To determine whether there is a linear relationship between the pounds of waste and the number of people in a home, we'll calculate the correlation coefficient [tex]\( r \)[/tex] and compare it to given critical values. Here we have the following data:
| Pounds of waste | 0.27 | 1.41 | 2.19 | 2.83 | 2.19 | 1.81 | 0.85 | 3.05 |
|-----------------|------|------|------|------|------|------|------|------|
| Number in home | 2 | 3 | 3 | 6 | 4 | 2 | 1 | 5 |
We'll follow these steps to come to the conclusion:
### Steps:
1. Calculate the Correlation Coefficient [tex]\( r \)[/tex]: The correlation coefficient is a measure of the linear relationship between two variables. It ranges from -1 to 1. A value of 1 implies a perfect positive linear relationship, -1 a perfect negative linear relationship, and 0 no linear relationship.
2. Interpret [tex]\( r \)[/tex] Based on Critical Values: Compare the calculated [tex]\( r \)[/tex] with certain critical values to determine the significance of the linear relationship.
### Result:
From the calculations, the correlation coefficient [tex]\( r \)[/tex] is approximately 0.842.
### Interpretations Based on Critical Values:
1. Critical Value 0.7067: If [tex]\( r \)[/tex] exceeds 0.7067, then we can conclude there is a linear relationship.
2. Critical Value 0.6319: If [tex]\( r \)[/tex] exceeds 0.6319 but is less than or equal to 0.7067, then there is also a linear relationship, but not as strong as the one exceeding 0.7067.
### Analysis:
- Given [tex]\( r = 0.842 \)[/tex], it exceeds both critical values 0.7067 and 0.6319. Therefore, we conclude there is a linear relationship because [tex]\( r \)[/tex] exceeds the critical value of 0.7067.
### Conclusion:
Based on our analysis, the correct conclusion is: "There is a linear relationship because [tex]\( r \)[/tex] exceeds the critical value of 0.7067."
This interpretation indicates a statistically significant positive linear relationship between the pounds of waste and the number of people in a home, indicating that as the number of people in a home increases, the pounds of waste also tend to increase.
