Question

find the equation for the least squares regression line of the data described below. paula is a programmer in silicon valley and spends most of her day typing and drinking coffee. she wants to know if drinking coffee makes her type faster. over the next few days, paula varies the number of cups of coffee she drinks, x, and measures the average number of words typed per minute on each day, y. cups of coffee words typed 0.66 102 0.98 103 1.48 104 2.65 109 2.83 110 3.35 109 5.43 109 round your answers to the nearest thousandth. y = x +

Explanation:

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

Let \(x_i\) be the number of cups of coffee and \(y_i\) be the number of words typed.
\(n = 7\)
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{0.66 + 0.98+1.48+2.65+2.83+3.35+5.43}{7}=\frac{16.38}{7}\approx2.34\)
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{102 + 103+104+109+110+109+109}{7}=\frac{746}{7}\approx106.571\)

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

\(S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\)
\((0.66 - 2.34)(102-106.571)+(0.98 - 2.34)(103 - 106.571)+(1.48 - 2.34)(104 - 106.571)+(2.65 - 2.34)(109 - 106.571)+(2.83 - 2.34)(110 - 106.571)+(3.35 - 2.34)(109 - 106.571)+(5.43 - 2.34)(109 - 106.571)\)
\(=(-1.68)(-4.571)+(-1.36)(-3.571)+(-0.86)(-2.571)+(0.31)(2.429)+(0.49)(3.429)+(1.01)(2.429)+(3.09)(2.429)\)
\(=7.679+4.857+2.211 + 0.753+1.680+2.453+7.496\)
\(=26.129\)
\(S_{xx}=\sum_{i = 1}^{n}(x_i-\bar{x})^2\)
\((0.66 - 2.34)^2+(0.98 - 2.34)^2+(1.48 - 2.34)^2+(2.65 - 2.34)^2+(2.83 - 2.34)^2+(3.35 - 2.34)^2+(5.43 - 2.34)^2\)
\(=(-1.68)^2+(-1.36)^2+(-0.86)^2+(0.31)^2+(0.49)^2+(1.01)^2+(3.09)^2\)
\(=2.822+1.849+0.739+0.096+0.240+1.020+9.548\)
\(=16.314\)
\(b_1=\frac{S_{xy}}{S_{xx}}=\frac{26.129}{16.314}\approx1.59\)

Step3: Calculate the y - intercept \(b_0\)

\(b_0=\bar{y}-b_1\bar{x}\)
\(b_0 = 106.571-1.59\times2.34\)
\(b_0=106.571 - 3.721\)
\(b_0\approx102.850\)

Answer:

\(y = 1.590x+102.850\)