Question

the annual profits for a company are given in the following table, where x represents the number of years since 2012, 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 2022, rounded to the nearest thousand dollars.

years since 2012 (x) profits (y)
(in thousands of dollars)
0 93
1 116
2 113
3 125
4 158
5 188

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

\(\bar{x}=\frac{0 + 1+2+3+4+5}{6}=\frac{15}{6} = 2.5\)
\(\bar{y}=\frac{93+116+113+125+158+188}{6}=\frac{793}{6}\approx132.2\)

Step2: Calculate the slope \(m\)

$$
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$$

$$
LATEXBLOCK1
$$

\(m=\frac{306.5}{17.5}\approx17.5\)

Step3: Calculate the y - intercept \(b\)

Using the formula \(y=mx + b\), substitute \(x = \bar{x}\), \(y=\bar{y}\) and \(m\approx17.5\)
\(132.2=17.5\times2.5+b\)
\(132.2 = 43.75+b\)
\(b=132.2-43.75 = 88.45\approx88.5\)

The linear regression equation is \(y = 17.5x+88.5\)

Step4: Find \(x\) for 2022

Since \(x\) represents the number of years since 2012, for 2022, \(x=2022 - 2012=10\)

Step5: Calculate the projected profit

Substitute \(x = 10\) into \(y = 17.5x+88.5\)
\(y=17.5\times10+88.5=175 + 88.5=263.5\approx264\)

Answer:

The linear regression equation is \(y = 17.5x + 88.5\). The projected profit for 2022 is \(264\) (in thousands of dollars).