Question

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.

x	y
6	6
9	9
10	11
14	12

a. y = 0.894x + 0.535
b. y = - 0.535x + 0.894
c. y = - 0.894x + 0.535
d. y = 0.535x + 0.894

Explanation:

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

Let \(x_1 = 5,x_2=6,x_3 = 9,x_4=10,x_5 = 14\) and \(y_1 = 4,y_2=6,y_3 = 9,y_4=11,y_5 = 12\).
The mean of \(x\), \(\bar{x}=\frac{5 + 6+9+10+14}{5}=\frac{44}{5}=8.8\)
The mean of \(y\), \(\bar{y}=\frac{4 + 6+9+11+12}{5}=\frac{42}{5}=8.4\)

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

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

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

We know that \(y=mx + b\), substituting \(\bar{x}\), \(\bar{y}\) and \(m\) into the equation:
\(8.4=0.894\times8.8 + b\)
\(b=8.4-0.894\times8.8\)
\(b=8.4 - 7.8672=0.533\approx0.535\)
The equation of the line of best - fit is \(y = 0.894x+0.535\)

Answer:

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