Question

logan 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. area (in square kilometers) visitors last year 4.63 28,563 6.91 68,094 7.11 25,468 13.73 183,775 14.45 148,482 15.38 97,972 find the equation for the least squares regression line of the data described below. round your answers to the nearest thousandth. y = x +

Explanation:

Step1: Recall the formula for the least - squares regression line

The equation of the least - squares regression line is $y = mx + b$, where $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $b=\overline{y}-m\overline{x}$, with $n$ being the number of data points, $\overline{x}=\frac{1}{n}\sum_{i = 1}^{n}x_i$ and $\overline{y}=\frac{1}{n}\sum_{i = 1}^{n}y_i$.
Let $n = 6$.
First, calculate the sums:
$\sum_{i = 1}^{6}x_i=4.63 + 6.91+7.11 + 13.73+14.45+15.38=62.21$
$\sum_{i = 1}^{6}y_i=28563+68094 + 25468+183775+148482+97972=552354$
$\sum_{i = 1}^{6}x_i^{2}=4.63^{2}+6.91^{2}+7.11^{2}+13.73^{2}+14.45^{2}+15.38^{2}$
$=21.4369+47.7481+50.5521+188.5129+208.8025+236.5444 = 753.5969$
$\sum_{i = 1}^{6}x_iy_i=4.63\times28563+6.91\times68094+7.11\times25468+13.73\times183775+14.45\times148482+15.38\times97972$
$=4.63\times28563 = 132246.69$
$=6.91\times68094=460529.54$
$=7.11\times25468 = 181077.48$
$=13.73\times183775=2522230.75$
$=14.45\times148482=2145564.9$
$=15.38\times97972=1507819.36$
$\sum_{i = 1}^{6}x_iy_i=132246.69+460529.54+181077.48+2522230.75+2145564.9+1507819.36=6949468.72$

$\overline{x}=\frac{\sum_{i = 1}^{6}x_i}{n}=\frac{62.21}{6}\approx10.3683$
$\overline{y}=\frac{\sum_{i = 1}^{6}y_i}{n}=\frac{552354}{6}=92059$

Step2: Calculate the slope $m$

$m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$
$=\frac{6\times6949468.72-62.21\times552354}{6\times753.5969 - 62.21^{2}}$
$=\frac{41696812.32-34341952.34}{4521.5814 - 3870.0841}$
$=\frac{7354859.98}{651.4973}\approx11289.16$

Step3: Calculate the y - intercept $b$

$b=\overline{y}-m\overline{x}$
$=92059-11289.16\times10.3683$
$=92059-116057.97$
$=- 24000.97$

The equation of the least - squares regression line is $y = 11289.16x-24000.97$

Answer:

$y = 11289.16x - 24000.97$