Question

the annual profits for a company are given in the following table, where x represents the number of years since 2009, 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 2021, rounded to the nearest thousand dollars.

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
Here \(n = 5\), \(\sum_{i=1}^{5}x_{i}=0 + 1+2 + 3+4=10\), so \(\bar{x}=\frac{10}{5}=2\).
\(\sum_{i = 1}^{5}y_{i}=47+64 + 66+89+98=364\), so \(\bar{y}=\frac{364}{5}=72.8\).

Step2: Calculate the slope \(m\)

The formula for the slope \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\((x_{1}-\bar{x})(y_{1}-\bar{y})=(0 - 2)(47-72.8)=(- 2)\times(-25.8)=51.6\)
\((x_{2}-\bar{x})(y_{2}-\bar{y})=(1 - 2)(64 - 72.8)=(-1)\times(-8.8)=8.8\)
\((x_{3}-\bar{x})(y_{3}-\bar{y})=(2 - 2)(66 - 72.8)=0\times(-6.8)=0\)
\((x_{4}-\bar{x})(y_{4}-\bar{y})=(3 - 2)(89 - 72.8)=1\times16.2 = 16.2\)
\((x_{5}-\bar{x})(y_{5}-\bar{y})=(4 - 2)(98 - 72.8)=2\times25.2=50.4\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=51.6+8.8 + 0+16.2+50.4=127\)

\((x_{1}-\bar{x})^{2}=(0 - 2)^{2}=4\)
\((x_{2}-\bar{x})^{2}=(1 - 2)^{2}=1\)
\((x_{3}-\bar{x})^{2}=(2 - 2)^{2}=0\)
\((x_{4}-\bar{x})^{2}=(3 - 2)^{2}=1\)
\((x_{5}-\bar{x})^{2}=(4 - 2)^{2}=4\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=4 + 1+0+1+4=10\)

So \(m=\frac{127}{10}=12.7\)

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

The formula for the y - intercept \(b=\bar{y}-m\bar{x}\)
\(b=72.8-12.7\times2=72.8 - 25.4=47.4\)

The linear regression equation is \(y = 12.7x+47.4\)

Step4: Find the value of \(x\) for 2021

Since \(x\) represents the number of years since 2009, for 2021, \(x=2021 - 2009=12\)

Step5: Predict the profit

Substitute \(x = 12\) into the equation \(y=12.7x + 47.4\)
\(y=12.7\times12+47.4=152.4+47.4=199.8\approx200\)

Answer:

The linear regression equation is \(y = 12.7x+47.4\) and the projected profit for 2021 is \(200\) (in thousands of dollars).