Question

find the equation for the least squares regression line of the data described below.
on todays episode of science time with sanjay, sanjay demonstrates that walking speed affects how wet a person will get in the rain. he simulates walking through the rain at various speeds.
sanjay keeps track of his walking speed (in meters per second), x. next, he measures how wet he is by taking the difference of his weight before and after walking through the rain (in kilograms), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
For \(x\) values \(x_1 = 0.5,x_2=1.0,x_3 = 1.5,x_4=1.9,x_5 = 1.9\), \(n = 5\).
\(\sum_{i=1}^{5}x_{i}=0.5 + 1.0+1.5+1.9+1.9=6.8\), \(\bar{x}=\frac{6.8}{5}=1.36\).
For \(y\) values \(y_1 = 1.6,y_2=0.5,y_3 = 1.3,y_4=1.9,y_5 = 0.5\), \(\sum_{i=1}^{5}y_{i}=1.6 + 0.5+1.3+1.9+0.5=5.8\), \(\bar{y}=\frac{5.8}{5}=1.16\).

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

The formula for the slope \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}\).
\((x_1-\bar{x})(y_1 - \bar{y})=(0.5 - 1.36)(1.6-1.16)=(- 0.86)\times0.44=-0.3784\)
\((x_2-\bar{x})(y_2 - \bar{y})=(1.0 - 1.36)(0.5 - 1.16)=(-0.36)\times(-0.66) = 0.2376\)
\((x_3-\bar{x})(y_3 - \bar{y})=(1.5 - 1.36)(1.3 - 1.16)=0.14\times0.14 = 0.0196\)
\((x_4-\bar{x})(y_4 - \bar{y})=(1.9 - 1.36)(1.9 - 1.16)=0.54\times0.74 = 0.3996\)
\((x_5-\bar{x})(y_5 - \bar{y})=(1.9 - 1.36)(0.5 - 1.16)=0.54\times(-0.66)=-0.3564\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=-0.3784 + 0.2376+0.0196 + 0.3996-0.3564=-0.078\)
\((x_1-\bar{x})^2=(0.5 - 1.36)^2=(-0.86)^2 = 0.7396\)
\((x_2-\bar{x})^2=(1.0 - 1.36)^2=(-0.36)^2=0.1296\)
\((x_3-\bar{x})^2=(1.5 - 1.36)^2=(0.14)^2 = 0.0196\)
\((x_4-\bar{x})^2=(1.9 - 1.36)^2=(0.54)^2=0.2916\)
\((x_5-\bar{x})^2=(1.9 - 1.36)^2=(0.54)^2=0.2916\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^2=0.7396+0.1296 + 0.0196+0.2916+0.2916=1.472\)
\(b=\frac{-0.078}{1.472}\approx - 0.053\)

Step3: Calculate the intercept \(a\)

The formula for the intercept \(a=\bar{y}-b\bar{x}\)
\(a = 1.16-(-0.053)\times1.36=1.16 + 0.07208=1.23208\approx1.232\)

Answer:

\(y=-0.053x + 1.232\)