Question

find the equation for the least squares regression line of the data described below.
the manager of a ski resort in the alps always worries there wont be enough snow to keep the resort open into the spring. she decided to see if there was a relationship between the temperature in january and the amount of snow in the spring.
for several years, she recorded the average temperature in january (in celsius), x. on march 1, she also measured the depth of the snow at the bottom of a particular ski slope (in centimeters), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

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

The \(x\) - values are \(x_1=-9,x_2 = - 8,x_3=-8,x_4=-6,x_5 = 0\).
The mean of \(x\) is \(\bar{x}=\frac{-9+( - 8)+(-8)+(-6)+0}{5}=\frac{-31}{5}=-6.2\)
The \(y\) - values are \(y_1 = 38,y_2=68,y_3 = 47,y_4=51,y_5 = 32\)
The mean of \(y\) is \(\bar{y}=\frac{38 + 68+47+51+32}{5}=\frac{236}{5}=47.2\)

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

The formula for the slope \(b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1-\bar{y})=(-9 + 6.2)(38 - 47.2)=(-2.8)(-9.2)=25.76\)
\((x_2-\bar{x})(y_2-\bar{y})=(-8 + 6.2)(68 - 47.2)=(-1.8)(20.8)=-37.44\)
\((x_3-\bar{x})(y_3-\bar{y})=(-8 + 6.2)(47 - 47.2)=(-1.8)(-0.2)=0.36\)
\((x_4-\bar{x})(y_4-\bar{y})=(-6 + 6.2)(51 - 47.2)=(0.2)(3.8)=0.76\)
\((x_5-\bar{x})(y_5-\bar{y})=(0 + 6.2)(32 - 47.2)=(6.2)(-15.2)=-94.24\)
\(\sum_{i = 1}^{5}(x_i-\bar{x})(y_i-\bar{y})=25.76-37.44 + 0.36+0.76-94.24=-104.8\)

\((x_1-\bar{x})^2=(-9 + 6.2)^2=(-2.8)^2 = 7.84\)
\((x_2-\bar{x})^2=(-8 + 6.2)^2=(-1.8)^2 = 3.24\)
\((x_3-\bar{x})^2=(-8 + 6.2)^2=(-1.8)^2 = 3.24\)
\((x_4-\bar{x})^2=(-6 + 6.2)^2=(0.2)^2 = 0.04\)
\((x_5-\bar{x})^2=(0 + 6.2)^2=38.44\)
\(\sum_{i = 1}^{5}(x_i-\bar{x})^2=7.84+3.24+3.24+0.04+38.44=52.8\)

\(b=\frac{-104.8}{52.8}\approx - 1.985\)

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

The formula for the \(y\) - intercept \(a=\bar{y}-b\bar{x}\)
\(a = 47.2-(-1.985)\times(-6.2)=47.2-12.307 = 34.893\)

Answer:

\(y=-1.985x + 34.893\)