If two regression lines between height (x) and weight (y) are 4y – 15x + 410 = 0 and 30x – 2y – 825 = 0, then what will be the correlation coefficient between height and weight?
Finding the Correlation Coefficient from Regression Lines
The problem asks us to determine the correlation coefficient between height (x) and weight (y) given two linear equations representing the regression lines. The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables.
The two given regression equations are:
4y – 15x + 410 = 030x – 2y – 825 = 0
The relationship between the correlation coefficient (r) and the slopes of the two regression lines, $b_{yx}$ (regression of y on x) and $b_{xy}$ (regression of x on y), is given by the formula: $$r^2 = b_{yx} \times b_{xy}$$ Also, the sign of the correlation coefficient (r) must be the same as the sign of the regression slopes.
Calculating Regression Slopes
To find the slopes, we need to rewrite each equation in the standard form for regression lines.
Let's assume the first equation is the regression of y on x, and the second equation is the regression of x on y. We will verify this assumption later.
Equation 1 (Assumed Regression of y on x):
4y – 15x + 410 = 0
To get the slope $b_{yx}$, we solve for y in terms of x:
$$4y = 15x - 410$$
$$y = \frac{15}{4}x - \frac{410}{4}$$
The slope of the regression of y on x is the coefficient of x, so:
$$b_{yx} = \frac{15}{4}$$
Equation 2 (Assumed Regression of x on y):
30x – 2y – 825 = 0
To get the slope $b_{xy}$, we solve for x in terms of y:
$$30x = 2y + 825$$
$$x = \frac{2}{30}y + \frac{825}{30}$$
$$x = \frac{1}{15}y + \frac{825}{30}$$
The slope of the regression of x on y is the coefficient of y, so:
$$b_{xy} = \frac{1}{15}$$
Let's list the slopes found under this assumption:
| Regression Line | Equation | Slope |
|---|---|---|
| y on x ($b_{yx}$) | 4y – 15x + 410 = 0 |
$\frac{15}{4}$ |
| x on y ($b_{xy}$) | 30x – 2y – 825 = 0 |
$\frac{1}{15}$ |
Now, we calculate $r^2$ using the formula $r^2 = b_{yx} \times b_{xy}$: $$r^2 = \frac{15}{4} \times \frac{1}{15}$$ $$r^2 = \frac{15}{60}$$ $$r^2 = \frac{1}{4}$$
To find r, we take the square root of $r^2$: $$r = \pm\sqrt{\frac{1}{4}}$$ $$r = \pm\frac{1}{2}$$
Since both calculated slopes, $b_{yx} = \frac{15}{4}$ and $b_{xy} = \frac{1}{15}$, are positive, the correlation coefficient (r) must also be positive.
Therefore, the correlation coefficient is: $$r = \frac{1}{2}$$
Verification: The product of the slopes $b_{yx} \times b_{xy}$ must be less than or equal to 1 for a valid correlation coefficient ($r^2 \le 1$). In our case, $\frac{15}{4} \times \frac{1}{15} = \frac{1}{4} \le 1$, which is valid. If we had assumed the opposite (Equation 1 is x on y and Equation 2 is y on x), we would get slopes $b_{xy} = \frac{4}{15}$ and $b_{yx} = 15$. Their product would be $\frac{4}{15} \times 15 = 4$. This gives $r^2 = 4$, which is greater than 1. A value of $r^2 > 1$ is not possible for a correlation coefficient, confirming our initial assumption about which equation represents which regression line was correct.
Final Correlation Coefficient
Based on the calculations, the correlation coefficient between height and weight is $\frac{1}{2}$.
Revision Table: Correlation from Regression Lines
| Concept | Description |
|---|---|
| Regression Lines | Linear equations describing the relationship between two variables, predicting one variable from the other. |
| Slope of y on x ($b_{yx}$) | Change in y for a one-unit change in x. Found by solving the equation for y. |
| Slope of x on y ($b_{xy}$) | Change in x for a one-unit change in y. Found by solving the equation for x. |
| Correlation Coefficient (r) | Measures the strength and direction of the linear relationship between two variables. Ranges from -1 to +1. |
| Relationship $r^2$ | $r^2 = b_{yx} \times b_{xy}$. The product of the slopes gives the square of the correlation coefficient. |
| Sign of r | The sign of r is the same as the sign of $b_{yx}$ and $b_{xy}$. |
Additional Information: Properties of Correlation and Regression
Understanding the connection between correlation and regression slopes is fundamental in statistics. Here are a few key points:
- The regression lines of y on x and x on y intersect at the point $(\bar{x}, \bar{y})$, which is the mean of x and the mean of y.
- The geometric mean of the two regression slopes is equal to the correlation coefficient: $\sqrt{b_{yx} \times b_{xy}} = |r|$.
- If the two regression lines are identical (which happens when r = +1 or r = -1), the relationship is perfectly linear.
- If the two regression lines are perpendicular (which happens when r = 0), there is no linear correlation between the variables.
- The slopes $b_{yx}$ and $b_{xy}$ must always have the same sign. Consequently, the correlation coefficient r must also have that same sign.
- The absolute value of the correlation coefficient $|r|$ is always less than or equal to 1 ($0 \le r^2 \le 1$). This property helps in identifying the correct assignment of slopes if it's not immediately obvious which equation corresponds to which regression line. The product of the slopes must not exceed 1.
These concepts are important for analyzing the linear relationship between variables in various fields like economics, biology, and social sciences.
Important Questions from Correlation and Regression
If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?
Which of the following statements is/are correct in respect of regression coefficients?
1. It measures the degree of linear relationship between two variables
2. It gives the value by which one variable changes for a unit change in the other variable.
Select the correct answer using the code given below.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 ?
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?
