Question

or
Find the equation for the least squares regression line of the data described below.
A botanist at a university is studying oak trees and the acorns they produce. He collected several acorns from the same tree and measured them.
For each acorn, he recorded its volume (in cubic centimeters), x, and its weight (in grams), y.
Volume (in cubic centimeters) Weight (in grams)
0.9 2
1.0 2
1.5 4
3.3 6
3.5 7
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the least squares regression line, we start with a set of data points and calculate the slope (\(m\)) and y-intercept (\(b\)) of the line in the form \(y = mx + b\).

The points from your data are:
\[
(0.9, 2), (1.0, 2), (1.5, 4), (3.3, 6), (3.5, 7)
\]

### Step 1: Calculate the Means

1. Calculate the means of \(x\) and \(y\):
\[
\bar{x} = \frac{0.9 + 1.0 + 1.5 + 3.3 + 3.5}{5} = \frac{11.2}{5} = 2.24
\]
\[
\bar{y} = \frac{2 + 2 + 4 + 6 + 7}{5} = \frac{21}{5} = 4.2
\]

### Step 2: Calculate \(m\) (Slope)

The formula for the slope (\(m\)) is:
\[
m = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}
\]

#### Calculate \(\sum (x - \bar{x})(y - \bar{y})\)

Using individual calculations:
- For \( (0.9, 2) \): \( (0.9 - 2.24)(2 - 4.2) = (-1.34)(-2.2) = 2.948 \)
- For \( (1.0, 2) \): \( (1.0 - 2.24)(2 - 4.2) = (-1.24)(-2.2) = 2.728 \)
- For \( (1.5, 4) \): \( (1.5 - 2.24)(4 - 4.2) = (-0.74)(-0.2) = 0.148 \)
- For \( (3.3, 6) \): \( (3.3 - 2.24)(6 - 4.2) = (1.06)(1.8) = 1.908 \)
- For \( (3.5, 7) \): \( (3.5 - 2.24)(7 - 4.2) = (1.26)(2.8) = 3.528 \)

Now, add them up:
\[
\sum (x - \bar{x})(y - \bar{y}) = 2.948 + 2.728 + 0.148 + 1.908 + 3.528 = 11.260
\]

#### Calculate \(\sum (x - \bar{x})^2\)

Using individual calculations:
- For \( 0.9 \): \( (0.9 - 2.24)^2 = (-1.34)^2 = 1.7956 \)
- For \( 1.0 \): \( (1.0 - 2.24)^2 = (-1.24)^2 = 1.5376 \)
- For \( 1.5 \): \( (1.5 - 2.24)^2 = (-0.74)^2 = 0.5476 \)
- For \( 3.3 \): \( (3.3 - 2.24)^2 = (1.06)^2 = 1.1236 \)
- For \( 3.5 \): \( (3.5 - 2.24)^2 = (1.26)^2 = 1.5876 \)

Now, add them up:
\[
\sum (x - \bar{x})^2 = 1.7956 + 1.5376 + 0.5476 + 1.1236 + 1.5876 = 6.5916
\]

#### Calculate \(m\):
\[
m = \frac{11.260}{6.5916} \approx 1.709
\]

### Step 3: Calculate \(b\) (Y-Intercept)

Using the means calculated earlier and the slope:
\[
b = \bar{y} - m \bar{x}
\]
\[
b = 4.2 - (1.709)(2.24) \approx 4.2 - 3.829 \approx 0.371
\]

### Final Regression Equation

So the least squares regression line is:
\[
y \approx 1.709x + 0.371
\]

### Rounding to Nearest Thousandth

* Final Equation:
\[
\boxed{y = 1.709x + 0.371}
\]