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 from the provided data 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 (in this case, 5)
- \( \sum xy \) is the sum of the product of each \( x \) and \( y \)
- \( \sum x \) is the sum of the \( x \) values
- \( \sum y \) is the sum of the \( y \) values
- \( \sum x^2 \) is the sum of the square of each \( x \)

Now, let's calculate the needed values:

1. **List the points**:

\[
(0, 12), (1, 14), (2, 16), (3, 18), (4, 20)
\]

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

Now we have:
- \( N = 5 \)
- \( \sum x = 10 \)
- \( \sum y = 80 \)
- \( \sum xy = 180 \)
- \( \sum x^2 = 30 \)

3. **Plug values into the slope formula**:

\[
m = \frac{5(180) - (10)(80)}{5(30) - (10)^2}
\]

Calculating the numerator:

\[
5(180) = 900
\]
\[
(10)(80) = 800
\]
So,
\[
900 - 800 = 100
\]

Calculating the denominator:

\[
5(30) = 150
\]
\[
(10)^2 = 100
\]
So,
\[
150 - 100 = 50
\]

Finally, calculate the slope:

\[
m = \frac{100}{50} = 2
\]

Thus, the slope of the least squares regression line is **2**.