Question
Find the equation for the least squares regression line of the data described below.
Fans of the science fiction television show Space Cruiser Zeta are hosting a convention. To figure out how much space the event will require, an employee at the convention center looked at the space requirements for previous conventions.
For each convention, the employee recorded the number of attendees, x, and the floor space required (in square meters), y.
Attendees Floor space (in square meters)
874 5,960
3,835 2,503
5,004 15,071
6,953 17,767
8,056 19,525
8,457 17,094
Round your answers to the nearest thousandth.
y=
x+
Answers
GPT-4o mini
To find the least squares regression line for the given data points, we first need to determine the values of the slope \( m \) and the y-intercept \( b \) of the line in the form \( y = mx + b \).
The least squares regression line can be computed using the following formulas:
1. \( m = \frac{N(\Sigma xy) - (\Sigma x)(\Sigma y)}{N(\Sigma x^2) - (\Sigma x)^2} \)
2. \( b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{N(\Sigma x^2) - (\Sigma x)^2} \)
Where:
- \( N \) is the number of data points,
- \( \Sigma xy \) is the sum of the product of each \( x \) and \( y \),
- \( \Sigma x \) is the sum of \( x \),
- \( \Sigma y \) is the sum of \( y \),
- \( \Sigma x^2 \) is the sum of \( x^2 \).
Given the data:
| Attendees (x) | Floor space (y) |
|-----------------|-----------------|
| 874 | 5960 |
| 3835 | 2503 |
| 5004 | 15071 |
| 6953 | 17767 |
| 8056 | 19525 |
| 8457 | 17094 |
### Step 1: Calculate sums
1. **Sum of x (\( \Sigma x \))**
\[
\Sigma x = 874 + 3835 + 5004 + 6953 + 8056 + 8457 = 33179
\]
2. **Sum of y (\( \Sigma y \))**
\[
\Sigma y = 5960 + 2503 + 15071 + 17767 + 19525 + 17094 = 76520
\]
3. **Sum of xy (\( \Sigma xy \))**
Calculating \( xy \) for each pair:
- \( 874 \times 5960 = 5209840 \)
- \( 3835 \times 2503 = 9608155 \)
- \( 5004 \times 15071 = 75496734 \)
- \( 6953 \times 17767 = 123756371 \)
- \( 8056 \times 19525 = 157620400 \)
- \( 8457 \times 17094 = 144383358 \)
So,
\(
\Sigma xy = 5209840 + 9608155 + 75496734 + 123756371 + 157620400 + 144383358 = 436463458
\)
4. **Sum of \( x^2 \) (\( \Sigma x^2 \))**
Calculating \( x^2 \) for each \( x \):
- \( 874^2 = 764876 \)
- \( 3835^2 = 14719625 \)
- \( 5004^2 = 25040016 \)
- \( 6953^2 = 48471609 \)
- \( 8056^2 = 64976036 \)
- \( 8457^2 = 71501349 \)
So,
\(
\Sigma x^2 = 764876 + 14719625 + 25040016 + 48471609 + 64976036 + 71501349 = 226068511
\)
### Step 2: Calculate \( m \) and \( b \)
1. **Calculate \( m \)**
Using our formula for \( m \):
\[
m = \frac{6(436463458) - (33179)(76520)}{6(226068511) - (33179)^2}
\]
Calculating the numerator:
\[
6 \cdot 436463458 = 2618779738
\]
\[
33179 \cdot 76520 = 2534740440
\]
\[
N \cdot \Sigma xy - \Sigma x \cdot \Sigma y = 2618779738 - 2534740440 = 8404938
\]
Calculating the denominator:
\[
6 \cdot 226068511 = 1356411066
\]
\[
(33179)^2 = 1108690641
\]
\[
N \cdot \Sigma x^2 - (\Sigma x)^2 = 1356411066 - 1108690641 = 247720425
\]
Now substituting to find \( m \):
\[
m = \frac{8404938}{247720425} \approx 0.0339
\]
2. **Calculate \( b \)**
Using our formula for \( b \):
\[
b = \frac{(76520)(226068511) - (33179)(436463458)}{6(226068511) - (33179)^2}
\]
Calculating the numerator:
\[
76520 \cdot 226068511 = 17322484077320
\]
\[
33179 \cdot 436463458 = 14477033518182
\]
\[
17322484077320 - 14477033518182 = 2845450569138
\]
Now substituting to find \( b \):
\[
b = \frac{2845450569138}{247720425} \approx 11486.177
\]
### Final Regression Line
Thus, the least squares regression line is:
\[
y = 0.034x + 11486.177
\]
### Rounding
Rounding the coefficients to the nearest thousandth:
\[
\text{Final equation: } y = 0.034x + 11486.177
\]
The least squares regression line can be computed using the following formulas:
1. \( m = \frac{N(\Sigma xy) - (\Sigma x)(\Sigma y)}{N(\Sigma x^2) - (\Sigma x)^2} \)
2. \( b = \frac{(\Sigma y)(\Sigma x^2) - (\Sigma x)(\Sigma xy)}{N(\Sigma x^2) - (\Sigma x)^2} \)
Where:
- \( N \) is the number of data points,
- \( \Sigma xy \) is the sum of the product of each \( x \) and \( y \),
- \( \Sigma x \) is the sum of \( x \),
- \( \Sigma y \) is the sum of \( y \),
- \( \Sigma x^2 \) is the sum of \( x^2 \).
Given the data:
| Attendees (x) | Floor space (y) |
|-----------------|-----------------|
| 874 | 5960 |
| 3835 | 2503 |
| 5004 | 15071 |
| 6953 | 17767 |
| 8056 | 19525 |
| 8457 | 17094 |
### Step 1: Calculate sums
1. **Sum of x (\( \Sigma x \))**
\[
\Sigma x = 874 + 3835 + 5004 + 6953 + 8056 + 8457 = 33179
\]
2. **Sum of y (\( \Sigma y \))**
\[
\Sigma y = 5960 + 2503 + 15071 + 17767 + 19525 + 17094 = 76520
\]
3. **Sum of xy (\( \Sigma xy \))**
Calculating \( xy \) for each pair:
- \( 874 \times 5960 = 5209840 \)
- \( 3835 \times 2503 = 9608155 \)
- \( 5004 \times 15071 = 75496734 \)
- \( 6953 \times 17767 = 123756371 \)
- \( 8056 \times 19525 = 157620400 \)
- \( 8457 \times 17094 = 144383358 \)
So,
\(
\Sigma xy = 5209840 + 9608155 + 75496734 + 123756371 + 157620400 + 144383358 = 436463458
\)
4. **Sum of \( x^2 \) (\( \Sigma x^2 \))**
Calculating \( x^2 \) for each \( x \):
- \( 874^2 = 764876 \)
- \( 3835^2 = 14719625 \)
- \( 5004^2 = 25040016 \)
- \( 6953^2 = 48471609 \)
- \( 8056^2 = 64976036 \)
- \( 8457^2 = 71501349 \)
So,
\(
\Sigma x^2 = 764876 + 14719625 + 25040016 + 48471609 + 64976036 + 71501349 = 226068511
\)
### Step 2: Calculate \( m \) and \( b \)
1. **Calculate \( m \)**
Using our formula for \( m \):
\[
m = \frac{6(436463458) - (33179)(76520)}{6(226068511) - (33179)^2}
\]
Calculating the numerator:
\[
6 \cdot 436463458 = 2618779738
\]
\[
33179 \cdot 76520 = 2534740440
\]
\[
N \cdot \Sigma xy - \Sigma x \cdot \Sigma y = 2618779738 - 2534740440 = 8404938
\]
Calculating the denominator:
\[
6 \cdot 226068511 = 1356411066
\]
\[
(33179)^2 = 1108690641
\]
\[
N \cdot \Sigma x^2 - (\Sigma x)^2 = 1356411066 - 1108690641 = 247720425
\]
Now substituting to find \( m \):
\[
m = \frac{8404938}{247720425} \approx 0.0339
\]
2. **Calculate \( b \)**
Using our formula for \( b \):
\[
b = \frac{(76520)(226068511) - (33179)(436463458)}{6(226068511) - (33179)^2}
\]
Calculating the numerator:
\[
76520 \cdot 226068511 = 17322484077320
\]
\[
33179 \cdot 436463458 = 14477033518182
\]
\[
17322484077320 - 14477033518182 = 2845450569138
\]
Now substituting to find \( b \):
\[
b = \frac{2845450569138}{247720425} \approx 11486.177
\]
### Final Regression Line
Thus, the least squares regression line is:
\[
y = 0.034x + 11486.177
\]
### Rounding
Rounding the coefficients to the nearest thousandth:
\[
\text{Final equation: } y = 0.034x + 11486.177
\]