Question

the annual profits for a company are given in the following table, where x represents the number of years since 1997, 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 2005, rounded to the nearest thousand dollars.

years since 1997 (x)	profits (y) (in thousands of dollars)
1	82
2	98
3	107
4	139
5	137

answer attempt 3 out of 3

regression equation:

Explanation:

Step1: Calculate sums

Let \(n = 6\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=0 + 1+2 + 3+4 + 5=15\)
\(\sum_{i = 1}^{n}y_{i}=81 + 82+98 + 107+139+137 = 644\)
\(\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=0 + 1+4 + 9+16+25 = 55\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=0\times81+1\times82 + 2\times98+3\times107+4\times139+5\times137\)
\(=0 + 82+196+321+556+685 = 1840\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)
Substitute \(n = 6\), \(\sum_{i = 1}^{n}x_{i}=15\), \(\sum_{i = 1}^{n}y_{i}=644\), \(\sum_{i = 1}^{n}x_{i}^{2}=55\), \(\sum_{i = 1}^{n}x_{i}y_{i}=1840\) into the formula:
\[

$$\begin{align*} m&=\frac{6\times1840-15\times644}{6\times55 - 15^{2}}\\ &=\frac{11040-9660}{330 - 225}\\ &=\frac{1380}{105}\\ &\approx13.1 \end{align*}$$

\]

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\)
Substitute \(n = 6\), \(\sum_{i = 1}^{n}x_{i}=15\), \(\sum_{i = 1}^{n}y_{i}=644\), \(m\approx13.1\) into the formula:
\[

$$\begin{align*} b&=\frac{644-13.1\times15}{6}\\ &=\frac{644 - 196.5}{6}\\ &=\frac{447.5}{6}\\ &\approx74.6 \end{align*}$$

\]

The regression equation is \(y = 13.1x+74.6\)

Step4: Find \(x\) for 2005

Since \(x\) represents the number of years since 1997, for 2005, \(x = 2005 - 1997=8\)

Step5: Predict profit

Substitute \(x = 8\) into the regression equation \(y=13.1x + 74.6\)
\(y=13.1\times8+74.6\)
\(y = 104.8+74.6\)
\(y=179.4\approx179\)

Answer:

Regression Equation: \(y = 13.1x+74.6\)
Projected profit for 2005: 179