Question

find the equation for the least squares regression line of the data described below.
paula has noticed that her bike ride to work takes longer on some days than others. she is curious to see how the morning temperature is related to the duration of her commute.
for the past several mornings, she measured the temperature (in celsius), x, and the time her commute had taken (in minutes), y.
round your answers to the nearest thousandth.
y = x +

Explanation:

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

Let \(x_i\) be the temperature values and \(y_i\) be the commute - time values.
The number of data points \(n = 7\).
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{-1.97-1.62 - 0.10+1.07+5.53+7.31+8.94}{7}=\frac{18.16}{7}\approx2.594\)
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{56.31 + 57.35+48.70+58.32+48.69+46.77+31.01}{7}=\frac{347.15}{7}\approx49.593\)

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

The formula for \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(-1.97 - 2.594)(56.31-49.593)=(-4.564)(6.717)\approx - 30.677\)
\((x_2-\bar{x})(y_2 - \bar{y})=(-1.62 - 2.594)(57.35 - 49.593)=(-4.214)(7.757)\approx-32.768\)
\((x_3-\bar{x})(y_3 - \bar{y})=(-0.10 - 2.594)(48.70 - 49.593)=(-2.694)(-0.893)\approx2.406\)
\((x_4-\bar{x})(y_4 - \bar{y})=(1.07 - 2.594)(58.32 - 49.593)=(-1.524)(8.727)\approx-13.290\)
\((x_5-\bar{x})(y_5 - \bar{y})=(5.53 - 2.594)(48.69 - 49.593)=(2.936)(-0.903)\approx - 2.651\)
\((x_6-\bar{x})(y_6 - \bar{y})=(7.31 - 2.594)(46.77 - 49.593)=(4.716)(-2.823)\approx-13.313\)
\((x_7-\bar{x})(y_7 - \bar{y})=(8.94 - 2.594)(31.01 - 49.593)=(6.346)(-18.583)\approx - 118.064\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=-30.677-32.768 + 2.406-13.290-2.651-13.313-118.064=-208.357\)

\((x_1-\bar{x})^2=(-1.97 - 2.594)^2=(-4.564)^2\approx20.830\)
\((x_2-\bar{x})^2=(-1.62 - 2.594)^2=(-4.214)^2\approx17.758\)
\((x_3-\bar{x})^2=(-0.10 - 2.594)^2=(-2.694)^2\approx7.258\)
\((x_4-\bar{x})^2=(1.07 - 2.594)^2=(-1.524)^2\approx2.323\)
\((x_5-\bar{x})^2=(5.53 - 2.594)^2=(2.936)^2\approx8.620\)
\((x_6-\bar{x})^2=(7.31 - 2.594)^2=(4.716)^2\approx22.241\)
\((x_7-\bar{x})^2=(8.94 - 2.594)^2=(6.346)^2\approx40.272\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=20.830 + 17.758+7.258+2.323+8.620+22.241+40.272=119.302\)

\(b_1=\frac{-208.357}{119.302}\approx - 1.746\)

Step3: Calculate the intercept \(b_0\)

The formula for \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 49.593-(-1.746)\times2.594=49.593 + 4.530=54.123\)

Answer:

\(y=-1.746x + 54.123\)