Calculating the Regression Equation of X on Y

This problem requires us to find the linear regression equation of X on Y using the provided statistical measures: means, standard deviations, and the correlation coefficient.

Given Information

We are given the following statistics for variables X and Y:

• Mean of X ($\bar{X}$): $\bar{X} = 10$
• Mean of Y ($\bar{Y}$): $\bar{Y} = 90$
• Standard deviation of X ($\sigma_X$): $\sigma_X = 3$
• Standard deviation of Y ($\sigma_Y$): $\sigma_Y = 12$
• Correlation coefficient between X and Y ($r_{XY}$): $r_{XY} = 0.8$

Our goal is to determine the regression equation that predicts X based on Y.

Formula for Regression Equation of X on Y

The general form of the regression equation of X on Y is given by:

$$(X - \bar{X}) = b_{XY} (Y - \bar{Y})$$

where $b_{XY}$ is the regression coefficient of X on Y. This coefficient measures the average change in X for a one-unit change in Y. The formula to calculate $b_{XY}$ using the correlation coefficient and standard deviations is:

$$b_{XY} = r_{XY} \frac{\sigma_X}{\sigma_Y}$$

Calculating the Regression Coefficient $b_{XY}$

Now, we substitute the given values into the formula for $b_{XY}$:

$$b_{XY} = 0.8 \times \frac{3}{12}$$

Simplify the fraction:

$$b_{XY} = 0.8 \times \frac{1}{4}$$

Convert the decimal and fraction to calculate the product:

$$b_{XY} = 0.8 \times 0.25$$

$$b_{XY} = 0.2$$

So, the regression coefficient of X on Y is 0.2.

Constructing the Regression Equation

Now we substitute the values of $\bar{X}$, $\bar{Y}$, and $b_{XY}$ into the regression equation formula $(X - \bar{X}) = b_{XY} (Y - \bar{Y})$:

$$(X - 10) = 0.2 (Y - 90)$$

Next, we simplify the equation to express X in terms of Y:

$$X - 10 = 0.2 \times Y - 0.2 \times 90$$

$$X - 10 = 0.2Y - 18$$

Add 10 to both sides of the equation to isolate X:

$$X = 0.2Y - 18 + 10$$

$$X = 0.2Y - 8$$

This equation can also be written as:

$$X = -8 + 0.2Y$$

Comparing with Options

Let's compare our calculated regression equation with the given options:

• Option 1: Y = 3.2X + 58 (This is a regression of Y on X, and the coefficients don't match.)
• Option 2: X = 3.2Y + 58 (This is a regression of X on Y, but the coefficients don't match our result.)
• Option 3: X = - 8 + 0.2Y (This exactly matches our calculated equation.)
• Option 4: Y = - 8 + 0.2X (This is a regression of Y on X, and the coefficients don't match.)

Our calculated regression equation of X on Y is $X = -8 + 0.2Y$, which corresponds to Option 3.

Revision Table: Regression Concepts

Additional Information on Linear Regression

Linear regression is a statistical method used to model the linear relationship between a dependent variable and one or more independent variables. In the case of two variables, X and Y, we can have two regression lines:

• Regression line of Y on X: Used to predict Y from X. The equation is typically written as $Y = c + b_{YX}X$. The coefficient $b_{YX}$ is calculated as $b_{YX} = r_{XY} \frac{\sigma_Y}{\sigma_X}$.
• Regression line of X on Y: Used to predict X from Y. The equation is typically written as $X = a + b_{XY}Y$. The coefficient $b_{XY}$ is calculated as $b_{XY} = r_{XY} \frac{\sigma_X}{\sigma_Y}$.

Unless the correlation is perfect ($r_{XY} = \pm 1$), the two regression lines are distinct and intersect at the point $(\bar{X}, \bar{Y})$. The correlation coefficient ($r_{XY}$) indicates the strength and direction of the linear relationship between X and Y. Its value ranges from -1 to +1. The sign of $r_{XY}$, $b_{XY}$, and $b_{YX}$ are always the same.

Understanding both regression lines is crucial for making predictions depending on which variable is considered independent and which is dependent in a given context.