -1y + 5
∴ bxy = -1
\({\rm{r}} = {\rm{\;}}\sqrt {{{\rm{b}}_{{\rm{xy}}}}{{\rm{b}}_{{\rm{yx}}}}} = \sqrt { - 0.5 \times - 1} = \sqrt {0.5} = {\rm{\;}} - 0.707\)
Correlation and Regression Question 6:
The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is
Answer (Detailed Solution Below)
Correlation and Regression Question 6 Detailed Solution
Concept:
Coefficient of Correlation: The geometric mean between two regression coefficient is equal to coefficient of correlation i.e.,
\({\rm{r}} = \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)
Some other properties of regression coefficient:
- The regression coefficient of y on x is represented by byx and x on y as bxy.
- Both regression coefficient must have same sign.
- If one regression coefficient is greater than unity than others will be lesser than unity.
- Arithmetic mean of both regression coefficient:
- \(\frac{{{{\rm{b}}_{{\rm{yx}}}} + {{\rm{b}}_{{\rm{xy}}}}}}{2} \ge {\rm{r}}\)
Calculation:
Given: byx = 0.2 & bxy = 1.8
To find: coefficient of correlation (r)
So, coefficient of correlation = r
\(= \sqrt {0.2 \times 1.8} \)
= √(0.36)
= 0.6
Correlation and Regression Question 7:
The regression coefficients of a bivariate distribution are -0.64 and -0.36. Then the correlation coefficient of the distribution is
Answer (Detailed Solution Below)
Correlation and Regression Question 7 Detailed Solution
Concept:
\({\rm{Coefficient\;of\;correlation\;}} = {\rm{\;r\;}} = {\rm{\;}}\sqrt {{{\rm{b}}_{{\rm{yx}}}} \times {{\rm{b}}_{{\rm{xy}}}}} \) ,
Where byx and bxy are regression coefficients or the slopes of the equation y on x and x on y are denoted as byx and bxy
Note: If bxy and byx are negative then the coefficient of correlation would be negative and if If bxy and byx are positive then the coefficient of correlation would be positive.
Calculation:
Given:
bxy = -0.64 and byx = -0.36
We know that,
\(\begin{array}{l} {\rm{Coefficient\;of\;correlation\;}} = {\rm{\;r\;}} = {\rm{\;}}\sqrt {{{\rm{b}}_{{\rm{yx}}}} \times {{\rm{b}}_{{\rm{xy}}}}} \\ {\rm{r\;}} = {\rm{\;}}\sqrt { - 0.64 \times - 0.36} = \; \pm 0.48 \end{array}\)
Here bxy and byx are negative then the coefficient of correlation would be negative
∴ r = -0.48
Correlation and Regression Question 8:
The correlation coefficient between X and Y is 0.7, and random variables Z and W are defined as Z = X - 5 and W = 3Y. Find the correlation coefficient between Z and W
Answer (Detailed Solution Below)
Correlation and Regression Question 8 Detailed Solution
Concept:
Correlation coefficients are used to measure how strong a relationship is between two variables.
The coefficient of correlation does not change with a change in origin or change in scale.
Calculation:
Given, r(X, Y) = 0.7
Z = X - 5 and W = 3Y
So, r(Z, W) = r(X - 5, 3Y)
= r(X, Y)
As the correlation coefficient does not change so the coefficient will remain the same for Z and W
So, the correlation coefficient between Z and W = 0.7
Correlation and Regression Question 9:
Let X and Y represent prices (in Rs) of a commodity in Kolkata and Mumbai respectively. It is given X̅ = 65, Y̅ = 67, σX = 2.5, σY = 3.5 and r(X, Y) = 0.8. What is the equation of regression of Y on X ?
Answer (Detailed Solution Below)
Correlation and Regression Question 9 Detailed Solution
Concept:
Equation of regression of Y on X:
The equation of regression of Y on X with given values of \(\rm \bar X, \bar Y,\sigma_{x}\mbox{ and }\sigma_{y}\) is given as follows:
\(\rm \rm \dfrac{y-\bar Y}{\sigma_y} =r_{xy} \dfrac{x-\bar X}{\sigma_x} \)
Calculation:
It is given that X̅ = 65, Y̅ = 67, σX = 2.5, σY = 3.5 and r(X, Y) = 0.8 .
Using the equation of regression of Y on X we write:
\(\rm \rm \rm \frac{y-\bar Y}{\sigma_y} =r_{xy} \frac{x-\bar X}{\sigma_x} \\ \left(\frac{y-67}{3.5}\right) =0.8 \times \left(\frac{x-65}{2.5}\right)\\ y - 67 = 1.12(x-65)\\y-67=1.12x-72.8\\y=1.12x-5.8\)
Therefore, the required equation is y = 1.12x - 5.8
Correlation and Regression Question 10:
If the regression coefficient of Y on X is -6, and the correlation coefficient between X and Y is \( - \frac{1}{2},\) then the regression coefficient of X on Y would be
Answer (Detailed Solution Below)
Correlation and Regression Question 10 Detailed Solution
Concept:
- Coefficient of correlation \({\rm{\;}} = {\rm{\;r\;}} = {\rm{\;}}\sqrt {{{\rm{b}}_{{\rm{yx}}}} \times {{\rm{b}}_{{\rm{xy}}}}} \)
Where, byx and bxy are regression coefficients or the slopes of the equation y on x and x on y are denoted as byx and bxy
Calculation:
Given:
Regression coefficient of Y on X = byx = -6
And correlation coefficient between X and Y = r = -1/2
We have to find the value of regression coefficient of X on Y = bxy
We know that Coefficient of correlation\({\rm{\;}} = {\rm{\;r\;}} = {\rm{\;}}\sqrt {{{\rm{b}}_{{\rm{yx}}}} \times {{\rm{b}}_{{\rm{xy}}}}} \)
Squaring both sides, we get
⇒ r2 = byx × bxy
\( \Rightarrow {\rm{\;}}{\left( {\frac{{ - 1}}{2}} \right)^2} = {\rm{\;}} - 6{\rm{\;}} \times {\rm{\;}}{{\rm{b}}_{{\rm{xy}}}}\)
∴ bxy = -1/24