To determine the correlation coefficient of the data and identify which value is most likely closest to it, we need to understand the relationship between the average temperature [tex]\( x \)[/tex] and the electricity costs [tex]\( y \)[/tex]. The correlation coefficient, denoted as [tex]\( r \)[/tex], measures the strength and direction of a linear relationship between two variables on a scatter plot.
Given the data pairs:
[tex]\[
(65,147), (59,141), (71,176), (78,189), (82,183), (85,211), (88,231), (91,227), (84,198), (79,188), (63,152), (59,126)
\][/tex]
Let's break down the steps to understand what the correlation coefficient tells us and how we can interpret the given choices:
1. Definition of the Correlation Coefficient (r):
- [tex]\( r \)[/tex] ranges from -1 to 1.
- [tex]\( r = 1 \)[/tex] indicates a perfect positive linear relationship.
- [tex]\( r = 0 \)[/tex] indicates no linear relationship.
- [tex]\( r = -1 \)[/tex] indicates a perfect negative linear relationship.
2. Examining the Data:
- By observing the data pairs, it appears that as the average temperature [tex]\( x \)[/tex] increases, the electricity costs [tex]\( y \)[/tex] also increase.
- This suggests a positive relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Possible Values of [tex]\( r \)[/tex]:
- [tex]\( r = -0.5 \)[/tex]: This indicates a moderate negative linear relationship, which doesn't align with our observation that both variables seem to increase together.
- [tex]\( r = 0.5 \)[/tex]: This indicates a moderate positive linear relationship. Given that the relationship appears quite strong, this might be an underestimate.
- [tex]\( r = 1 \)[/tex]: This indicates a perfect positive linear relationship. While the relationship appears strong, perfect correlations are rare in real-world data.
4. Determining the Closest Value:
- Since the correlation appears to be strongly positive, but not necessarily perfect, we should choose a value that represents this strong, but imperfect, positive association.
Considering the provided values:
- Given the observation and the numerical result confirming the strength and direction of the relationship, the closest offered value to the actual correlation coefficient (which is approximately 0.968) is likely to be [tex]\( r = 1 \)[/tex], acknowledging that it's the nearest given choice despite not being exact.
Therefore, the value most likely closest to the correlation coefficient of the data is:
[tex]\[ \boxed{r = 1} \][/tex]
