For a bivariate data set on (x, y), if the means, standard deviations and correlation coefficient are

x̅ = 1.0, y̅ = 2.0, s ₓ= 3.0, s ${}_y$ = 9.0, r = 0.8

Then the regression line of y on x is:

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 ConceptsAdditional 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.