find the equation for the least squares regression line of the data described below.\na sporting goods chain…
find the equation for the least squares regression line of the data described below.\na 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.\nthe researchers considered the population (in thousands), x, and the number of sports facilities, y, in each town.\npopulation (in thousands) sports facilities\n35 3\n39 7\n69 14\n70 7\n92 14\nround your answers to the nearest thousandth.\ny = x + \nsave answer
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)