To determine the proportion of variance explained (denoted as [tex]\( R^2 \)[/tex]), we need to follow several steps that include computing intermediate statistical measures. Here is the step-by-step solution:
1. Calculate the mean of pounds of waste:
Given the data:
[tex]\( \text{Pounds of waste} = [0.27, 1.41, 2.19, 2.83, 2.19, 1.81, 0.85, 3.05] \)[/tex]
[tex]\[
\text{mean\_pounds\_of\_waste} = \frac{0.27 + 1.41 + 2.19 + 2.83 + 2.19 + 1.81 + 0.85 + 3.05}{8} = 1.825
\][/tex]
2. Calculate the variance of pounds of waste:
Using the mean previously calculated,
[tex]\[
\text{variance\_pounds\_of\_waste} = 0.9026
\][/tex]
3. Calculate the mean of the number in home:
Given the data:
[tex]\( \# \text{in home} = [2, 3, 3, 6, 4, 2, 1, 5] \)[/tex]
[tex]\[
\text{mean\_num\_in\_home} = \frac{2 + 3 + 3 + 6 + 4 + 2 + 1 + 5}{8} = 3.25
\][/tex]
4. Calculate the variance of the number in home:
Using the mean previously calculated,
[tex]\[
\text{variance\_num\_in\_home} = 2.7857
\][/tex]
5. Calculate the covariance between pounds of waste and the number in the home:
The covariance quantifies the degree to which two variables vary together,
[tex]\[
\text{covariance} = 1.3357
\][/tex]
6. Calculate the correlation coefficient (r):
[tex]\[
r = \frac{\text{covariance}}{\text{std\_dev\_pounds\_of\_waste} \times \text{std\_dev\_num\_in\_home}}
\][/tex]
Where:
[tex]\[
\text{std\_dev\_pounds\_of\_waste} = \sqrt{\text{variance\_pounds\_of\_waste}} = \sqrt{0.9026}
\][/tex]
[tex]\[
\text{std\_dev\_num\_in\_home} = \sqrt{\text{variance\_num\_in\_home}} = \sqrt{2.7857}
\][/tex]
Therefore,
[tex]\[
r = 0.8424
\][/tex]
7. Calculate [tex]\( R^2 \)[/tex] (the proportion of variance explained):
The [tex]\( R^2 \)[/tex] value is simply the square of the correlation coefficient [tex]\( r \)[/tex],
[tex]\[
R^2 = r^2 = (0.8424)^2 = 0.7096
\][/tex]
8. Round [tex]\( R^2 \)[/tex] to three decimal places:
[tex]\[
R^2 \approx 0.710
\][/tex]
Therefore, the proportion of variance explained, when rounded to three decimal places, is [tex]\( \mathbf{0.710} \)[/tex].
