Question

Question
Use the table to answer the question.

x y
0 12
1 14
2 16
3 18
4 20
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?(1 point)
The slope of the least squares regression line is

Answer 1

To find the slope of the least squares regression line based on the points in the table, 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 products of \( x \) and \( y \),
- \( \sum x \) is the sum of \( x \) values,
- \( \sum y \) is the sum of \( y \) values,
- \( \sum x^2 \) is the sum of the squares of \( x \) values.

Given the table:

\[
\begin{align*}
x & \quad y \\
0 & \quad 12 \\
1 & \quad 14 \\
2 & \quad 16 \\
3 & \quad 18 \\
4 & \quad 20 \\
\end{align*}
\]

Let's calculate each component:

1. **Number of points \( n \)**: There are 5 points.

2. **Calculate \( \sum x \), \( \sum y \), \( \sum xy \), and \( \sum x^2 \)**:
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 12 + 14 + 16 + 18 + 20 = 80 \)
- \( \sum xy = (0 \cdot 12) + (1 \cdot 14) + (2 \cdot 16) + (3 \cdot 18) + (4 \cdot 20) = 0 + 14 + 32 + 54 + 80 = 180 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)

3. **Plugging these values into the slope formula**:

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

Thus, the slope of the least squares regression line is \( \boxed{2} \).