Question

the annual profits for a company are given in the following table, where x represents the number of years since 1995, and y represents the profit in thousands of dollars. write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. using this equation, estimate the calendar year in which the profits would reach 207 thousand dollars.

years since 1995 (x)	profits (y) (in thousands of dollars)
1	69
2	76
3	112
4	131
5	126

Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sums for \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \).

Given the data:

• \( x \): 0, 1, 2, 3, 4, 5
• \( y \): 49, 69, 76, 112, 131, 126

\( \sum x = 0 + 1 + 2 + 3 + 4 + 5 = 15 \)

\( \sum y = 49 + 69 + 76 + 112 + 131 + 126 = 563 \)

\( \sum xy = (0\times49)+(1\times69)+(2\times76)+(3\times112)+(4\times131)+(5\times126) = 0 + 69 + 152 + 336 + 524 + 630 = 1711 \)

\( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 0 + 1 + 4 + 9 + 16 + 25 = 55 \)

\( n = 6 \) (number of data points)

Step2: Calculate slope (\( m \)) and y-intercept (\( b \))

The formula for the slope \( m \) of the linear regression line is:

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

Substitute the values:

\( m = \frac{6\times1711 - 15\times563}{6\times55 - 15^2} = \frac{10266 - 8445}{330 - 225} = \frac{1821}{105} \approx 17.34 \)

The formula for the y-intercept \( b \) is:

\( b = \frac{\sum y - m\sum x}{n} \)

Substitute the values:

\( b = \frac{563 - 17.34\times15}{6} = \frac{563 - 260.1}{6} = \frac{302.9}{6} \approx 50.48 \)

So the linear regression equation is \( y = 17.34x + 50.48 \)

Step3: Estimate the year when \( y = 207 \)

Set \( y = 207 \) in the equation and solve for \( x \):

\( 207 = 17.34x + 50.48 \)

Subtract 50.48 from both sides:

\( 207 - 50.48 = 17.34x \)

\( 156.52 = 17.34x \)

Divide both sides by 17.34:

\( x = \frac{156.52}{17.34} \approx 9.03 \)

Since \( x \) is the number of years since 1995, the year is \( 1995 + 9 = 2004 \) (we round \( x \) to the nearest whole number as it represents years)

Answer:

The linear regression equation is \( y = 17.34x + 50.48 \) and the calendar year when profits reach 207 thousand dollars is 2004.