Question

find the equation for the least squares regression line of the data described below.
clay countys parks and recreation department is considering establishing a new park. as part of the decision - making process, the department asked an intern to conduct a park usage study.
the study considered the area (in square kilometers), x, and the number of visitors last year, y, of each county park.
round your answers to the nearest thousandth.
y = x +

Explanation:

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

Let \(x_1 = 11.94\), \(x_2=12.15\), \(x_3 = 17.89\), \(x_4=18.59\), \(x_5 = 19.01\), \(x_6=19.75\)
\(\bar{x}=\frac{11.94 + 12.15+17.89+18.59+19.01+19.75}{6}=\frac{109.33}{6}\approx18.222\)

Let \(y_1 = 26775\), \(y_2 = 39960\), \(y_3=162735\), \(y_4 = 74791\), \(y_5=82542\), \(y_6=179248\)
\(\bar{y}=\frac{26775+39960 + 162735+74791+82542+179248}{6}=\frac{566051}{6}\approx94341.833\)

Step2: Calculate \(S_{xx}\) and \(S_{xy}\)

\(S_{xx}=\sum_{i = 1}^{6}(x_i-\bar{x})^2\)
\((11.94 - 18.222)^2+(12.15 - 18.222)^2+(17.89 - 18.222)^2+(18.59 - 18.222)^2+(19.01 - 18.222)^2+(19.75 - 18.222)^2\)
\(=(- 6.282)^2+(-6.072)^2+(-0.332)^2+(0.368)^2+(0.788)^2+(1.528)^2\)
\(=39.464 + 36.869+0.110+0.135+0.621+2.335\approx79.534\)

\(S_{xy}=\sum_{i = 1}^{6}(x_i-\bar{x})(y_i-\bar{y})\)
\((11.94 - 18.222)(26775 - 94341.833)+(12.15 - 18.222)(39960 - 94341.833)+(17.89 - 18.222)(162735 - 94341.833)+(18.59 - 18.222)(74791 - 94341.833)+(19.01 - 18.222)(82542 - 94341.833)+(19.75 - 18.222)(179248 - 94341.833)\)
\(=(-6.282)(-67566.833)+(-6.072)(-54381.833)+(-0.332)(68343.167)+(0.368)(-19550.833)+(0.788)(-11807.833)+(1.528)(84906.167)\)
\(=424907.216+330117.490-22690.931-7194.707-9304.573+129736.624\approx845570.129\)

Step3: Calculate the slope \(b\) and the intercept \(a\)

The slope \(b=\frac{S_{xy}}{S_{xx}}=\frac{845570.129}{79.534}\approx10631.530\)
The intercept \(a=\bar{y}-b\bar{x}=94341.833-10631.530\times18.222\)
\(=94341.833 - 193718.750=-99376.917\)

Answer:

\(y = 10631.530x-99376.917\)