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

Question

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 $\boxed{{\displaystyle 2}}$.