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 =

⇒ 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.

The correct answer is 0.3.
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.,

CALCULATIONS:

    (On squaring both the sides)

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 = .

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.
• 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.
• 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 = 

σ_x = √var X

Calculation:

Given, r = 0.8

Cov (X, Y) = 121

Var Y = 64

 r = 

∴ 0.8 = , 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.

CALCULATIONS:

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

 And 

           (On squaring both sides)

                      ⇒ 

∴ 

Concept:

Correlation coefficeient of x and y is given by, 

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,  

Hence, option (1) is correct.

Concept:

Formulas used:

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, 

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 

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 = - 

We know that, r = √(b_yx × b_xy)

∴ -  = √(- 8 × bxy)

Squaring both sides we get,

 = -8 × b_xy

∴ b_xy = -