Question

The annual profit of a company depends on its annual marketing expenditure. The information of preceding 3 years' annual profit and marketing expenditure is given in the table. Based on linear regression, the estimated profit (in units) of the 4superscript{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

Correct Answer: 30

 To estimate the annual profit using linear regression, we need to find the linear relationship between marketing expenditure (x) and annual profit (y). The data is:

Year	Expenditure (x)	Profit (y)
1	3	22
2	4	27
3	6	36

We use the formula for the regression line: y = mx + c.

Calculate the slope (m) and intercept (c):

\( m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \)

• \( \sum x = 3 + 4 + 6 = 13 \)
• \( \sum y = 22 + 27 + 36 = 85 \)
• \( \sum xy = (3 \times 22) + (4 \times 27) + (6 \times 36) = 66 + 108 + 216 = 390 \)
• \( \sum x^2 = 3^2 + 4^2 + 6^2 = 9 + 16 + 36 = 61 \)
• \( n = 3 \)

Substituting in the formula:

\( m = \frac{3(390) - 13(85)}{3(61) - 13^2} = \frac{1170 - 1105}{183 - 169} = \frac{65}{14} \approx 4.64 \)

Calculate the intercept (c):

\( c = \frac{\sum y - m \sum x}{n} = \frac{85 - 4.64 \times 13}{3} = \frac{85 - 60.32}{3} \approx 8.23 \)

Regression line: \( y = 4.64x + 8.23 \).

Substitute x = 5 to estimate the profit:

\( y = 4.64(5) + 8.23 = 23.2 + 8.23 = 31.43 \)

The estimated profit for a marketing expenditure of 5 units is 31.43 units.

The range given is 30.30. The estimated profit of 31.43 falls within the acceptable range.