Question

the annual profits for a company are given in the following table, where x represents the number of years since 2009, 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, find the projected profit (in thousands of dollars) for 2021, rounded to the nearest thousand dollars.

years since 2009 (x)	profits (y) (in thousands of dollars)
1	64
2	66
3	89
4	98

Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((0, 47)\), \((1, 64)\), \((2, 66)\), \((3, 89)\), \((4, 98)\)

Number of data points \(n = 5\)

Sum of \(x\) values: \(\sum x = 0 + 1 + 2 + 3 + 4 = 10\)

Sum of \(y\) values: \(\sum y = 47 + 64 + 66 + 89 + 98 = 364\)

Sum of \(x \cdot y\) values: \(0 \cdot 47 + 1 \cdot 64 + 2 \cdot 66 + 3 \cdot 89 + 4 \cdot 98 = 0 + 64 + 132 + 267 + 392 = 855\)

Sum of \(x^2\) values: \(0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear regression line is:
\[
m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}
\]
Substituting the values:
\[
m = \frac{5 \cdot 855 - 10 \cdot 364}{5 \cdot 30 - 10^2} = \frac{4275 - 3640}{150 - 100} = \frac{635}{50} = 12.7
\]

Step3: Calculate y-intercept \(b\)

The formula for the y-intercept \(b\) is:
\[
b = \frac{\sum y - m\sum x}{n}
\]
Substituting the values:
\[
b = \frac{364 - 12.7 \cdot 10}{5} = \frac{364 - 127}{5} = \frac{237}{5} = 47.4
\]

So the linear regression equation is \(y = 12.7x + 47.4\)

Step4: Find \(x\) for 2021

2021 is \(2021 - 2009 = 12\) years since 2009, so \(x = 12\)

Step5: Calculate projected profit

Substitute \(x = 12\) into the equation:
\[
y = 12.7 \cdot 12 + 47.4 = 152.4 + 47.4 = 199.8
\]
Rounding to the nearest thousand dollars (since \(y\) is in thousands of dollars), we get \(200\) (because \(199.8\) is closer to \(200\) than to \(199\)).

Answer:

The linear regression equation is \(y = 12.70x + 47.40\) (rounded to nearest hundredth), and the projected profit for 2021 is \(\boxed{200}\) (in thousands of dollars).