Question

#_______ previous answer: 0.4
determine a linear model for the following data:

Explanation:

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

$$ \bar{x}=\frac{6 + 27+14 + 38+63+32}{6}=\frac{180}{6}=30 $$

$$ \bar{y}=\frac{76 + 43+70 + 34+17+46}{6}=\frac{286}{6}\approx47.67 $$

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

$$ \text{Numerator}=\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=(6 - 30)(76 - 47.67)+(27 - 30)(43 - 47.67)+(14 - 30)(70 - 47.67)+(38 - 30)(34 - 47.67)+(63 - 30)(17 - 47.67)+(32 - 30)(46 - 47.67) $$

$$ =-24\times28.33-3\times(-4.67)-16\times22.33 + 8\times(-13.67)+33\times(-30.67)+2\times(-1.67) $$

$$ =-680 - (-14) - 357.28-109.36-1012.11 - 3.34=-2148.09 $$

$$ \text{Denominator}=\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=(6 - 30)^{2}+(27 - 30)^{2}+(14 - 30)^{2}+(38 - 30)^{2}+(63 - 30)^{2}+(32 - 30)^{2} $$

$$ =(-24)^{2}+(-3)^{2}+(-16)^{2}+8^{2}+33^{2}+2^{2}=576 + 9+256+64+1089+4 = 1998 $$

$$ m=\frac{-2148.09}{1998}\approx - 1.08 $$

Step3: Calculate the intercept \(b\)

$$ b=\bar{y}-m\bar{x}=47.67-(-1.08)\times30=47.67 + 32.4=80.07 $$

Step4: Write the linear model

The linear model is \(y=-1.08x + 80.07\)

Answer:

\(y=-1.08x + 80.07\)