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 for the given data points, we can use the formula for the slope \( m \) of the best fitting line in linear regression:

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

where \( N \) is the number of data points, \( \sum xy \) is the sum of the product of corresponding \( x \) and \( y \) values, \( \sum x \) is the sum of the \( x \) values, and \( \sum y \) is the sum of the \( y \) values.

Given the table:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 12 \\
1 & 14 \\
2 & 16 \\
3 & 18 \\
4 & 20 \\
\hline
\end{array}
\]

Now we will compute the required sums:

1. \( N = 5 \) (the number of points)
2. \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
3. \( \sum y = 12 + 14 + 16 + 18 + 20 = 80 \)
4. \( \sum xy = (0 \cdot 12) + (1 \cdot 14) + (2 \cdot 16) + (3 \cdot 18) + (4 \cdot 20) = 0 + 14 + 32 + 54 + 80 = 180 \)
5. \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)

Now substituting these values into the slope formula:

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

Calculating the numerator:

\[
5 \cdot 180 = 900
\]
\[
10 \cdot 80 = 800
\]
So, the numerator is:

\[
900 - 800 = 100
\]

Now for the denominator:

\[
5 \cdot 30 = 150
\]
\[
(10)^2 = 100
\]
So, the denominator is:

\[
150 - 100 = 50
\]

Now compute the slope \( m \):

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

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