Question

8.2.3 quiz: fitting linear models to data
what is the equation of the line of best fit for the following data? round the slope and y - intercept of the line to three decimal places.
a. ( y = 0.535x + 0.894 )
b. ( y = - 0.894x + 0.535 )
c. ( y = 0.894x + 0.535 )
d. ( y = - 0.535x + 0.894 )

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

\(\bar{x}=\frac{5 + 6+9+10+14}{5}=\frac{44}{5} = 8.8\)
\(\bar{y}=\frac{4 + 6+9+11+12}{5}=\frac{42}{5}=8.4\)

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

Numerator: \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\((5 - 8.8)(4 - 8.4)+(6 - 8.8)(6 - 8.4)+(9 - 8.8)(9 - 8.4)+(10 - 8.8)(11 - 8.4)+(14 - 8.8)(12 - 8.4)\)
\(=(-3.8)(-4.4)+(-2.8)(-2.4)+(0.2)(0.6)+(1.2)(2.6)+(5.2)(3.6)\)
\(=16.72+6.72 + 0.12+3.12+18.72=45.4\)
Denominator: \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\)
\((5 - 8.8)^{2}+(6 - 8.8)^{2}+(9 - 8.8)^{2}+(10 - 8.8)^{2}+(14 - 8.8)^{2}\)
\(=(-3.8)^{2}+(-2.8)^{2}+(0.2)^{2}+(1.2)^{2}+(5.2)^{2}\)
\(=14.44+7.84+0.04+1.44+27.04 = 50.8\)
Slope \(m=\frac{45.4}{50.8}\approx0.894\)

Step3: Calculate the \(y\) - intercept \(b\)

Using the formula \(b=\bar{y}-m\bar{x}\)
\(b = 8.4-0.894\times8.8\)
\(b=8.4 - 7.8672=0.533\approx0.535\)

Answer:

C. \(y = 0.894x+0.535\)