Correlation and Regression MCQ Quiz - Objective Question with Answer for Correlation and Regression - Download Free PDF

Explanation:

Lines of regression of x on y

⇒x - 3y + 4 = 0

⇒x = 3y – 4

⇒ b_xy = 3

Line of regression of y on x

⇒ 2x – 7y+8 = 0

⇒ y =\(\frac{2}{7}x + \frac{8}{7}\)

⇒ b_yx = 2/7

Now

⇒ b _xy + 7b _yx  =  3 + 7× 2 /7 = 5

∴ Option (d) is correct

The correct answer is - 0 (zero)

Key Points

• Regression Lines at Right Angle
  • When two regression lines intersect at a right angle, the angle between them is 90 degrees.
  • This occurs only if the coefficient of correlation (r) is zero.
• Coefficient of Correlation
  • The coefficient of correlation (r) measures the strength and direction of a linear relationship between two variables.
  • If r is zero, it indicates no linear relationship between the variables.

Additional Information

• Properties of Regression Lines
  • Regression lines are used to estimate the relationship between two variables.
  • The regression line of Y on X is given by Y = a + bX.
  • The regression line of X on Y is given by X = a' + b'Y.
  • If these lines are perpendicular, the product of their slopes is -1.
• Interpretation of Correlation Coefficient
  • A correlation coefficient of +1 indicates a perfect positive linear relationship.
  • A correlation coefficient of -1 indicates a perfect negative linear relationship.
  • A correlation coefficient of 0 indicates no linear relationship.
  • Values between 0 and ±1 indicate varying degrees of linear relationship strength.

\( \Rightarrow \text{Cor}(X,Y)=\dfrac{1}{\sqrt{1} \cdot \sqrt{4}}=\dfrac{1}{2} \)

Key Points 

• The correlation coefficient between 2X and 3Y is also 0.3.
• When a variable is multiplied by a constant, such as multiplying X by 2 and Y by 3, it does not affect the correlation coefficient between the two variables. The correlation coefficient is not affected by the change of scale or origin.
• The correlation coefficient measures the strength and direction of the linear relationship between two variables, and scaling or multiplying the variables by a constant does not change that relationship.

Hence if the correlation between X and Y is 0.3, the correlation coefficient between 2X and 3Y would still be 0.3. 

Explanation

The regression coefficient of  X on Y = b_xy

⇒ ∑xy/∑y²

The regression coefficient Y on X = b_yx

⇒ ∑xy/x²

Correlation coefficient is denoted by r

⇒ r = √(b_xy ⋅b _yx)

∴ The coefficient of correlation is the Geometric mean of the correlation coefficients.

CONCEPT:

Correlation coefficient is the geometric mean between regression coefficients i.e.,

\({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)

CALCULATIONS:

\({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)

\(0.8 = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}} \times 0.32} \)    (On squaring both the sides)

\({{\rm{b}}_{{\rm{yx}}}} = \frac{{0.64}}{{0.32}} = 2\)

Calculation

The coefficient of rank correlation = 1 – 6∑d²/n(n² – 1)

D = r₁ – r₂

r₁, r₂ ------ are ranks

According to spearman’s rank correlation coefficient ρ = 1 – 6∑d²/n(n² – 1)

⇒ 1 – 6 × 214/10(100 – 1)

⇒ 1 – 6 × 214/990

⇒ 1 – 1.297

∴ The value of rank correlation is approx. – 0.3

Concept:

Co-efficient of Correlation (r):
In simple linear regression analysis, the co-efficient of correlation is a statistic which indicates an association between the independent variable and the dependent variable. The co-efficient of correlation is represented by "r" and its value lies between -1.00 and +1.00.

• When the co-efficient of correlation is positive, such as +0.80, it means the dependent variable is increasing/decreasing when the independent variable is increasing/decreasing. A negative value indicates an inverse association; the dependent variable is increasing/decreasing when the independent variable is decreasing/increasing.
• A co-efficient of correlation of +0.8 or -0.8 indicates a strong correlation between the independent variable and the dependent variable. An r of +0.20 or -0.20 indicates a weak correlation between the variables. When the co-efficient of correlation is 0.00, there is no correlation.
• r = \(\rm \frac{\sum\left(x_i-\bar x\right)\left(y_i-\bar y\right)}{\sqrt{\sum\left(x_i-\bar x\right)^2\sum\left(y_i-\bar y\right)^2}}\).

Calculation:

From the properties/nature of the co-efficient of correlation, we know that the correlation coefficient is independent of the choice of origin and scale.

CONCEPT:

When r = ± 1, then

• The two regression lines become identical i.e., they coincide.
• \({b_{yx}} = \frac{1}{{{b_{xy}}}}\)
• Perfect linear co-relationship is observed and the angle between the two regression lines becomes 0°.
• For a particular value of x we shall obtain a specific value of y.

EXPLANATION:

When r = ± 1, then

• The two regression lines become identical i.e., they coincide.
• \({b_{yx}} = \frac{1}{{{b_{xy}}}}\)
• Perfect linear co-relationship is observed and the angle between the two regression lines becomes 0°.
• For a particular value of x we shall obtain a specific value of y.

Concept:

Some useful formulas are:

The correlation coefficient, r = \(\rm Cov (X, Y)\over σ_x\ \times\ σ_y\)

σ_x = √var X

Calculation:

Given, r = 0.8

Cov (X, Y) = 121

Var Y = 64

 r = \(\rm Cov (X, Y)\over σ_x\ \times\ σ_y\)

∴ 0.8 = \(121\over σ_x\ \times\ √64\), since Var X = 64, then σ_y = √64 = 8

∴ σ_x = 18.91

∴ √var X = 18.91

∴ var X = 357.59

CONCEPT:

Correlation coefficient is the geometric mean between regression coefficients i.e.\({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)

CALCULATIONS:

Given equations are x = 2y + 4 and y = kx + 6.

\({{\rm{b}}_{{\rm{yx}}}} = k\) And \({{\rm{b}}_{{\rm{xy}}}} = 2\)

\({\rm{r}} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}}{{\rm{b}}_{{\rm{xy}}}}} \)

\(\frac{1}{2} = \pm \sqrt {{{\rm{b}}_{{\rm{yx}}}} \times 2} \)           (On squaring both sides)

\(2{{\rm{b}}_{{\rm{yx}}}} = \frac{1}{4}\)                      ⇒ \({{\rm{b}}_{{\rm{yx}}}} = \frac{1}{8}\)

∴ \({{\rm{b}}_{{\rm{yx}}}} = k = \frac{1}{8}\)

Concept:

Correlation coefficeient of x and y is given by, \(\rm r = \frac{cov(x, y)}{\sqrt{V(x)\times V(y)}}\)

Where cov(x, y) = covariance between x and y, V(x) = variance of x and V(y) = variance of y

Calculation:

Here, covariance(x, y) = 12, V(x) = 64, V(y) = 36

Correlation coefficeient,  \(\rm r = \frac{cov(x, y)}{\sqrt{V(x)\times V(y)}}\)

\(=\rm \frac{12}{\sqrt{64\times36}}\)

\(=\rm \frac{12}{{8\times6}}\)

\(=\rm \frac{1}{{4}}\)

Hence, option (1) is correct.

Concept:

Formulas used:

\({\rm{r}}\left( {{\rm{x}},{\rm{y}}} \right) = \frac{{{\rm{\;Cov}}\left( {{\rm{x}},{\rm{y}}} \right)}}{{{\rm{\sigma }}\left( {\rm{x}} \right){\rm{\sigma }}\left( {\rm{y}} \right)}}\)

Where,

r (x, y) is the Correlation coefficient between x and y

Cov(x, y) Covariance of x and y

σ(x), σ(y) is the standard deviation of x, y respectively

Calculation:

Given:

Correlation coefficient between x and y, r(x, y) = 0.8014

Covariance of x and y, Cov(x, y) = ?

standard deviation y, σ(y) = (25)^1/2 = 5

standard deviation x, σ(x) = (16)^1/2 = 4

We know that, \({\rm{r}}\left( {{\rm{x}},{\rm{y}}} \right) = \frac{{{\rm{\;Cov}}\left( {{\rm{x}},{\rm{y}}} \right)}}{{{\rm{\sigma }}\left( {\rm{x}} \right){\rm{\sigma }}\left( {\rm{y}} \right)}}\)

\(Cov(x,y)=r(x,y)× \sigma(x)\sigma(y)\)

Cov (x, y) = 0.8014 × 5 × 4 = 16.028

Explanation

The regression coefficient of  X on Y = b_xy

⇒ ∑xy/∑y²

The regression coefficient Y on X = b_yx

⇒ ∑xy/x²

Correlation coefficient is denoted by r

⇒ r = √(b_xy ⋅b _yx)

∴ The coefficient of correlation is the Geometric mean of the correlation coefficients.

Concept:

Coefficient of correlation \(\rm = r = \sqrt{(b_{yx}×b_{xy} ) }\)

Where b_yx and b_xy are regression coefficients or the slopes of the equation y on x and x on y are denoted as b_yx and b_xy

Calculation:

Given b_yx = - 8

r = - \(1\over 4\)

We know that, r = √(b_yx × b_xy)

∴ - \(1\over 4\) = √(- 8 × bxy)

Squaring both sides we get,

\(1\over 16\) = -8 × b_xy

∴ b_xy = - \(1\over 128\)