Part A: Finding the Correlation Coefficient
To determine the correlation coefficient between the number of hours spent practicing per week and the save percentage, we first perform a linear regression analysis. This involves fitting a straight line to the data points such that the difference between the actual data points and the predicted data points from the line is minimized.
The correlation coefficient ([tex]\( r \)[/tex]) is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where:
- [tex]\( r = 1 \)[/tex] indicates a perfect positive linear relationship,
- [tex]\( r = -1 \)[/tex] indicates a perfect negative linear relationship, and
- [tex]\( r = 0 \)[/tex] indicates no linear relationship.
After performing the linear regression analysis on the given data, we find the correlation coefficient to be [tex]\( r = 0.837 \)[/tex].
Therefore, the correlation coefficient, rounded to the nearest thousandth, is 0.837. This indicates a strong positive linear relationship between the number of hours spent practicing and the save percentage.
Part B: Identifying the Slope and its Interpretation
Next, we identify the slope of the linear regression line. The slope is denoted by "m" in the line equation [tex]\( y = mx + b \)[/tex], where "y" is the dependent variable (save percentage), "x" is the independent variable (hours of practice), "m" is the slope, and "b" is the y-intercept.
The slope represents the change in the save percentage for each additional hour of practice. In this context, the slope indicates how much the save percentage is expected to increase for each extra hour of practice per week.
From the linear regression analysis, we determine that the slope [tex]\( m \)[/tex] is approximately 0.008.
So, to the nearest thousandth, the slope is 0.008.
Interpretation of the Slope:
In this context, the slope of 0.008 means that for each additional hour of practice per week, the save percentage increases by 0.008 (or 0.8%). This shows that increased practice time is positively associated with improving the goalie's save percentage.
