raquel throws darts at a coordinate grid centered at the origin. her goal is to create a line of darts. her…

Explanation:

Step1: Calculate the slope

To find the line of best fit, we can first calculate the slope ( m ) using the formula ( m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}} ), where ( \bar{x}=\frac{\sum x_{i}}{n} ) and ( \bar{y}=\frac{\sum y_{i}}{n} )

First, find ( \bar{x} ) and ( \bar{y} ):

( x )-values: (- 5,1,4,-8,0,9) ( \sum x=-5 + 1+4-8 + 0+9=1) ( n = 6), so ( \bar{x}=\frac{1}{6}\approx0.17)

( y )-values: (0,-3,5,-6,2,6) ( \sum y=0-3 + 5-6 + 2+6=4) ( \bar{y}=\frac{4}{6}\approx0.67)

Now calculate ( \sum(x_{i}-\bar{x})(y_{i}-\bar{y}) ) and ( \sum(x_{i}-\bar{x})^{2} )

For ( (-5,0) ): ( (x-\bar{x})=-5 - 0.17=-5.17) ( (y-\bar{y})=0 - 0.67=-0.67) ( (x - \bar{x})(y - \bar{y})=(-5.17)\times(-0.67)\approx3.46) ( (x - \bar{x})^{2}=(-5.17)^{2}\approx26.73)

For ( (1,-3) ): ( (x-\bar{x})=1 - 0.17 = 0.83) ( (y-\bar{y})=-3 - 0.67=-3.67) ( (x - \bar{x})(y - \bar{y})=0.83\times(-3.67)\approx - 3.05) ( (x - \bar{x})^{2}=0.83^{2}\approx0.69)

For ( (4,5) ): ( (x-\bar{x})=4 - 0.17 = 3.83) ( (y-\bar{y})=5 - 0.67 = 4.33) ( (x - \bar{x})(y - \bar{y})=3.83\times4.33\approx16.6) ( (x - \bar{x})^{2}=3.83^{2}\approx14.67)

For ( (-8,-6) ): ( (x-\bar{x})=-8 - 0.17=-8.17) ( (y-\bar{y})=-6 - 0.67=-6.67) ( (x - \bar{x})(y - \bar{y})=(-8.17)\times(-6.67)\approx54.5) ( (x - \bar{x})^{2}=(-8.17)^{2}\approx66.75)

For ( (0,2) ): ( (x-\bar{x})=0 - 0.17=-0.17) ( (y-\bar{y})=2 - 0.67 = 1.33) ( (x - \bar{x})(y - \bar{y})=(-0.17)\times1.33\approx - 0.23) ( (x - \bar{x})^{2}=(-0.17)^{2}\approx0.03)

For ( (9,6) ): ( (x-\bar{x})=9 - 0.17 = 8.83) ( (y-\bar{y})=6 - 0.67 = 5.33) ( (x - \bar{x})(y - \bar{y})=8.83\times5.33\approx47.06) ( (x - \bar{x})^{2}=8.83^{2}\approx78.0)

Now sum up the products and squares:

( \sum(x - \bar{x})(y - \bar{y})\approx3.46-3.05 + 16.6+54.5-0.23 + 47.06\approx118.34)

( \sum(x - \bar{x})^{2}\approx26.73 + 0.69+14.67+66.75+0.03+78.0\approx186.87)

Then ( m=\frac{118.34}{186.87}\approx0.63\approx0.6)

Step2: Check the y - intercept

We can also check the y - intercept by plugging in the point ( (0,2) ) (or approximate using the mean values). The mean of ( y ) is approximately ( 0.67 ) and the mean of ( x ) is approximately ( 0.17 ). Using the equation ( y=mx + b ), ( 0.67=0.6\times0.17 + b ), ( 0.67 = 0.102+b ), ( b\approx0.57\approx0.6)

Now check the options:

• Option 1: ( y = 0.6x+0.6 ) (slope ( 0.6 ), y - intercept ( 0.6 ))
• Option 2: ( y=0.1x + 0.8 ) (slope ( 0.1 ), not matching)
• Option 3: ( y = 0.8x+0.1 ) (slope ( 0.8 ), not matching)
• Option 4: ( y=0.5x + 0.6 ) (slope ( 0.5 ), close but our calculated slope is closer to ( 0.6 ))

Answer:

( y = 0.6x + 0.6 ) (the first option)