To estimate the score of Luke for week 12, we need to determine first the equation for the line of best fit.
To get the equation, here are the steps:
1. Solve for the average value of the x-coordinates and y-coordinates.
Our x-coordinates will be the number of week while the y-coordinates will be the score in each week.
Average for the x-coordinates is:
[tex]\text{average (x)}=\frac{1+2+3+4+5+6}{6}=\frac{21}{6}=3.5[/tex]
Average for the y-coordinate is:
[tex]\text{average(y)}=\frac{95+92+89+88+86+84}{6}=\frac{534}{6}=89[/tex]
2. Subtract each x and y coordinates to its corresponding averages.
3. Multiply the difference of x and its average to the difference of y and its average. (5th column)
4. Square the difference of x and its average. (6th column)
5. Solve for the slope of the equation. The formula is:
[tex]m=\frac{\sum ^{}_{}(x-ave_x)(y-ave_y)}{\sum ^{}_{}(x-ave_x)^2\text{ }}[/tex][tex]\begin{gathered} m=\frac{-15-4.5+0-0.5-4.5-12.5}{6.25+2.25+0.25+0.25+2.25+6.25} \\ m=-\frac{37}{17.5} \\ m=-2.11429 \end{gathered}[/tex]
The slope of the line is m = -2.11429.
6. Solve for the y-intercept. The formula is:
[tex]b=ave_y-m(ave_x)[/tex]
Since we already have ave y = 89, ave x = 3.5, and m = -2.11429, let's plug them in to the formula above.
[tex]\begin{gathered} b=89-(-2.11429)(3.5) \\ b=89+7.4 \\ b=96.4 \end{gathered}[/tex]
The y-intercept is 96.4.
Hence, the equation of the line of best fit is y = -2.11429x + 96.4.
To estimate the score of Luke for week 12, plug in x = 12 in the equation above and solve for y.
[tex]\begin{gathered} y=-2.11429x+96.4 \\ y=-2.11429(12)+96.4 \\ y=-25.37148+96.4 \\ y=71.02852 \\ y\approx71 \end{gathered}[/tex]
Therefore, the score of Luke for week 12 is estimatedly to be 71.
