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 tenth. using this equation, find the projected profit (in thousands of dollars) for 2017, rounded to the nearest thousand dollars.

years since 2006 (x)	profits (y) (in thousands of dollars)
1	185
2	157
3	200

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Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((0, 135)\), \((1, 185)\), \((2, 157)\), \((3, 200)\)

Let \(n = 4\) (number of data points).

Calculate \(\sum x\), \(\sum y\), \(\sum xy\), and \(\sum x^2\):

\(\sum x = 0 + 1 + 2 + 3 = 6\)

\(\sum y = 135 + 185 + 157 + 200 = 677\)

\(\sum xy = (0\times135) + (1\times185) + (2\times157) + (3\times200) = 0 + 185 + 314 + 600 = 1099\)

\(\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 = 0 + 1 + 4 + 9 = 14\)

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:

\(m = \frac{4\times1099 - 6\times677}{4\times14 - 6^2}\)

First, calculate numerator: \(4\times1099 = 4396\), \(6\times677 = 4062\), so numerator is \(4396 - 4062 = 334\)

Denominator: \(4\times14 = 56\), \(6^2 = 36\), so denominator is \(56 - 36 = 20\)

Thus, \(m = \frac{334}{20} = 16.7\) (rounded to nearest tenth)

Step3: Calculate y-intercept \(b\)

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

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

Substitute the values:

\(b = \frac{677 - 16.7\times6}{4}\)

First, calculate \(16.7\times6 = 100.2\)

Then, \(677 - 100.2 = 576.8\)

Thus, \(b = \frac{576.8}{4} = 144.2\) (rounded to nearest tenth)

So the linear regression equation is \(y = 16.7x + 144.2\)

Step4: Find x for 2017

2017 is \(2017 - 2006 = 11\) years since 2006, so \(x = 11\)

Step5: Predict profit for 2017

Substitute \(x = 11\) into the regression equation:

\(y = 16.7\times11 + 144.2\)

Calculate \(16.7\times11 = 183.7\)

Then, \(y = 183.7 + 144.2 = 327.9\)

Rounded to nearest thousand dollars (since y is in thousands), it's 328.

Answer:

The linear regression equation is \(y = 16.7x + 144.2\) and the projected profit for 2017 is 328 (in thousands of dollars).