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)
0 | 96
1 | 88
2 | 113
3 | 155
4 | 183
5 | 217
copy values for calculator

Explanation:

Step1: Calculate mean of x and y

First, find the mean of \( x \) values: \( x = [0,1,2,3,4,5] \), so \( \bar{x}=\frac{0 + 1+2+3+4+5}{6}=\frac{15}{6}=2.5 \)
Then, find the mean of \( y \) values: \( y = [96,88,113,155,183,217] \), so \( \bar{y}=\frac{96 + 88+113+155+183+217}{6}=\frac{852}{6}=142 \)

Step2: Calculate slope (m)

The formula for slope \( m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2} \)
Calculate \( (x_i-\bar{x})(y_i-\bar{y}) \) for each \( i \):

• For \( i = 1 \): \( (0 - 2.5)(96 - 142)=(-2.5)(-46)=115 \)
• For \( i = 2 \): \( (1 - 2.5)(88 - 142)=(-1.5)(-54)=81 \)
• For \( i = 3 \): \( (2 - 2.5)(113 - 142)=(-0.5)(-29)=14.5 \)
• For \( i = 4 \): \( (3 - 2.5)(155 - 142)=(0.5)(13)=6.5 \)
• For \( i = 5 \): \( (4 - 2.5)(183 - 142)=(1.5)(41)=61.5 \)
• For \( i = 6 \): \( (5 - 2.5)(217 - 142)=(2.5)(75)=187.5 \)

Sum of these: \( 115+81 + 14.5+6.5+61.5+187.5 = 466 \)

Calculate \( (x_i-\bar{x})^2 \) for each \( i \):

• For \( i = 1 \): \( (0 - 2.5)^2 = 6.25 \)
• For \( i = 2 \): \( (1 - 2.5)^2 = 2.25 \)
• For \( i = 3 \): \( (2 - 2.5)^2 = 0.25 \)
• For \( i = 4 \): \( (3 - 2.5)^2 = 0.25 \)
• For \( i = 5 \): \( (4 - 2.5)^2 = 2.25 \)
• For \( i = 6 \): \( (5 - 2.5)^2 = 6.25 \)

Sum of these: \( 6.25+2.25+0.25+0.25+2.25+6.25 = 17.5 \)

So, \( m=\frac{466}{17.5}\approx26.63 \) (rounded to nearest hundredth)

Step3: Calculate y - intercept (b)

Using \( y = mx + b \), and \( \bar{y}=m\bar{x}+b \)
So, \( b=\bar{y}-m\bar{x} \)
Substitute \( \bar{y}=142 \), \( m\approx26.63 \), \( \bar{x}=2.5 \)
\( b = 142-26.63\times2.5=142 - 66.575 = 75.425\approx75.43 \) (rounded to nearest hundredth)

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

Step4: Find x for 2014

2014 - 2006 = 8, so \( x = 8 \)

Step5: Predict y for x = 8

Substitute \( x = 8 \) into the equation: \( y=26.63\times8 + 75.43=213.04+75.43 = 288.47\approx288 \) (rounded to nearest thousand dollars)

Answer:

The linear regression equation is \( y = 26.63x + 75.43 \) and the projected profit for 2014 is 288 (in thousands of dollars).