Understanding Bivariate Data and Regression Lines

A bivariate data set involves observations on two variables for each individual or data point. In this question, we are given a very simple bivariate data set consisting of just two points: (-1, 1) and (3, 2).

The line of regression of y on x is essentially a line that describes how the variable y is expected to change as the variable x changes. When you only have two data points, the line of regression of y on x is simply the straight line that passes through these two points. There is no complex fitting required as the line through two distinct points is unique.

Finding the Equation of the Line Through Two Points

To find the equation of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), we first need to calculate the slope (m) of the line using the formula:

\( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Once we have the slope, we can use the point-slope form of the linear equation, which is:

\( y - y_1 = m(x - x_1) \)

Alternatively, we can find the y-intercept (c) using the slope-intercept form \(y = mx + c\) by substituting one of the points into the equation.

Calculating the Slope for the Given Bivariate Data

Our two points are \((-1, 1)\) and \((3, 2)\).

Let \((x_1, y_1) = (-1, 1)\) and \((x_2, y_2) = (3, 2)\).

Now, we calculate the slope:

\( m = \frac{2 - 1}{3 - (-1)} \)

\( m = \frac{1}{3 + 1} \)

\( m = \frac{1}{4} \)

So, the slope of the line of regression is \(\frac{1}{4}\).

Deriving the Equation of the Line of Regression

We have the slope \(m = \frac{1}{4}\) and we can use either of the two points. Let's use the point \((-1, 1)\) and the point-slope form \(y - y_1 = m(x - x_1)\).

Substitute the values:

\( y - 1 = \frac{1}{4}(x - (-1)) \)

\( y - 1 = \frac{1}{4}(x + 1) \)

To remove the fraction, multiply both sides of the equation by 4:

\( 4(y - 1) = 4 \times \frac{1}{4}(x + 1) \)

\( 4y - 4 = x + 1 \)

Now, rearrange the terms to get the equation in the standard form \(Ax + By + C = 0\):

\( 0 = x - 4y + 1 + 4 \)

\( 0 = x - 4y + 5 \)

Or, writing it conventionally:

\( x - 4y + 5 = 0 \)

This is the equation of the line of regression of y on x for the given bivariate data set.

Comparing with the Options

Let's compare our derived equation \(x - 4y + 5 = 0\) with the given options:

1. x - 4y + 5 = 0
2. 3x + 2y - 1 = 0
3. x + 4y + 1 = 0
4. 5x - 4y + 1 = 0

Our equation matches option 1.

Revision Table: Key Concepts

Additional Information on Regression

While for two points the regression line is simply the line connecting them, for more than two points, the line of regression (either y on x or x on y) is calculated using methods like the method of least squares. This method minimizes the sum of the squared errors (the vertical distances from the points to the line for y on x regression). The formulas for the slope (b) and y-intercept (a) of the regression line \(y = a + bx\) for n points are:

• Slope, \( b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \)
• Y-intercept, \( a = \frac{\sum y - b(\sum x)}{n} \) or \( a = \bar{y} - b\bar{x} \)

Where \(\bar{x}\) and \(\bar{y}\) are the means of x and y respectively, and n is the number of data points. For two points, these complex formulas simplify to the line through the points.