Question

years since 2005, 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 the equation, find the projected profit (in thousands of dollars) for 2016, rounded to the nearest thousand dollars.

years since 2005 (x) profits (y) (in thousands of dollars)
0 79
1 109
2 113
3 111

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answer attempt 2 out of 2

regression equation:

final answer:

Explanation:

Step1: Calculate sums

Let \(n = 4\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=0 + 1+2 + 3=6\)
\(\sum_{i = 1}^{n}y_{i}=79 + 109+113 + 111 = 412\)
\(\sum_{i = 1}^{n}x_{i}^{2}=0^{2}+1^{2}+2^{2}+3^{2}=0 + 1+4 + 9 = 14\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=0\times79+1\times109+2\times113+3\times111=0 + 109+226+333 = 668\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression line 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 \(n = 4\), \(\sum_{i = 1}^{n}x_{i}=6\), \(\sum_{i = 1}^{n}y_{i}=412\), \(\sum_{i = 1}^{n}x_{i}^{2}=14\), \(\sum_{i = 1}^{n}x_{i}y_{i}=668\) into the formula:
\[

$$\begin{align*} m&=\frac{4\times668-6\times412}{4\times14 - 6^{2}}\\ &=\frac{2672-2472}{56 - 36}\\ &=\frac{200}{20}\\ & = 10 \end{align*}$$

\]

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 \(n = 4\), \(m = 10\), \(\sum_{i = 1}^{n}x_{i}=6\), \(\sum_{i = 1}^{n}y_{i}=412\) into the formula:
\[

$$\begin{align*} b&=\frac{412-10\times6}{4}\\ &=\frac{412 - 60}{4}\\ &=\frac{352}{4}\\ &=88 \end{align*}$$

\]

Step4: Write the regression equation

The linear regression equation is \(y=mx + b\), so \(y = 10x+88\)

Step5: Calculate \(x\) for 2016

Since \(x\) is the number of years since 2005, for 2016, \(x=2016 - 2005=11\)

Step6: Predict the profit

Substitute \(x = 11\) into the regression equation \(y = 10x+88\)
\(y=10\times11 + 88=110+88 = 198\)

Answer:

Regression Equation: \(y = 10.00x+88.00\)
Final Answer: 198