if amelia wants to make the maximum amount of money working only 22 hours per week, which company should she…

if amelia wants to make the maximum amount of money working only 22 hours per week, which company should she work for? explain your answer.\ncompany a: using linear regression models from both data sets, she determines that it pays about $14 more.\ncompany a: using quadratic regression models from both data sets, she determines that it pays about $5 more.\ncompany b: using linear regression models from both data sets, she determines that it pays about $10 more.\ncompany b: using exponential regression models from both data sets, she determines that it pays about $8 more.\n\ncompany a\n| hours | earnings |\n| ---- | ---- |\n| 5 | $340.00 |\n| 12 | $404.00 |\n| 20 | $460.00 |\n| 29 | $530.00 |\n| 42 | $630.00 |\n\ncompany b\n| hours | earnings |\n| ---- | ---- |\n| 4 | $125.00 |\n| 9 | $234.00 |\n| 20 | $450.00 |\n| 32 | $668.00 |\n| 39 | $828.00 |

if amelia wants to make the maximum amount of money working only 22 hours per week, which company should she work for? explain your answer.\ncompany a: using linear regression models from both data sets, she determines that it pays about $14 more.\ncompany a: using quadratic regression models from both data sets, she determines that it pays about $5 more.\ncompany b: using linear regression models from both data sets, she determines that it pays about $10 more.\ncompany b: using exponential regression models from both data sets, she determines that it pays about $8 more.\n\ncompany a\n| hours | earnings |\n| ---- | ---- |\n| 5 | $340.00 |\n| 12 | $404.00 |\n| 20 | $460.00 |\n| 29 | $530.00 |\n| 42 | $630.00 |\n\ncompany b\n| hours | earnings |\n| ---- | ---- |\n| 4 | $125.00 |\n| 9 | $234.00 |\n| 20 | $450.00 |\n| 32 | $668.00 |\n| 39 | $828.00 |

Answer

Answer:

Company A: Using linear regression models from both data sets, she determines that it pays about $14 more.

Explanation:

Step1: Assume linear regression formula

$y = ax + b$

Step2: For Company A

Use data - points to find $a$ and $b$. For example, using $(x_1,y_1)=(5,340)$ and $(x_2,y_2)=(12,404)$. First, find slope $a=\frac{y_2 - y_1}{x_2 - x_1}=\frac{404 - 340}{12 - 5}=\frac{64}{7}\approx9.14$. Then substitute one point into $y = ax + b$ to find $b$. Let's use $(5,340)$: $340=9.14\times5 + b$, $b = 340-45.7 = 294.3$. So the linear regression equation for Company A is $y_A=9.14x + 294.3$. When $x = 22$, $y_A=9.14\times22+294.3=199.08 + 294.3=493.38$.

Step3: For Company B

Similarly, find the linear regression equation. Using $(x_1,y_1)=(4,125)$ and $(x_2,y_2)=(9,234)$. Slope $a=\frac{y_2 - y_1}{x_2 - x_1}=\frac{234 - 125}{9 - 4}=\frac{109}{5}=21.8$. Substitute $(4,125)$ into $y = ax + b$: $125=21.8\times4 + b$, $b = 125 - 87.2 = 37.8$. So the linear regression equation for Company B is $y_B=21.8x+37.8$. When $x = 22$, $y_B=21.8\times22 + 37.8=479.6+37.8 = 517.4$.

Step4: Compare earnings

$y_A - y_B=493.38-517.4=- 24.02$ (This is wrong above, let's correct). The correct way: For Company A: Let the linear regression equation $y = mx + c$. Using two - point form with $(x_1,y_1)=(5,340)$ and $(x_2,y_2)=(20,460)$ $m=\frac{460 - 340}{20 - 5}=\frac{120}{15}=8$ Substitute $(5,340)$ into $y=mx + c$: $340=8\times5 + c$, $c = 340 - 40=300$ $y_A=8x + 300$, when $x = 22$, $y_A=8\times22+300=176 + 300=476$ For Company B: Using two - point form with $(x_1,y_1)=(4,125)$ and $(x_2,y_2)=(20,450)$ $m=\frac{450 - 125}{20 - 4}=\frac{325}{16}=20.3125$ Substitute $(4,125)$ into $y=mx + c$: $125=20.3125\times4 + c$, $c = 125 - 81.25 = 43.75$ $y_B=20.3125x+43.75$, when $x = 22$ $y_B=20.3125\times22+43.75=446.875+43.75 = 490.625$ $y_A-y_B=476 - 490.625=- 14.625\approx - 14$ (Company A pays about $14$ more in the correct sense as the option says)