Question

find the equation for the least squares regression line of the data described below. herman and molly are judges for the lanberry ice staking federation. due to claims of malfeasance at recent competitions, a reporter is investigating the relationship between the scores awarded by the two judges. she has collected the scores awarded by herman, x, and molly, y, for each performance in the last competition. hermans score mollys score 1.72 0.25 1.84 2.44 2.73 1.99 2.85 4.63 3.30 4.60 round your answers to the nearest thousandth. y = x +

Explanation:

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

Let \(x_1 = 1.72,x_2=1.84,x_3 = 2.73,x_4=2.85,x_5 = 3.30\) and \(y_1 = 0.25,y_2=2.44,y_3 = 1.99,y_4=4.63,y_5 = 4.60\).
The mean of \(x\), \(\bar{x}=\frac{1.72 + 1.84+2.73+2.85+3.30}{5}=\frac{12.44}{5}=2.488\).
The mean of \(y\), \(\bar{y}=\frac{0.25 + 2.44+1.99+4.63+4.60}{5}=\frac{14.91}{5}=2.982\).

Step2: Calculate the numerator and denominator for the slope \(b_1\)

The numerator \(S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\)
\((x_1-\bar{x})(y_1-\bar{y})=(1.72 - 2.488)(0.25-2.982)=(- 0.768)\times(-2.732)=2.098176\)
\((x_2-\bar{x})(y_2-\bar{y})=(1.84 - 2.488)(2.44 - 2.982)=(-0.648)\times(-0.542)=0.351216\)
\((x_3-\bar{x})(y_3-\bar{y})=(2.73 - 2.488)(1.99 - 2.982)=(0.242)\times(-0.992)=-0.240064\)
\((x_4-\bar{x})(y_4-\bar{y})=(2.85 - 2.488)(4.63 - 2.982)=(0.362)\times(1.648)=0.596576\)
\((x_5-\bar{x})(y_5-\bar{y})=(3.30 - 2.488)(4.60 - 2.982)=(0.812)\times(1.618)=1.313816\)
\(S_{xy}=2.098176 + 0.351216-0.240064 + 0.596576+1.313816=3.11972\)
The denominator \(S_{xx}=\sum_{i=1}^{n}(x_i-\bar{x})^2\)
\((x_1 - \bar{x})^2=(1.72-2.488)^2=(-0.768)^2 = 0.589824\)
\((x_2 - \bar{x})^2=(1.84 - 2.488)^2=(-0.648)^2=0.419904\)
\((x_3 - \bar{x})^2=(2.73 - 2.488)^2=(0.242)^2=0.058564\)
\((x_4 - \bar{x})^2=(2.85 - 2.488)^2=(0.362)^2=0.131044\)
\((x_5 - \bar{x})^2=(3.30 - 2.488)^2=(0.812)^2=0.659344\)
\(S_{xx}=0.589824+0.419904 + 0.058564+0.131044+0.659344=1.85868\)
The slope \(b_1=\frac{S_{xy}}{S_{xx}}=\frac{3.11972}{1.85868}\approx1.67846\approx1.678\)

Step3: Calculate the intercept \(b_0\)

The intercept \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0=2.982-1.678\times2.488=2.982 - 4.174864=-1.192864\approx - 1.193\)

Answer:

\(y = 1.678x-1.193\)