Test: Correlation - 2 - Commerce MCQ

Concept:
The correlation coefficient is statistical measure of the strength of the relationship between the relative moments of two variables.
The values range between  -1.0  and  1.0.
It is determined by dividing the covariance by the product of the two variables standard deviation. Thus,
Coefficient of correlation Does not depends on scale and origin.

Calculation:
In the given question, we are given that Coefficient of correlation between the ages of husband and wife at the time of marriage for a set of 100 couples was 0.7
and According to the question, If couples survive to celebrate their silver jubilee, then 
Since, Coefficient of correlation doesn't changes on scale and time
Thus, the Coefficient of correlation at that point will be same as it was earlier 
⇒ The Coefficient of correlation = 0.7

Concept:
Regression coefficients 

Regression coefficients

Calculation:

We know that 


∴ The regression coefficients b_yx and b_xy are -0.216 and -0.129, respectively.

Concept:
The regression coefficients bxy & byx are the change occurring in x for a unit change in y and y for a unit change in x respectively

Formulae:
Regression coefficient of X on Y, 
Coefficient of correlations 

Calculation:
X - X̅ = bxy(Y - Y̅)
⇒ X - 4 = 0.929(Y - 11)
⇒ X - 4 = 0.929Y - 10.219
⇒ X = 0.929Y - 6.219
∴ Regression equation of X on Y is X = 0.929Y - 6.219

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.
• 

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:
Correlation coefficient is the geometric mean between regression coefficients

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

 (On squaring both sides)
  

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

Concept:
Some useful formulas are:
The correlation coefficient,

σ_x = √var X

Calculation:
Given, r = 0.8
Cov (X, Y) = 121
Var Y = 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,

Calculation:

Concept:
Covariance is defined for the pair of random variables which is associated or related to each other. It is the average product of individual deviation from the corresponding means.

Calculation:

n = 9, ΣX_i = 45, ΣY, = 44

Concept:
Coefficient of correlation = 
Here, 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 bxy

Calculation:
Given two lines of regression,
x + 3y = 11
⇒ y = 11/3 − x/3
So, b_yx = − 1/3
Again,
2x + y = 7
⇒ x = 7/2 - y/2
So, bxy = −1/2
Coefficient of correlation

Since b_xy and b_yx are negative.
r = −0.408
Note: If b_xy and b_yx are negative then coefficient of correlation would be negative and if If b_xy and b_yx are positive then coefficient of correlation would be positive.

Variance is independent of change of origin as the change in origin is uniformly added to all the values and hence the mean also and hence, when  is calculated, it remains unaffected. But, change of scale alters all values unevenly and hence, variation changes. 
The coefficient of variance is standard deviation by mean which doesn't depend on the unit of observation.
So, Statement 1 is correct Statement 2 is incorrect.

Concept:
Correlation Coefficient: The correlation coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables.

Coefficient of correlation: The correlation coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables.
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 
Note: If b_xy and b_yx 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 that: b_yx = -3/2 and b_xy = -1/6
The correlation coefficient 

Since both byx and bxy are negative, so the correlation coefficient will also be negative.
Hence r = -1/2

Concept:

The line of regression of y on x is given by:  where b_yx is called the regression coefficient of y on x.
Similarly, the line of regression of x on y is given by:  where bxy is called the regression coefficient of x on y.
The correlation coefficient r² = b_yx × b_xy
The two lines of regression intersect each other at 

Calculation:
Given:
x̅ = 25, y̅ = 120, b_xy = 2, y = 130, x = ?
The line of regression of x on y is given by: 
x - 25 = 2 × (130 - 120)
x - 25 = 20
x = 45

Concept:
The regression coefficients b_xy & _byx are the change occurring in x for a unit change in y and y for a unit change in x respectively
Formulae
Regression coefficient of X on Y, b_xy  
Coefficient of correlations 

Calculation:
Given:
Regression equation of Y on X is Y= 0.929X + 7.284.
So, b_yx = 0.929
Regression equation of X on Y is X = 0.929Y - 6.219.
So, b_xy = 0.929
We know that correlation coefficient, 

∴ Correlation coefficient is less than 1.

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

Calculation:

 (On squaring both the sides)