Direction: Consider the following for the next two (02) items that follow. Two regression lines are given as 3x - 4y + 8 = 0 and 4x - 3y - 1 = 0.
Consider the following statements: 1. The coefficient of correlations r is \(\rm \frac{3}{4}\) . 2. The means of x and y are 3 and 4 respectively. Which of the above statements is/are correct?
1 only
The problem provides two regression lines and asks us to evaluate two statements regarding the coefficient of correlation and the means of x and y.
Understanding Regression Lines and Statistics
Regression lines are used in statistics to model the relationship between two variables. The intersection point of the two regression lines (regression line of y on x and regression line of x on y) represents the means of the two variables, \(\bar{x}\) and \(\bar{y}\). The slope of these lines is related to the regression coefficients, which in turn are related to the coefficient of correlation (r).
Finding the Means (\(\bar{x}\), \(\bar{y}\))
The means of x and y are the coordinates of the point where the two regression lines intersect. We can find this point by solving the given system of linear equations:
- \(3x - 4y + 8 = 0 \implies 3x - 4y = -8\) (Equation 1)
- \(4x - 3y - 1 = 0 \implies 4x - 3y = 1\) (Equation 2)
We can solve this system using the elimination method. Multiply Equation 1 by 4 and Equation 2 by 3 to make the coefficients of x equal:
- \(4 \times (3x - 4y) = 4 \times (-8) \implies 12x - 16y = -32\) (Equation 3)
- \(3 \times (4x - 3y) = 3 \times (1) \implies 12x - 9y = 3\) (Equation 4)
Now, subtract Equation 4 from Equation 3:
\[ (12x - 16y) - (12x - 9y) = -32 - 3 \]
\[ 12x - 16y - 12x + 9y = -35 \]
\[ -7y = -35 \]
\[ y = \frac{-35}{-7} = 5 \]
Substitute the value of y = 5 into Equation 2:
\[ 4x - 3(5) = 1 \]
\[ 4x - 15 = 1 \]
\[ 4x = 1 + 15 \]
\[ 4x = 16 \]
\[ x = \frac{16}{4} = 4 \]
Thus, the means are \(\bar{x} = 4\) and \(\bar{y} = 5\). Statement 2 says the means of x and y are 3 and 4 respectively. This is incorrect.
Finding the Coefficient of Correlation (r)
The slopes of the regression lines are the regression coefficients. Let's assume one equation is the regression of y on x and the other is the regression of x on y.
From Equation 1 (\(3x - 4y + 8 = 0\)), express y in terms of x:
\[ 4y = 3x + 8 \implies y = \frac{3}{4}x + 2 \]
The slope of this line (assuming it's y on x) is the regression coefficient of y on x, denoted as \(b_{yx}\). So, \(b_{yx} = \frac{3}{4}\).
From Equation 2 (\(4x - 3y - 1 = 0\)), express x in terms of y:
\[ 4x = 3y + 1 \implies x = \frac{3}{4}y + \frac{1}{4} \]
The slope of this line (assuming it's x on y) is the regression coefficient of x on y, denoted as \(b_{xy}\). So, \(b_{xy} = \frac{3}{4}\).
The coefficient of correlation, r, is the geometric mean of the two regression coefficients: \(r^2 = b_{yx} \times b_{xy}\). The sign of r is the same as the sign of the regression coefficients.
\[ r^2 = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16} \]
\[ r = \pm\sqrt{\frac{9}{16}} = \pm\frac{3}{4} \]
Since both \(b_{yx} = \frac{3}{4}\) and \(b_{xy} = \frac{3}{4}\) are positive, the correlation coefficient r must also be positive.
\[ r = +\frac{3}{4} \]
We can check if this assignment of regression lines was correct. For valid regression coefficients, their product \(b_{yx} \times b_{xy}\) must be less than or equal to 1 (which implies \(r^2 \le 1\)). In our case, \(r^2 = \frac{9}{16} \le 1\), so the assignment is consistent.
Statement 1 says the coefficient of correlation r is \(\frac{3}{4}\). This is correct.
Evaluating the Statements
Based on our calculations:
- Statement 1: The coefficient of correlations r is \(\frac{3}{4}\). This is Correct.
- Statement 2: The means of x and y are 3 and 4 respectively. This is Incorrect, as the means are 4 and 5 respectively.
Therefore, only statement 1 is correct.
| Statement | Details | Correct/Incorrect |
|---|---|---|
| 1. Coefficient of correlation \(r = \frac{3}{4}\) | Calculated \(r = \frac{3}{4}\) from \(b_{yx} = \frac{3}{4}\) and \(b_{xy} = \frac{3}{4}\). | Correct |
| 2. Means \(\bar{x}=3\), \(\bar{y}=4\) | Calculated means by solving simultaneous equations: \(\bar{x}=4\), \(\bar{y}=5\). | Incorrect |
Conclusion
Statement 1 is correct, and Statement 2 is incorrect.
Similar Questions
If the regression coefficient of x on y and y on x are \(- \frac{1}{2}\) and \(- \frac{1}{8}\) respectively, then what is the correlation coefficient between x and y?
It is given that X̅ = 10, Y̅ = 90, σ X= 3, σ Y= 12 and r XY = 0.8. The regression equation of X on Y is
For 10 observations on price (x) and supply (y), the following data was obtained:
∑x = 130, ∑y = 220, ∑ x 2= 2288, ∑y 2= 5506 and ∑ xy = 3467.
What is the line of regression of y on x?If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?
A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?
If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?
Consider the following statements:
1. The regression line of y on x is \(\rm y = \frac{3}{4}x+2\)
2. The regression line of x on y is \(\rm x = \frac{3}{4}y+\frac{1}{4}\)
Which of the above statements is/are correct?
Let X and Y represent prices (in Rs) of a commodity in Kolkata and Mumbai respectively. It is given X̅ = 65, Y̅ = 67, σ X = 2.5, σ Y = 3.5 and r(X, Y) = 0.8. What is the equation of regression of Y on X ?
For the variables x and y, the two regression lines are 6x + y = 30 and 3x + 2y = 25. What are the values of x̅, y̅ and r respectively?
If two variables X and Y are independent, then what is the correlation coefficient between them?
Important Questions from Correlation and Regression
If the regression coefficient of x on y and y on x are \(- \frac{1}{2}\) and \(- \frac{1}{8}\) respectively, then what is the correlation coefficient between x and y?
Variable Y regresses with variable X with the conditions that \(\overline X = 5.50\) , \(\overline Y = 3.50\) and b = 1.50 in the linear regression model (Y = a + bX), where \(\overline Y\) and \(\overline X\) are means of the respective variables and b refers to gradient of line of Y w. r. t X. Which one of the following values of parameter 'a' of the model is correct?
It is given that X̅ = 10, Y̅ = 90, σ X= 3, σ Y= 12 and r XY = 0.8. The regression equation of X on Y is
For 10 observations on price (x) and supply (y), the following data was obtained:
∑x = 130, ∑y = 220, ∑ x 2= 2288, ∑y 2= 5506 and ∑ xy = 3467.
What is the line of regression of y on x?If, in a series. there are m items whose ranks are common, then which factor is added for correction for each repeating value in both the series?
