Question

find the equation for the least squares regression line of the data described below.
a sporting goods chain is planning to open a new store in taylor county. in order to decide on an optimal location, their market researchers gathered data on sports facilities in towns across the county.
the researchers considered the population (in thousands), x, and the number of sports facilities, y, in each town.
population (in thousands) sports facilities
35 3
39 7
69 14
70 7
92 14
round your answers to the nearest thousandth.
y = x +
save answer

Explanation:

Step1: Calculate the sums

Let \(n = 5\).
The \(x\) - values are \(x_1=35,x_2 = 39,x_3=69,x_4 = 70,x_5=92\).
The \(y\) - values are \(y_1 = 3,y_2=7,y_3=14,y_4 = 7,y_5=14\).
\(\sum_{i = 1}^{n}x_i=35 + 39+69+70+92=305\)
\(\sum_{i = 1}^{n}y_i=3 + 7+14+7+14=45\)
\(\sum_{i = 1}^{n}x_i^2=35^2+39^2+69^2+70^2+92^2=1225+1521+4761+4900+8464=20871\)
\(\sum_{i = 1}^{n}x_iy_i=35\times3+39\times7+69\times14+70\times7+92\times14=105 + 273+966+490+1288=3122\)

Step2: Calculate the slope \(m\)

The formula for the slope \(m\) of the least - squares regression line is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\)
Substitute \(n = 5,\sum_{i = 1}^{n}x_i = 305,\sum_{i = 1}^{n}y_i=45,\sum_{i = 1}^{n}x_i^2=20871,\sum_{i = 1}^{n}x_iy_i=3122\) into the formula:
\[

$$\begin{align*} m&=\frac{5\times3122-305\times45}{5\times20871-(305)^2}\\ &=\frac{15610 - 13725}{104355-93025}\\ &=\frac{1885}{11330}\\ &\approx0.166 \end{align*}$$

\]

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

The formula for the \(y\) - intercept \(b\) is \(b=\overline{y}-m\overline{x}\), where \(\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\) and \(\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\)
\(\overline{x}=\frac{305}{5}=61\), \(\overline{y}=\frac{45}{5}=9\)
\(b = 9-0.166\times61=9 - 10.126=- 1.126\)

Answer:

\(y = 0.166x-1.126\)