If the regression line of Y on X is Y = 30 - 0.9X and the standard deviations are S x= 2 and S y= 9, then the value of the correlation coefficient r xy is :
-0.2
Understanding the Regression Line and Correlation Coefficient
This question asks us to find the correlation coefficient between two variables, X and Y, given the equation of the regression line of Y on X and the standard deviations of X and Y. The regression line provides information about the linear relationship between the variables, and the correlation coefficient quantifies the strength and direction of this linear relationship.
Key Concepts: Regression Line of Y on X
The regression line of Y on X is typically represented by the equation:
\( Y = a + b_{YX}X \)
Here:
- \( Y \) is the dependent variable (the one being predicted).
- \( X \) is the independent variable (the one used for prediction).
- \( a \) is the Y-intercept, the value of Y when X is 0.
- \( b_{YX} \) is the slope of the regression line, representing the change in Y for a one-unit change in X.
Relationship between Slope, Correlation Coefficient, and Standard Deviations
There is a direct relationship linking the slope of the regression line (\( b_{YX} \)), the correlation coefficient (\( r_{xy} \)), the standard deviation of Y (\( S_y \)), and the standard deviation of X (\( S_x \)). The formula is:
\( b_{YX} = r_{xy} \left( \frac{S_y}{S_x} \right) \)
This formula is crucial for solving this problem.
Analyzing the Given Information
From the question, we are provided with the following information:
- Equation of the regression line of Y on X: \( Y = 30 - 0.9X \)
- Standard deviation of X: \( S_x = 2 \)
- Standard deviation of Y: \( S_y = 9 \)
Comparing the given regression equation \( Y = 30 - 0.9X \) with the standard form \( Y = a + b_{YX}X \), we can identify the slope of the regression line of Y on X.
The slope \( b_{YX} \) is the coefficient of X, which is -0.9.
So, we have:
- \( b_{YX} = -0.9 \)
- \( S_x = 2 \)
- \( S_y = 9 \)
Calculating the Correlation Coefficient rxy
We can now use the relationship formula to find the correlation coefficient \( r_{xy} \):
\( b_{YX} = r_{xy} \left( \frac{S_y}{S_x} \right) \)
We need to rearrange the formula to solve for \( r_{xy} \):
\( r_{xy} = b_{YX} \left( \frac{S_x}{S_y} \right) \)
Now, substitute the known values into this formula:
\( r_{xy} = (-0.9) \left( \frac{2}{9} \right) \)
Performing the calculation:
\( r_{xy} = -0.9 \times \frac{2}{9} \)
\( r_{xy} = - \frac{0.9 \times 2}{9} \)
\( r_{xy} = - \frac{1.8}{9} \)
\( r_{xy} = -0.2 \)
Thus, the value of the correlation coefficient \( r_{xy} \) is -0.2.
Verification and Interpretation of the Correlation Coefficient
The calculated correlation coefficient is -0.2. This value is between -1 and +1, which is expected for a correlation coefficient. The negative sign indicates a negative linear relationship between X and Y. As X increases, Y tends to decrease, which is consistent with the negative slope (-0.9) of the regression line.
Summary of Calculation Steps
- Identify the slope (\( b_{YX} \)) of the regression line of Y on X from the given equation.
- Note down the standard deviations of X (\( S_x \)) and Y (\( S_y \)).
- Use the formula relating \( b_{YX} \), \( r_{xy} \), \( S_y \), and \( S_x \): \( b_{YX} = r_{xy} \left( \frac{S_y}{S_x} \right) \).
- Rearrange the formula to solve for \( r_{xy} \): \( r_{xy} = b_{YX} \left( \frac{S_x}{S_y} \right) \).
- Substitute the known values into the rearranged formula and calculate \( r_{xy} \).
| Parameter | Value |
|---|---|
| Regression Line Slope (\( b_{YX} \)) | -0.9 |
| Standard Deviation of X (\( S_x \)) | 2 |
| Standard Deviation of Y (\( S_y \)) | 9 |
| Formula for \( r_{xy} \) | \( r_{xy} = b_{YX} \left( \frac{S_x}{S_y} \right) \) |
| Calculation | \( r_{xy} = -0.9 \left( \frac{2}{9} \right) = -0.2 \) |
Revision Table: Key Statistics Concepts
| Concept | Description | Related Symbol(s) |
|---|---|---|
| Regression Line | A line that best describes the linear relationship between two variables, used for prediction. | \( Y = a + bX \) |
| Slope of Regression Line (Y on X) | Indicates how much Y is expected to change when X increases by one unit. Direction (positive/negative) shows the type of relationship. | \( b_{YX} \) |
| Standard Deviation | A measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean. | \( S_x, S_y \) |
| Correlation Coefficient | A measure of the strength and direction of a linear relationship between two variables. Ranges from -1 (perfect negative) to +1 (perfect positive). 0 indicates no linear relationship. | \( r_{xy} \) |
Additional Information: Properties of Correlation and Regression
Here are some additional points about correlation and regression analysis that are helpful to know:
- The correlation coefficient \( r_{xy} \) and the slope of the regression line \( b_{YX} \) always have the same sign. If one is positive, the other is positive; if one is negative, the other is negative. This indicates that they both reflect the direction of the linear relationship.
- The value of the correlation coefficient \( r_{xy} \) is unitless, meaning it is not affected by the units of measurement of X and Y.
- Correlation does not imply causation. A strong correlation between X and Y does not necessarily mean that X causes Y or Y causes X. There might be other factors involved.
- There are two regression lines for any pair of variables X and Y: the regression line of Y on X (\( Y = a + b_{YX}X \)) and the regression line of X on Y (\( X = c + b_{XY}Y \)). The slopes \( b_{YX} \) and \( b_{XY} \) are generally not equal unless the correlation is perfect (\( r_{xy} = \pm 1 \)).
- The formula relating the two slopes and the correlation coefficient is \( r_{xy}^2 = b_{YX} \times b_{XY} \).
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A. If means of both the variables are equal
B. If one variable is measured in interval scale and another is measured in ordinal scale
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D. If data are obtained in interval or ratio scale for both the variables
E. If direction of relationship between two variables is known
Choose the most appropriate answer from the options given below :
