Question

the annual profits for a company are given in the following table, where x represents the number of years since 2006, 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 2015, rounded to the nearest thousand dollars.

years since 2006 (x)	profits (y) (in thousands of dollars)
1	106
2	99
3	139
4	201
5	213

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answer attempt 1 out of 2
regression equation:
final answer: thousand dollars

Explanation:

Step1: Calculate necessary sums

We have the data points: \((0, 71)\), \((1, 106)\), \((2, 99)\), \((3, 139)\), \((4, 201)\), \((5, 213)\)

First, calculate \(n = 6\) (number of data points).

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

Calculate \(\sum y\): \(71 + 106 + 99 + 139 + 201 + 213 = 829\)

Calculate \(\sum xy\): \((0\times71)+(1\times106)+(2\times99)+(3\times139)+(4\times201)+(5\times213)\)
\(= 0 + 106 + 198 + 417 + 804 + 1065 = 2590\)

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

Step2: Calculate slope \(m\)

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

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

Substitute the values:

$$m=\frac{6\times2590 - 15\times829}{6\times55 - 15^2}$$

$$m=\frac{15540 - 12435}{330 - 225}$$

$$m=\frac{3105}{105}\approx29.57$$

Step3: Calculate y-intercept \(b\)

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

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

Substitute the values:

$$b=\frac{829 - 29.57\times15}{6}$$

$$b=\frac{829 - 443.55}{6}$$

$$b=\frac{385.45}{6}\approx64.24$$

So the linear regression equation is \(y = 29.57x + 64.24\)

Step4: Find \(x\) for 2015

2015 - 2006 = 9, so \(x = 9\)

Step5: Predict profit for \(x = 9\)

Substitute \(x = 9\) into the regression equation:

$$y = 29.57\times9 + 64.24$$

$$y = 266.13 + 64.24$$

$$y = 330.37\approx330$$

(rounded to the nearest thousand dollars)

Answer:

Regression Equation: \(y = 29.57x + 64.24\)
Final Answer: \(330\) thousand dollars