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 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 ?
The coefficient of correlation between ages of husband and wife at the time of marriage for a given set of 100 couples was noted to be 0.7. Assume that all these couples survive to celebrate the silver jubilee of their marriage. The coefficient of correlation at that point of time will be
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?
If two variables X and Y are independent, then what is the correlation coefficient between them?
In which one of the following cases would you expect to get a negative correlation?
If the regression coefficient of Y on X is -6, and the correlation coefficient between X and Y is \( - \frac{1}{2},\) then the regression coefficient of X on Y would be
The set of bivariate observation (x 1, y 1) (x 2y 2) …., (x n, y n) are such that all the values are distinct and all the observations fall on a straight line with non-zero slope. Then the possible value of the correlation coefficient between x and y are
The correlation coefficient computed form a set of 30 observations is 0.8 then the percentage of variation not explained by linear regression is
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?
Important Questions from Correlation and Regression
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?
Given below are two statements:
Statement l: One of the assumptions under OLS method states that the regression model is linear in the parameters, though it may or may not be linear in the variables.
Statement ll: The variance of the error, or disturbance, term is not the same regardless of the value of the explanatory variable under OLS.
In light of the above statements, choose the most appropriate answer from the options given below
If two regression coefficients are -0.8 and -0.2, then the value of coefficient of correlation is
Which of the following statements relating to Correlation and Regression are true?
(a) The Coefficient of Correlation is independent of change of origin and scale.
(b) The Coefficient of Correlation between the two variables is the arithmetic average of the two Regression Coefficients.
(c) The probable error of the Coefficient Correlation is 0.6745 times its standard error.
(d) Coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line.
Code:
If two regression coefficients are 0.8 and 1.2, which one of the following is the value of coefficient of correlation?
