Question

6.4.3 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 = 4.105x + 1.560 )
b. ( y = 1.560x + 4.105 )
c. ( y = 4.105x - 1.560 )
d. ( y = 1.560x - 4.105 )

Explanation:

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

\(\bar{x}=\frac{4 + 6+8+11+13}{5}=\frac{42}{5}=8.4\)
\(\bar{y}=\frac{3 + 4+9+12+17}{5}=\frac{45}{5}=9\)

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

Numerator: \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(4 - 8.4)(3 - 9)+(6 - 8.4)(4 - 9)+(8 - 8.4)(9 - 9)+(11 - 8.4)(12 - 9)+(13 - 8.4)(17 - 9)\)
\(=(- 4.4)\times(-6)+(-2.4)\times(-5)+(-0.4)\times0+(2.6)\times3+(4.6)\times8\)
\(=26.4 + 12+0+7.8+36.8=83\)

Denominator: \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(4 - 8.4)^{2}+(6 - 8.4)^{2}+(8 - 8.4)^{2}+(11 - 8.4)^{2}+(13 - 8.4)^{2}\)
\(=(-4.4)^{2}+(-2.4)^{2}+(-0.4)^{2}+(2.6)^{2}+(4.6)^{2}\)
\(=19.36+5.76 + 0.16+6.76+21.16=53.2\)

Slope \(m=\frac{83}{53.2}\approx1.560\)

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

Using the formula \(b=\bar{y}-m\bar{x}\), substitute \(m = 1.560\) and \(\bar{x}=8.4\), \(\bar{y}=9\)
\(b=9-1.560\times8.4=9 - 13.104=- 4.104\approx - 4.105\)

The equation of the line of best - fit is \(y = 1.560x-4.105\)

Answer:

D. \(y = 1.560x-4.105\)