Question

what is the slope of the least - squares regression line for these data? carry you answer to at least two decimal places. (if necessary, consult a list of formulas.

Explanation:

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

Let \(n = 12\) (number of data points).
\(\bar{x}=\frac{25.5 + 48.6+13.9+31.1+34.9+45.6+49.3+48.0+14.3+25.6+42.4+18.3}{12}\)
\(=\frac{377.5}{12}\approx31.46\)
\(\bar{y}=\frac{72.5 + 62.0+72.9+62.0+66.0+57.5+60.6+53.6+75.3+72.3+66.1+72.6}{12}\)
\(=\frac{813.4}{12}\approx67.78\)

Step2: Calculate the numerator and denominator for the slope formula

The formula for the slope \(b\) of the least - squares regression line is \(b=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
First, calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) and \((x_{i}-\bar{x})^{2}\) for each data point:
For \(x = 25.5,y = 72.5\): \((25.5 - 31.46)(72.5-67.78)=(- 5.96)\times4.72=-28.13\)
\((25.5 - 31.46)^{2}=(-5.96)^{2}=35.52\)
For \(x = 48.6,y = 62.0\): \((48.6 - 31.46)(62.0 - 67.78)=17.14\times(-5.78)=-99.07\)
\((48.6 - 31.46)^{2}=17.14^{2}=293.78\)
For \(x = 13.9,y = 72.9\): \((13.9 - 31.46)(72.9 - 67.78)=(-17.56)\times5.12=-89.91\)
\((13.9 - 31.46)^{2}=(-17.56)^{2}=308.35\)
For \(x = 31.1,y = 62.0\): \((31.1 - 31.46)(62.0 - 67.78)=(-0.36)\times(-5.78)=2.08\)
\((31.1 - 31.46)^{2}=(-0.36)^{2}=0.13\)
For \(x = 34.9,y = 66.0\): \((34.9 - 31.46)(66.0 - 67.78)=3.44\times(-1.78)=-6.12\)
\((34.9 - 31.46)^{2}=3.44^{2}=11.83\)
For \(x = 45.6,y = 57.5\): \((45.6 - 31.46)(57.5 - 67.78)=14.14\times(-10.28)=-145.36\)
\((45.6 - 31.46)^{2}=14.14^{2}=199.94\)
For \(x = 49.3,y = 60.6\): \((49.3 - 31.46)(60.6 - 67.78)=17.84\times(-7.18)=-128.19\)
\((49.3 - 31.46)^{2}=17.84^{2}=318.27\)
For \(x = 48.0,y = 53.6\): \((48.0 - 31.46)(53.6 - 67.78)=16.54\times(-14.18)=-234.54\)
\((48.0 - 31.46)^{2}=16.54^{2}=273.57\)
For \(x = 14.3,y = 75.3\): \((14.3 - 31.46)(75.3 - 67.78)=(-17.16)\times7.52=-129.04\)
\((14.3 - 31.46)^{2}=(-17.16)^{2}=294.46\)
For \(x = 25.6,y = 72.3\): \((25.6 - 31.46)(72.3 - 67.78)=(-5.86)\times4.52=-26.49\)
\((25.6 - 31.46)^{2}=(-5.86)^{2}=34.34\)
For \(x = 42.4,y = 66.1\): \((42.4 - 31.46)(66.1 - 67.78)=10.94\times(-1.68)=-18.38\)
\((42.4 - 31.46)^{2}=10.94^{2}=119.68\)
For \(x = 18.3,y = 72.6\): \((18.3 - 31.46)(72.6 - 67.78)=(-13.16)\times4.82=-63.43\)
\((18.3 - 31.46)^{2}=(-13.16)^{2}=173.18\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=-28.13-99.07 - 89.91+2.08-6.12-145.36-128.19-234.54-129.04-26.49-18.38-63.43=-1166.5\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=35.52 + 293.78+308.35+0.13+11.83+199.94+318.27+273.57+294.46+34.34+119.68+173.18=2062.05\)

\(b=\frac{-1166.5}{2062.05}\approx - 0.57\)

Answer:

\(-0.57\)