To determine the linear correlation between pounds of waste and household size, we need to follow these steps:
1. Understand the Data:
- We have two sets of data:
- Pounds of waste: [tex]\([0.27, 1.41, 2.19, 2.83, 2.19, 1.81, 0.85, 3.05]\)[/tex]
- Number of people in the home: [tex]\([2, 3, 3, 6, 4, 2, 1, 5]\)[/tex]
2. Calculate the Correlation Coefficient:
- The correlation coefficient is a statistical measure that describes the strength and direction of a linear relationship between two variables.
- It can be calculated using the formula for Pearson's correlation coefficient, which is:
[tex]\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the number of data points.
- [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are variables representing each set of data.
- [tex]\( \sum xy \)[/tex] is the sum of the product of pairs.
- [tex]\( \sum x \)[/tex] and [tex]\( \sum y \)[/tex] are the sums of the individual data sets.
- [tex]\( \sum x^2 \)[/tex] and [tex]\( \sum y^2 \)[/tex] are the sums of the squares of the individual data sets.
3. Result:
- Applying the formula to the given data, we compute the correlation coefficient.
4. Round the Result:
- After calculating the correlation coefficient, we round it to three decimal places to find the linear correlation between pounds of waste and household size.
The correlation coefficient for the given data is:
[tex]\[
0.842
\][/tex]
Thus, the linear correlation between pounds of waste and household size, rounded to three decimal places, is [tex]\(\boxed{0.842}\)[/tex].
