Question

8.4.3 test (cst): data and mathematical modeling
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 = - 13.634x + 0.927
b. y = 0.927x - 13.634
c. y = 13.634x - 0.927
d. y = - 0.927x + 13.634

Explanation:

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

\(\bar{x}=\frac{1 + 5+6+12+15}{5}=\frac{39}{5} = 7.8\)
\(\bar{y}=\frac{14 + 11+4+2+1}{5}=\frac{32}{5}=6.4\)

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

Numerator: \(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 7.8)(14 - 6.4)+(5 - 7.8)(11 - 6.4)+(6 - 7.8)(4 - 6.4)+(12 - 7.8)(2 - 6.4)+(15 - 7.8)(1 - 6.4)\)
\(=(- 6.8)\times7.6+(-2.8)\times4.6+(-1.8)\times(-2.4)+4.2\times(-4.4)+7.2\times(-5.4)\)
\(=-51.68-12.88 + 4.32-18.48-38.88=-117.5\)
Denominator: \(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=(1 - 7.8)^{2}+(5 - 7.8)^{2}+(6 - 7.8)^{2}+(12 - 7.8)^{2}+(15 - 7.8)^{2}\)
\(=(-6.8)^{2}+(-2.8)^{2}+(-1.8)^{2}+4.2^{2}+7.2^{2}\)
\(=46.24 + 7.84+3.24+17.64+51.84 = 126.8\)
Slope \(m=\frac{-117.5}{126.8}\approx - 0.927\)

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

Using the formula \(b=\bar{y}-m\bar{x}\), substitute \(m=-0.927\) and \(\bar{x}=7.8\), \(\bar{y}=6.4\)
\(b = 6.4-(-0.927)\times7.8=6.4 + 7.2306=13.6306\approx13.634\)

Step4: Write the equation of the line

The equation of the line is \(y=mx + b\), so \(y=-0.927x + 13.634\)

Answer:

D. \(y=-0.927x + 13.634\)