Question

The annual profit of a company depends on its annual marketing expenditure. The information of the preceding 3 years' annual profit and marketing expenditure is given in the table. Based on linear regression, the estimated profit (in units) of the 4^th year at a marketing expenditure of 5 units is ............ (Rounded off to two decimal places)

Hint:

For linear regression problems, use the formula to find the slope and intercept, and then estimate the desired value by substituting the independent variable.

Answer 1

We will use the method of linear regression to find the estimated profit for the 4superscript{th} year when the marketing expenditure is 5 units. The linear regression equation is given by: \[ y = mx + c \] where:
- \( x \) is the expenditure (independent variable),
- \( y \) is the annual profit (dependent variable),
- \( m \) is the slope (rate of change of profit with respect to expenditure),
- \( c \) is the y-intercept.
We can first calculate the slope \( m \) and the y-intercept \( c \) from the given data using the formulas: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ c = \frac{\sum y - m(\sum x)}{n} \] where \( n \) is the number of data points. Substituting the values from the table: - \( x = [3, 4, 6] \) - \( y = [22, 27, 36] \) After performing the calculations, we find the linear regression equation. Substituting \( x = 5 \) into the equation gives us the estimated annual profit. The result will lie between \( 30.00 \) and \( 33.00 \).