Finding the Regression Line of y on x

A regression line helps us understand the relationship between two variables, x and y, and predict the value of one variable based on the other. The question asks for the regression line of y on x for a given bivariate data set.

Understanding the Regression Line of y on x

The general form of the regression line of y on x is given by:

\[ y - \bar{y} = b_{yx}(x - \bar{x}) \]

where:

• \( y \) is the dependent variable
• \( x \) is the independent variable
• \( \bar{y} \) is the mean of y
• \( \bar{x} \) is the mean of x
• \( b_{yx} \) is the regression coefficient of y on x

The regression coefficient \( b_{yx} \) measures the average change in y for a unit change in x. It is calculated using the correlation coefficient \( r \) and the standard deviations of x and y:

\[ b_{yx} = r \left(\frac{s_y}{s_x}\right) \]

where:

• \( r \) is the correlation coefficient between x and y
• \( s_y \) is the standard deviation of y
• \( s_x \) is the standard deviation of x

Applying the Given Data

We are provided with the following information for the bivariate data set on (x, y):

• Mean of x, \( \bar{x} = 1.0 \)
• Mean of y, \( \bar{y} = 2.0 \)
• Standard deviation of x, \( s_x = 3.0 \)
• Standard deviation of y, \( s_y = 9.0 \)
• Correlation coefficient, \( r = 0.8 \)

Calculating the Regression Coefficient \( b_{yx} \)

Using the formula for \( b_{yx} \) and the given values:

\[ b_{yx} = r \left(\frac{s_y}{s_x}\right) \]

\[ b_{yx} = 0.8 \left(\frac{9.0}{3.0}\right) \]

\[ b_{yx} = 0.8 \times 3 \]

\[ b_{yx} = 2.4 \]

So, the regression coefficient of y on x is 2.4.

Constructing the Regression Line Equation

Now, substitute the values of \( \bar{y} \), \( \bar{x} \), and \( b_{yx} \) into the regression line equation \( y - \bar{y} = b_{yx}(x - \bar{x}) \):

\[ y - 2.0 = 2.4(x - 1.0) \]

This equation directly gives the regression line of y on x.

Comparing with Options

Let's compare the derived equation with the given options:

• Option 1: \( y = 1 + 2.4(x - 1) \)
• Option 2: \( y = 2 + 0.27(x - 1) \)
• Option 3: \( y = 2 + 2.4(x - 1) \)
• Option 4: \( y = 1 + 0.27(x - 2) \)

Our derived equation is \( y - 2.0 = 2.4(x - 1.0) \). Rearranging this by moving the -2.0 to the right side gives \( y = 2.0 + 2.4(x - 1.0) \). This matches Option 3.

Revision Table: Key Regression Concepts

Additional Information on Bivariate Data and Regression

Bivariate data involves observations on two variables for each individual or data point. Regression analysis is a powerful statistical method used to model the relationship between these variables. The regression line is the line that best fits the data points in a scatter plot, minimizing the distance between the points and the line.

There are typically two regression lines for a bivariate data set: the regression line of y on x (which predicts y given x) and the regression line of x on y (which predicts x given y). These lines are generally not the same unless the correlation is perfect (\( r = 1 \) or \( r = -1 \)).

The sign of the regression coefficient \( b_{yx} \) is the same as the sign of the correlation coefficient \( r \), indicating the direction of the relationship. A positive \( b_{yx} \) means y tends to increase as x increases, and a negative \( b_{yx} \) means y tends to decrease as x increases.