Question

find the equation for the least squares regression line of the data described below. coach leonard is responsible for recruiting male athletes to join the european masters track and field team. to improve his recruitment strategies, he wants to investigate the connection between an athletes height and 3000 - meter run time. coach leonard has recorded the heights of the men on the track and field team (in centimeters), x, and their best 3000 - meter times (in minutes), y. round your answers to the nearest thousandth. y = x +

Explanation:

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

Let \(x\) be the height values \(157,164,170,172,178,180,183,187\) and \(y\) be the time values \(8.80,7.77,8.51,8.37,7.92,7.54,7.60,7.38\).
The mean of \(x\), \(\bar{x}=\frac{157 + 164+170+172+178+180+183+187}{8}=\frac{1411}{8}=176.375\)
The mean of \(y\), \(\bar{y}=\frac{8.80 + 7.77+8.51+8.37+7.92+7.54+7.60+7.38}{8}=\frac{64.89}{8}=8.11125\)

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

The formula for \(b_1=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}\)
\((x_1-\bar{x})(y_1-\bar{y})=(157 - 176.375)(8.80 - 8.11125)=(- 19.375)\times0.68875=-13.353515625\)
\((x_2-\bar{x})(y_2-\bar{y})=(164 - 176.375)(7.77 - 8.11125)=(-12.375)\times(-0.34125)=4.222265625\)
\((x_3-\bar{x})(y_3-\bar{y})=(170 - 176.375)(8.51 - 8.11125)=(-6.375)\times0.39875=-2.5416015625\)
\((x_4-\bar{x})(y_4-\bar{y})=(172 - 176.375)(8.37 - 8.11125)=(-4.375)\times0.25875=-1.130859375\)
\((x_5-\bar{x})(y_5-\bar{y})=(178 - 176.375)(7.92 - 8.11125)=(1.625)\times(-0.19125)=-0.3103515625\)
\((x_6-\bar{x})(y_6-\bar{y})=(180 - 176.375)(7.54 - 8.11125)=(3.625)\times(-0.57125)=-2.0717578125\)
\((x_7-\bar{x})(y_7-\bar{y})=(183 - 176.375)(7.60 - 8.11125)=(6.625)\times(-0.51125)=-3.384140625\)
\((x_8-\bar{x})(y_8-\bar{y})=(187 - 176.375)(7.38 - 8.11125)=(10.625)\times(-0.73125)=-7.763671875\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})=-13.353515625 + 4.222265625-2.5416015625-1.130859375-0.3103515625-2.0717578125-3.384140625-7.763671875=-26.2336328125\)

\((x_1-\bar{x})^2=(157 - 176.375)^2=(-19.375)^2 = 375.390625\)
\((x_2-\bar{x})^2=(164 - 176.375)^2=(-12.375)^2=153.140625\)
\((x_3-\bar{x})^2=(170 - 176.375)^2=(-6.375)^2 = 40.640625\)
\((x_4-\bar{x})^2=(172 - 176.375)^2=(-4.375)^2=19.140625\)
\((x_5-\bar{x})^2=(178 - 176.375)^2=(1.625)^2=2.640625\)
\((x_6-\bar{x})^2=(180 - 176.375)^2=(3.625)^2 = 13.140625\)
\((x_7-\bar{x})^2=(183 - 176.375)^2=(6.625)^2=43.980625\)
\((x_8-\bar{x})^2=(187 - 176.375)^2=(10.625)^2=112.890625\)
\(\sum_{i=1}^{n}(x_i-\bar{x})^2=375.390625+153.140625 + 40.640625+19.140625+2.640625+13.140625+43.980625+112.890625=760.965625\)

\(b_1=\frac{-26.2336328125}{760.965625}\approx - 0.034\)

Step3: Calculate the intercept \(b_0\)

The formula for \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0=8.11125-(-0.034)\times176.375=8.11125 + 6.00675=14.118\)

Answer:

\(y=-0.034x + 14.118\)