To find the correlation coefficient for the given data, we need to follow these steps:
1. First, list the data in two arrays:
[tex]\[
\text{Grades} = [5, 6, 2, 1, 4, 2, 4, 1, 4, 3, 5]
\][/tex]
[tex]\[
\text{Number of pets} = [0, 4, 2, 3, 5, 1, 2, 2, 0, 5, 2]
\][/tex]
2. Use a statistical method to calculate the correlation coefficient.
The correlation coefficient is a measure of the linear relationship between two variables. It ranges from -1 to 1 where:
- 1 indicates a perfect positive linear relationship,
- -1 indicates a perfect negative linear relationship,
- 0 indicates no linear relationship.
3. The formula for the Pearson correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}}}
\][/tex]
Where:
- [tex]\( x_i \)[/tex] and [tex]\( y_i \)[/tex] are the individual sample points,
- [tex]\( \bar{x} \)[/tex] and [tex]\( \bar{y} \)[/tex] are the means of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] samples,
- [tex]\( \sum \)[/tex] denotes the sum over all the data points.
4. Calculate the correlation coefficient for the given data:
After performing these calculations, we find that the correlation coefficient is approximately [tex]\( 0.018472121975124454 \)[/tex].
5. Round the correlation coefficient to the nearest hundredth:
The correlation coefficient rounded to the nearest hundredth is [tex]\( 0.02 \)[/tex].
6. Interpret the result:
The correlation coefficient of [tex]\( 0.02 \)[/tex] suggests that there is a very weak positive linear relationship between the grade and the number of pets these students have at home.
So, the correlation coefficient for this data, rounded to the nearest hundredth, is [tex]\( 0.02 \)[/tex].
