Correlation Analysis MCQ Quiz in मराठी - Objective Question with Answer for Correlation Analysis - मोफत PDF डाउनलोड करा

Given

Let θ be the angle made by the line of regression of Y on X.

σY = 2σX and the correlation coefficient between X and Y  = γ =  0.3

Calculation

Let L_YX be line of regression of Y on X

Let the angle made by L_yx with x – axis be θ.

∴ Tangent of the angle made by L_yx with x – axis is tan θ.

But slope of any line with x – axis is tan θ

⇒ Slope of L_yx = b_yx

⇒ b_yx = tan θ

⇒ b_yx = γ(σ_y/σ_x)

⇒ tan θ = 0.3 × 2(σ_x/σ_x)

⇒ tan θ = 0.3 × 2

∴ The value of θ is tan^-1(0.6)

Explanation

The coefficient of regression is given as = β_xIy and β_γIx

We know that the coefficient of correlation r = ± √(b_xy × b_yx) where b_xy and b_yx are coefficient of regression

⇒ Similarly from the above options there is only 1 option that satisfied this equation 

∴ The coefficient of correlation is 

Concept:

General Formula for multiple correlation coefficient

Given

 r12 = +0.80, r13 = -0.40 and r23 = -0.56

Calculation

= 0.64  0.16 - 2 × 0.8 × (- 0.40)(-0.56)/[1 - (0.56)²]/[1 - (0.56)²

⇒ (0.80 - 0.3584)/(1 - 0.3136)

∴  = 0.6434

Given

Multiple correlation coefficient = R_1.23

Explanation

For multiple correlation coefficient,

⇒ R_1.23 ≥ r₁₂ × r₁₃ × r₂₃

From given expression we can say that

∴ The multiple correlation coefficient R_1,23 as compared to any simple correlation coefficients between the distinct variable X₁ ,X₂, and X₃ is not less than any r₁₂, r₁₃ and r₂₃

The correct answer is Both (A) and (R) are correct and (R) is the correct explanation of (A).

Key Points Assertion (A): If the securities with less than the perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly.

• When securities with less than perfect negative correlation (or even perfect negative correlation) are combined in a portfolio, the overall portfolio risk can be significantly reduced.
• Negative correlation means that the securities tend to move in opposite directions. When one security is performing poorly, the other security is likely to perform well, and vice versa.
• By holding a combination of negatively correlated securities, the fluctuations in their values tend to offset each other, leading to lower portfolio volatility and reduced overall risk.

Reason (R): The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive.

• The reason provided correctly explains why combining negatively correlated securities reduces portfolio risk.
• When calculating the total portfolio risk, the covariance (or correlation) between the different securities' price movements is taken into account.
• Negative correlation has the effect of reducing the computed value of total portfolio risk because the covariance term, which contributes to overall risk, is negative or close to zero.
• This means that the combined risk of the portfolio is lower than the sum of the individual risks of the securities.

Explanation

The correlation Coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables. The value range between -1 to 1

A calculated number greater than 1 or less than -1 means thta there was an error in the correlation measurement

Correlation 1 means a perfect positive correlation

Correlation -1 means a perfect negative correlation

0 correlation shows no linear relationship between the movement of the two variables

∴ The value of simple correlation coefficient in the interval of [-1, 1]

We are given that X, Y, Z  are uncorrelated and have variances  respectively,

Then by the definition of the correlation coefficient

Lets compute the numerator and denominator of the above expression individually

Numerator:

cov(X + Y, Y + Z) = cov(X, Y) + cov(X, Z) + cov(Y, Y) + cov(Y, Z)

cov(X + Y, Y + Z) = 0 + 0 + σ_y² + 0

cov(X + Y, Y + Z) = σy2

Denominator :

Var(X + Y) = Var(X) + Var(Y)   (Given X and Y are uncorrelated)

Var(X + Y) = σx2 + σy2

Var(Y + Z) = Var(Y) + Var(Z)   (Given Y and Z are uncorrelated)

Var(Y + Z) = σy2 + σ_z2

Finally, we have the following expression for the correlation coefficient

Hence correct option will be none of these.

Note:

If given σx2 = σy2 = σz2 = σ²  (Equal variences) then

Correlation coefficient is given as,

Given              

Formula

Correlation coefficient = n∑xy – (∑x × ∑ y)/√{[n∑x² – (∑x)²] × [n∑y² – (∑y)²}

n∑xy = arithmetic mean of the summition of product of x and y

x and y = arithmetic average of x and y series

n∑x² and n∑y² = arithmetic mean of sum of sauare of item of x and y

Calculation

Putting these value in formula

⇒ r = 5 × 93 – 21 × 17/√{[5 × 109 – (21)²] × [5 × 93 - (17)²}

⇒ r = 108/√[104 ×  176]

⇒ r = 108/135

⇒ r = 0.8

∴ The correlation coefficient between x and y are 0.8

Karl Pearson's coefficient of correlation:

• It is used to find out the linear relationship between two related variables and it is denoted by 'r'.
• It is also known as the Pearsonian coefficient of correlation and is mostly used quantitative method in practice.
• The correlation of coefficient is independent of its origin and scale.
• If the relationship between two variables X and Y is to be obtained then by origin, it means subtracting any non-zero constant from the value of X and Y the value of 'r' remains unchanged. By scale it means, there is no effect on the value of 'r' if the value of X and Y is divided or multiplied by any constant. 
• The geometric mean of two regression coefficients is equal to the coefficient of correlation.
• The geometric mean formula is the square root of the product of their regression coefficients.
• Symbolically it is represented as:

           r =   

• Where, r = coefficient of correlation, byx = the regression coefficient of y on x, bxy = the regression coefficient of x on y.

Therefore, from the above explanation, it is clear that Karl Pearson’s coefficient of correlation between two variables is the square root of the product of their regression coefficients. 

Formula

If θ be the angle between the regression lines then,

Tan θ = [(1 - r²)/r] × (σ_x ×  σ_y)/(σ_x² + σ_y²)

Explanation

According to question

⇒ σ_x = σ_y

⇒ Tanθ = [(1 - r²/r] × (σ_x²/(2σ_x)

⇒ Tanθ = [(1 - r²/r] × 1/2

⇒ Tan θ = (1 - r²/2r

⇒ Tanθ = Sinθ/cosθ

⇒ Tanθ = Sinθ × cosθ 

⇒ Sinθ = Tanθ /secθ 

⇒ Sinθ = Tanθ/√(1 + tan²θ)

⇒ Sinθ = [(1 - r²/r]/√(1 + (1 - r²/2r)²)

⇒ Sinθ = (1 - r²)/√[(1 - r²)² + 4r²]

∴