Question

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. She collected several acorns from the same tree and measured them.
For each acorn, she recorded its volume (in cubic centimeters), x, and its weight (in grams), y.
Volume (in cubic centimeters) Weight (in grams)
2.92 3.33
2.98 3.42
3.05 4.70
3.49 3.51
3.89 5.06
4.98 7.98
Round your answers to the nearest thousandth.
y=
x+

Answer 1

To find the equation for the least squares regression line, we first need to compute the necessary statistics such as means, sums, and sum of products. The regression line has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

We need to calculate the following:

1. The means of \( x \) and \( y \):
\[
\overline{x} = \frac{\sum x}{n}, \quad \overline{y} = \frac{\sum y}{n}
\]
2. The slope \( m \):
\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]
3. The y-intercept \( b \):
\[
b = \overline{y} - m\overline{x}
\]

Let's calculate the necessary sums for this dataset.

### Data summary
- Volume (x): [2.92, 2.98, 3.05, 3.49, 3.89, 4.98]
- Weight (y): [3.33, 3.42, 4.70, 3.51, 5.06, 7.98]

### Calculations

1. **Sums and means**:

\[
n = 6
\]
\[
\sum x = 2.92 + 2.98 + 3.05 + 3.49 + 3.89 + 4.98 = 21.31
\]
\[
\sum y = 3.33 + 3.42 + 4.70 + 3.51 + 5.06 + 7.98 = 27.00
\]
\[
\sum xy = (2.92 \cdot 3.33) + (2.98 \cdot 3.42) + (3.05 \cdot 4.70) + (3.49 \cdot 3.51) + (3.89 \cdot 5.06) + (4.98 \cdot 7.98) = 9.7336 + 10.1936 + 14.355 + 12.25149 + 19.6874 + 39.6996 = 106.4201
\]
\[
\sum x^2 = (2.92)^2 + (2.98)^2 + (3.05)^2 + (3.49)^2 + (3.89)^2 + (4.98)^2 = 8.5344 + 8.8804 + 9.3025 + 12.1801 + 15.1521 + 24.8004 = 79.8499
\]

2. **Compute means**:

\[
\overline{x} = \frac{21.31}{6} \approx 3.55167
\]
\[
\overline{y} = \frac{27.00}{6} = 4.50
\]

3. **Calculate slope \( m \)**:
\[
m = \frac{6(106.4201) - (21.31)(27.00)}{6(79.8499) - (21.31)^2}
\]
Calculate \( (21.31)(27.00) \):
\[
21.31 \times 27.00 \approx 575.37
\]
So,
\[
m = \frac{638.5206 - 575.37}{479.0994 - 453.4161} \approx \frac{63.1506}{25.6833} \approx 2.459
\]

4. **Calculate y-intercept \( b \)**:
\[
b = 4.50 - (2.459)(3.55167) \approx 4.50 - 8.747 \approx -4.247
\]

### Final Equation
Now we round our values to the nearest thousandth:

\[
y \approx 2.459x - 4.247
\]

Thus, the equation of the least squares regression line is:

\[
y = 2.459x - 4.247
\]