Question

Find the equation for the least squares regression line of the data described below.
Stem and Leaf Agriculture wants to add a new compound to its bags of sunflower fertilizer. The company produced several bags containing varying amounts of the compound to see how it would affect sunflower growth.
Next, Stem and Leaf collected data on the amount of the compound added to each bag (in grams), x, and the weekly growth of the sunflowers treated with each bag (in centimeters), y.
Amount of compound Sunflower growth
108 27.00
111 8.88
114 25.52
135 31.96
200 38.49
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line, we need to calculate the slope (m) and y-intercept (b) of the line in the form \(y = mx + b\).

First, let's organize the data:

| Amount of compound (x) | Sunflower growth (y) |
|------------------------|----------------------|
| 108 | 27.00 |
| 111 | 8.88 |
| 114 | 25.52 |
| 135 | 31.96 |
| 200 | 38.49 |

Now, we can calculate some necessary sums:

1. **Count of Data Points (n)**:
\[ n = 5 \]

2. **Sum of x values (\(\sum x\))**:
\[ \sum x = 108 + 111 + 114 + 135 + 200 = 668 \]

3. **Sum of y values (\(\sum y\))**:
\[ \sum y = 27.00 + 8.88 + 25.52 + 31.96 + 38.49 = 131.85 \]

4. **Sum of x squared values (\(\sum x^2\))**:
\[
\sum x^2 = 108^2 + 111^2 + 114^2 + 135^2 + 200^2 = 11664 + 12321 + 12996 + 18225 + 40000 = 95006
\]

5. **Sum of xy values (\(\sum xy\))**:
\[
\sum xy = (108 \times 27) + (111 \times 8.88) + (114 \times 25.52) + (135 \times 31.96) + (200 \times 38.49)
\]
\[
= 2916 + 985.68 + 2903.88 + 4315.6 + 7698 = 18819.16
\]

Now, we can calculate the slope (m) and y-intercept (b):

### Slope (m):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
Plugging in the sums:
\[
m = \frac{5(18819.16) - (668)(131.85)}{5(95006) - (668)^2}
\]
Calculating:
\[
= \frac{94095.8 - 88081.8}{475030 - 446224}
\]
\[
= \frac{6014}{28706}
\]
\[
\approx 0.209
\]

### Y-intercept (b):
\[
b = \frac{\sum y - m(\sum x)}{n}
\]
Substituting in the values:
\[
b = \frac{131.85 - (0.209)(668)}{5}
\]
\[
= \frac{131.85 - 139.812}{5}
\]
\[
= \frac{-7.962}{5}
\]
\[
\approx -1.592
\]

### Final equation:
Thus, the equation of the least squares regression line is:
\[
y \approx 0.209x - 1.592
\]

Rounding to the nearest thousandth, we have:
\[
y = 0.209x - 1.592
\]