Question

Devin creates a table showing five points. He uses a spreadsheet calculator to calculate the slope and y intercept of the least squares regression line based on the points in the table what is the slope of this line
Table
x y
0 12
1 14
2 16
3 18
4 20

Answer 1

To find the slope of the least squares regression line for the given points, we can use the formula for the slope \( m \):

\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]

Where:
- \( N \) is the number of points,
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \),
- \( \sum x \) is the sum of all \( x \) values,
- \( \sum y \) is the sum of all \( y \) values,
- \( \sum x^2 \) is the sum of the squares of each \( x \) value.

Given your data points:
- \( (0, 12) \)
- \( (1, 14) \)
- \( (2, 16) \)
- \( (3, 18) \)
- \( (4, 20) \)

We first calculate the required sums:

1. **Calculating \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \)**:

\[
\sum x = 0 + 1 + 2 + 3 + 4 = 10
\]

\[
\sum y = 12 + 14 + 16 + 18 + 20 = 80
\]

\[
\sum xy = (0 \cdot 12) + (1 \cdot 14) + (2 \cdot 16) + (3 \cdot 18) + (4 \cdot 20) = 0 + 14 + 32 + 54 + 80 = 180
\]

\[
\sum x^2 = (0^2) + (1^2) + (2^2) + (3^2) + (4^2) = 0 + 1 + 4 + 9 + 16 = 30
\]

2. **Calculating \( N \)**:

\[
N = 5
\]

3. **Now plug into the slope formula**:

\[
m = \frac{5(180) - (10)(80)}{5(30) - (10)^2}
\]
\[
m = \frac{900 - 800}{150 - 100}
\]
\[
m = \frac{100}{50} = 2
\]

Therefore, the slope of the least squares regression line based on the given points is \( \boxed{2} \).