Question

find the equation for the least - squares regression line of the data described below. lester works with a property developer building houses close to the coastline in italy. his boss thinks that demand for the houses will be based primarily on their size. lester wants to show his boss that proximity to the ocean is also a big factor to consider. so, he looks at several houses of the same size in the area. he records the distance of each house from the ocean (in kilometers), x. he also notes the number of people who offered to buy each house, y, when it was last put up for sale. round your answers to the nearest thousandth. y = x +

Explanation:

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

Let \(x_1 = 7,x_2=12,x_3 = 17,x_4=17,x_5 = 18\) and \(y_1 = 14,y_2=6,y_3 = 6,y_4=7,y_5 = 12\).
The mean of \(x\), \(\bar{x}=\frac{7 + 12+17+17+18}{5}=\frac{71}{5}=14.2\)
The mean of \(y\), \(\bar{y}=\frac{14 + 6+6+7+12}{5}=\frac{45}{5}=9\)

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

The numerator \(S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\)
\((7-14.2)(14 - 9)+(12-14.2)(6 - 9)+(17-14.2)(6 - 9)+(17-14.2)(7 - 9)+(18-14.2)(12 - 9)\)
\(=(- 7.2)\times5+(-2.2)\times(-3)+2.8\times(-3)+2.8\times(-2)+3.8\times3\)
\(=-36 + 6.6-8.4 - 5.6+11.4\)
\(=-31.6\)
The denominator \(S_{xx}=\sum_{i = 1}^{n}(x_i-\bar{x})^2\)
\((7 - 14.2)^2+(12-14.2)^2+(17-14.2)^2+(17-14.2)^2+(18-14.2)^2\)
\(=(-7.2)^2+(-2.2)^2+2.8^2+2.8^2+3.8^2\)
\(=51.84+4.84 + 7.84+7.84+14.44\)
\(=86.8\)
The slope \(b_1=\frac{S_{xy}}{S_{xx}}=\frac{-31.6}{86.8}\approx - 0.364\)

Step3: Calculate the intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}\)
\(b_0=9-(-0.364)\times14.2\)
\(b_0=9 + 5.169\)
\(b_0 = 14.169\)

Answer:

\(y=-0.364x + 14.169\)