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 tenth. using this equation, estimate the calendar year in which the profits would reach 420 thousand dollars.\n\n| years since 2012 (x) | profits (y) (in thousands of dollars) |\n| ---- | ---- |\n| 0 | 113 |\n| 1 | 142 |\n| 2 | 162 |\n| 3 | 161 |\n| 4 | 171 |\n| 5 | 210 |

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, estimate the calendar year in which the profits would reach 420 thousand dollars.\n\n| years since 2012 (x) | profits (y) (in thousands of dollars) |\n| ---- | ---- |\n| 0 | 113 |\n| 1 | 142 |\n| 2 | 162 |\n| 3 | 161 |\n| 4 | 171 |\n| 5 | 210 |

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{113 + 142+162+161+171+210}{6}=\frac{959}{6}\approx159.8)

Step2: Calculate the slope (m)

[ \begin{align*} m&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\ \sum_{i = 1}^{6}(x_i-\bar{x})(y_i - \bar{y})&=(0 - 2.5)(113-159.8)+(1 - 2.5)(142 - 159.8)+(2-2.5)(162-159.8)+(3 - 2.5)(161-159.8)+(4-2.5)(171-159.8)+(5 - 2.5)(210-159.8)\ &=(- 2.5)(-46.8)+(-1.5)(-17.8)+(-0.5)(2.2)+(0.5)(1.2)+(1.5)(11.2)+(2.5)(50.2)\ &=117 + 26.7-1.1 + 0.6+16.8+125.5\ &=285.5 \end{align*} ] [ \begin{align*} \sum_{i=1}^{6}(x_i-\bar{x})^2&=(0 - 2.5)^2+(1 - 2.5)^2+(2-2.5)^2+(3 - 2.5)^2+(4-2.5)^2+(5 - 2.5)^2\ &=6.25+2.25 + 0.25+0.25+2.25+6.25\ &=17.5 \end{align*} ] (m=\frac{285.5}{17.5}\approx16.3)

Step3: Calculate the y - intercept (b)

We know that (y=mx + b), substituting (\bar{x}) and (\bar{y}) and (m) into the equation: (159.8=16.3\times2.5+b) (159.8 = 40.75+b) (b=159.8 - 40.75=119.1)

The linear - regression equation is (y = 16.3x+119.1)

Step4: Estimate the year when (y = 420)

Set (y = 420) in the equation (y = 16.3x+119.1) (420=16.3x+119.1) (16.3x=420 - 119.1) (16.3x=300.9) (x=\frac{300.9}{16.3}\approx18.5)

The year is (2012 + 19=2031) (we round up (x) to the next whole number since we are talking about years)

Answer:

The linear - regression equation is (y = 16.3x+119.1). The profits would reach 420 thousand dollars in the year 2031.