Question
Find the equation for the least squares regression line of the data described below.
Faced with budget cuts, a state commission is looking into closing some of its public libraries. To help minimize the negative impact of the closures, the commission gathered data on local school and library facilities.
The number of schools, x, and the number of libraries, y, were recorded for each town.
Schools Libraries
2 3
3 4
4 3
8 4
14 8
15 8
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the equation for the least squares regression line \(y = mx + b\), we need to calculate the slope \(m\) and the y-intercept \(b\).
Given your data points:
- (2, 3)
- (3, 4)
- (4, 3)
- (8, 4)
- (14, 8)
- (15, 8)
We can calculate:
1. The means of x and y:
\[
\bar{x} = \frac{2 + 3 + 4 + 8 + 14 + 15}{6} = \frac{46}{6} = 7.667
\]
\[
\bar{y} = \frac{3 + 4 + 3 + 4 + 8 + 8}{6} = \frac{30}{6} = 5
\]
2. The slope \(m\):
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
First, we calculate the necessary components:
- Calculate \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[
\begin{align*}
(2 - 7.667)(3 - 5) & = (-5.667)(-2) = 11.334 \\
(3 - 7.667)(4 - 5) & = (-4.667)(-1) = 4.667 \\
(4 - 7.667)(3 - 5) & = (-3.667)(-2) = 7.334 \\
(8 - 7.667)(4 - 5) & = (0.333)(-1) = -0.333 \\
(14 - 7.667)(8 - 5) & = (6.333)(3) = 19.000 \\
(15 - 7.667)(8 - 5) & = (7.333)(3) = 22.000 \\
\end{align*}
\]
- Sum:
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 11.334 + 4.667 + 7.334 - 0.333 + 19.000 + 22.000 = 64.000
\]
- Calculate \( (x_i - \bar{x})^2 \):
\[
\begin{align*}
(2 - 7.667)^2 & = 32.097 \\
(3 - 7.667)^2 & = 21.778 \\
(4 - 7.667)^2 & = 13.478 \\
(8 - 7.667)^2 & = 0.111 \\
(14 - 7.667)^2 & = 43.221 \\
(15 - 7.667)^2 & = 54.444 \\
\end{align*}
\]
- Sum:
\[
\sum (x_i - \bar{x})^2 = 32.097 + 21.778 + 13.478 + 0.111 + 43.221 + 54.444 = 164.129
\]
- Now calculate \(m\):
\[
m = \frac{64.000}{164.129} \approx 0.390
\]
3. Calculate the intercept \(b\):
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 5 - (0.390)(7.667) \approx 5 - 2.998 = 2.002
\]
4. Therefore, the least squares regression line is:
\[
y = 0.390x + 2.002
\]
Rounded to the nearest thousandth, the equation is:
\[
y = 0.390x + 2.002
\]
Given your data points:
- (2, 3)
- (3, 4)
- (4, 3)
- (8, 4)
- (14, 8)
- (15, 8)
We can calculate:
1. The means of x and y:
\[
\bar{x} = \frac{2 + 3 + 4 + 8 + 14 + 15}{6} = \frac{46}{6} = 7.667
\]
\[
\bar{y} = \frac{3 + 4 + 3 + 4 + 8 + 8}{6} = \frac{30}{6} = 5
\]
2. The slope \(m\):
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]
First, we calculate the necessary components:
- Calculate \( (x_i - \bar{x})(y_i - \bar{y}) \):
\[
\begin{align*}
(2 - 7.667)(3 - 5) & = (-5.667)(-2) = 11.334 \\
(3 - 7.667)(4 - 5) & = (-4.667)(-1) = 4.667 \\
(4 - 7.667)(3 - 5) & = (-3.667)(-2) = 7.334 \\
(8 - 7.667)(4 - 5) & = (0.333)(-1) = -0.333 \\
(14 - 7.667)(8 - 5) & = (6.333)(3) = 19.000 \\
(15 - 7.667)(8 - 5) & = (7.333)(3) = 22.000 \\
\end{align*}
\]
- Sum:
\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = 11.334 + 4.667 + 7.334 - 0.333 + 19.000 + 22.000 = 64.000
\]
- Calculate \( (x_i - \bar{x})^2 \):
\[
\begin{align*}
(2 - 7.667)^2 & = 32.097 \\
(3 - 7.667)^2 & = 21.778 \\
(4 - 7.667)^2 & = 13.478 \\
(8 - 7.667)^2 & = 0.111 \\
(14 - 7.667)^2 & = 43.221 \\
(15 - 7.667)^2 & = 54.444 \\
\end{align*}
\]
- Sum:
\[
\sum (x_i - \bar{x})^2 = 32.097 + 21.778 + 13.478 + 0.111 + 43.221 + 54.444 = 164.129
\]
- Now calculate \(m\):
\[
m = \frac{64.000}{164.129} \approx 0.390
\]
3. Calculate the intercept \(b\):
\[
b = \bar{y} - m\bar{x}
\]
\[
b = 5 - (0.390)(7.667) \approx 5 - 2.998 = 2.002
\]
4. Therefore, the least squares regression line is:
\[
y = 0.390x + 2.002
\]
Rounded to the nearest thousandth, the equation is:
\[
y = 0.390x + 2.002
\]