Question
Find the equation for the least squares regression line of the data described below.
Ernesto's landscape architecture firm won a contract to design a new public playground in Seville. To decide how many swings to include, Ernesto collected information about other playgrounds in the city.
For each playground, he recorded the area (in square meters), x, and the number of swings, y.
Area (in square meters) Swings
11 9
16 4
22 5
41 12
47 12
58 12
Round your answers to the nearest thousandth.
Ernesto's landscape architecture firm won a contract to design a new public playground in Seville. To decide how many swings to include, Ernesto collected information about other playgrounds in the city.
For each playground, he recorded the area (in square meters), x, and the number of swings, y.
Area (in square meters) Swings
11 9
16 4
22 5
41 12
47 12
58 12
Round your answers to the nearest thousandth.
Answers
GPT-4o mini
To find the least squares regression line, we will follow the steps to calculate the necessary parameters. The regression line can be represented by the equation:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
First, we'll gather the data points:
\[
\begin{align*}
(11, 9) \\
(16, 4) \\
(22, 5) \\
(41, 12) \\
(47, 12) \\
(58, 12)
\end{align*}
\]
### Step 1: Calculate the means of x and y
\[
\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}
\]
Calculating \( \sum x \) and \( \sum y \):
\[
\sum x = 11 + 16 + 22 + 41 + 47 + 58 = 195
\]
\[
\sum y = 9 + 4 + 5 + 12 + 12 + 12 = 54
\]
Now, calculate the means:
\[
\bar{x} = \frac{195}{6} = 32.5
\]
\[
\bar{y} = \frac{54}{6} = 9
\]
### Step 2: Calculate the slope (m)
The formula for the slope \( m \) of the regression line is:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
#### Calculate necessary summations
1. \( \sum xy \):
\[
\sum xy = (11 \cdot 9) + (16 \cdot 4) + (22 \cdot 5) + (41 \cdot 12) + (47 \cdot 12) + (58 \cdot 12)
= 99 + 64 + 110 + 492 + 564 + 696 = 2025
\]
2. \( \sum x^2 \):
\[
\sum x^2 = 11^2 + 16^2 + 22^2 + 41^2 + 47^2 + 58^2
= 121 + 256 + 484 + 1681 + 2209 + 3364 = 8115
\]
3. Number of data points \( n = 6 \).
Now we can substitute these values into the slope formula:
\[
m = \frac{6(2025) - (195)(54)}{6(8115) - (195)^2}
\]
Calculating each part:
- \( 6(2025) = 12150 \)
- \( (195)(54) = 10530 \)
- \( 6(8115) = 48690 \)
- \( (195)^2 = 38025 \)
Substituting back into the formula:
\[
m = \frac{12150 - 10530}{48690 - 38025} = \frac{1620}{10665} \approx 0.151
\]
### Step 3: Calculate the y-intercept (b)
The formula for \( b \):
\[
b = \bar{y} - m\bar{x}
\]
Substituting the values:
\[
b = 9 - (0.151 \cdot 32.5) \approx 9 - 4.9075 \approx 4.093
\]
### Final Equation
Thus, the least squares regression line is:
\[
y = 0.151x + 4.093
\]
Rounding to the nearest thousandth, we get:
\[
\boxed{y = 0.151x + 4.093}
\]
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
First, we'll gather the data points:
\[
\begin{align*}
(11, 9) \\
(16, 4) \\
(22, 5) \\
(41, 12) \\
(47, 12) \\
(58, 12)
\end{align*}
\]
### Step 1: Calculate the means of x and y
\[
\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}
\]
Calculating \( \sum x \) and \( \sum y \):
\[
\sum x = 11 + 16 + 22 + 41 + 47 + 58 = 195
\]
\[
\sum y = 9 + 4 + 5 + 12 + 12 + 12 = 54
\]
Now, calculate the means:
\[
\bar{x} = \frac{195}{6} = 32.5
\]
\[
\bar{y} = \frac{54}{6} = 9
\]
### Step 2: Calculate the slope (m)
The formula for the slope \( m \) of the regression line is:
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
#### Calculate necessary summations
1. \( \sum xy \):
\[
\sum xy = (11 \cdot 9) + (16 \cdot 4) + (22 \cdot 5) + (41 \cdot 12) + (47 \cdot 12) + (58 \cdot 12)
= 99 + 64 + 110 + 492 + 564 + 696 = 2025
\]
2. \( \sum x^2 \):
\[
\sum x^2 = 11^2 + 16^2 + 22^2 + 41^2 + 47^2 + 58^2
= 121 + 256 + 484 + 1681 + 2209 + 3364 = 8115
\]
3. Number of data points \( n = 6 \).
Now we can substitute these values into the slope formula:
\[
m = \frac{6(2025) - (195)(54)}{6(8115) - (195)^2}
\]
Calculating each part:
- \( 6(2025) = 12150 \)
- \( (195)(54) = 10530 \)
- \( 6(8115) = 48690 \)
- \( (195)^2 = 38025 \)
Substituting back into the formula:
\[
m = \frac{12150 - 10530}{48690 - 38025} = \frac{1620}{10665} \approx 0.151
\]
### Step 3: Calculate the y-intercept (b)
The formula for \( b \):
\[
b = \bar{y} - m\bar{x}
\]
Substituting the values:
\[
b = 9 - (0.151 \cdot 32.5) \approx 9 - 4.9075 \approx 4.093
\]
### Final Equation
Thus, the least squares regression line is:
\[
y = 0.151x + 4.093
\]
Rounding to the nearest thousandth, we get:
\[
\boxed{y = 0.151x + 4.093}
\]