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 2014, rounded to the nearest thousand dollars.

years since 2006 (x)	profits (y) (in thousands of dollars)
1	88
2	113
3	155
4	183
5	217

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answer
attempt 2 out of 3

regression equation:

final answer: thousand dollars

Explanation:

Step1: Calculate necessary sums

We have the data points \((x_i, y_i)\): \((0, 96)\), \((1, 88)\), \((2, 113)\), \((3, 155)\), \((4, 183)\), \((5, 217)\).

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

Calculate \(\sum x_i\): \(0 + 1 + 2 + 3 + 4 + 5=\frac{5\times(5 + 1)}{2}=15\)

Calculate \(\sum y_i\): \(96+88 + 113+155+183+217 = 852\)

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

Calculate \(\sum x_i y_i\): \(0\times96+1\times88+2\times113+3\times155+4\times183+5\times217\)
\(=0 + 88+226+465+732+1085 = 2596\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the linear regression line \(y=mx + b\) is:

$$m=\frac{n\sum x_i y_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2}$$

Substitute the values:

$$m=\frac{6\times2596-15\times852}{6\times55 - 15^2}=\frac{15576-12780}{330 - 225}=\frac{2796}{105}\approx26.63$$

Step3: Calculate y - intercept \(b\)

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

$$b=\frac{\sum y_i-m\sum x_i}{n}$$

Substitute the values:

$$b=\frac{852-26.63\times15}{6}=\frac{852 - 399.45}{6}=\frac{452.55}{6}\approx75.43$$

So the linear regression equation is \(y = 26.63x+75.43\)

Step4: Find \(x\) for 2014

Since \(x\) is the number of years since 2006, for 2014, \(x=2014 - 2006 = 8\)

Step5: Predict \(y\) for \(x = 8\)

Substitute \(x = 8\) into the regression equation \(y=26.63x + 75.43\)

$$y=26.63\times8+75.43=213.04 + 75.43=288.47\approx288$$

Answer:

Regression Equation: \(y = 26.63x+75.43\)
Final Answer: \(288\) thousand dollars