the annual profits for a company are given in the following table, where x represents the number of years…

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 hundredth. using this equation, find the projected profit (in thousands of dollars) for 2021, rounded to the nearest thousand dollars.\n\n| years since 2012 (x) | profits (y) (in thousands of dollars) |\n| ---- | ---- |\n| 0 | 122 |\n| 1 | 137 |\n| 2 | 133 |\n| 3 | 158 |\n| 4 | 167 |\n| 5 | 175 |

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 hundredth. using this equation, find the projected profit (in thousands of dollars) for 2021, rounded to the nearest thousand dollars.\n\n| years since 2012 (x) | profits (y) (in thousands of dollars) |\n| ---- | ---- |\n| 0 | 122 |\n| 1 | 137 |\n| 2 | 133 |\n| 3 | 158 |\n| 4 | 167 |\n| 5 | 175 |

Answer

Explanation:

Step1: Calculate the means of x and y

Let $n = 6$. $\bar{x}=\frac{0 + 1+2+3+4+5}{6}=\frac{15}{6}=2.5$ $\bar{y}=\frac{122 + 137+133+158+167+175}{6}=\frac{892}{6}\approx148.67$

Step2: Calculate the numerator and denominator for the slope (m)

The formula for the slope $m$ of the regression - line is $m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}$

$(x_1-\bar{x})(y_1 - \bar{y})=(0 - 2.5)(122-148.67)=(-2.5)(-26.67) = 66.675$ $(x_2-\bar{x})(y_2 - \bar{y})=(1 - 2.5)(137-148.67)=(-1.5)(-11.67)=17.505$ $(x_3-\bar{x})(y_3 - \bar{y})=(2 - 2.5)(133-148.67)=(-0.5)(-15.67)=7.835$ $(x_4-\bar{x})(y_4 - \bar{y})=(3 - 2.5)(158-148.67)=(0.5)(9.33)=4.665$ $(x_5-\bar{x})(y_5 - \bar{y})=(4 - 2.5)(167-148.67)=(1.5)(18.33)=27.495$ $(x_6-\bar{x})(y_6 - \bar{y})=(5 - 2.5)(175-148.67)=(2.5)(26.33)=65.825$

$\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=66.675 + 17.505+7.835+4.665+27.495+65.825=189.00$

$(x_1-\bar{x})^2=(0 - 2.5)^2 = 6.25$ $(x_2-\bar{x})^2=(1 - 2.5)^2 = 2.25$ $(x_3-\bar{x})^2=(2 - 2.5)^2 = 0.25$ $(x_4-\bar{x})^2=(3 - 2.5)^2 = 0.25$ $(x_5-\bar{x})^2=(4 - 2.5)^2 = 2.25$ $(x_6-\bar{x})^2=(5 - 2.5)^2 = 6.25$

$\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=6.25+2.25 + 0.25+0.25+2.25+6.25 = 17.5$

$m=\frac{189.00}{17.5}\approx10.80$

Step3: Calculate the y - intercept (b)

The formula for the y - intercept $b$ is $b=\bar{y}-m\bar{x}$ $b = 148.67-10.80\times2.5=148.67 - 27=121.67\approx121.67$

The linear regression equation is $y = 10.80x+121.67$

Step4: Find the value of x for 2021

Since $x$ represents the number of years since 2012, for 2021, $x=2021 - 2012=9$

Step5: Predict the profit

Substitute $x = 9$ into the regression equation: $y=10.80\times9+121.67=97.2+121.67 = 218.87\approx219$

Answer:

The linear regression equation is $y = 10.80x+121.67$, and the projected profit for 2021 is $219$ thousand dollars.