Alright, let's delve into the problem step by step:
Part A: Correlation Coefficient
First, we need to determine the correlation coefficient for the linear regression given the data of hours spent practicing per week (x) and the percentage of goals saved (y).
We are given the following data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline \text{Hours } (x) & 5 & 8 & 10 & 6 & 4 & 10 & 13 & 8 \\
\hline \text{Save Percentage } (y) & 0.875 & 0.892 & 0.931 & 0.883 & 0.846 & 0.918 & 0.92 & 0.927 \\
\hline
\end{array}
\][/tex]
When we calculate the correlation coefficient using these data points, we find the numerical relationship between the hours of practice and the save percentage. This coefficient, denoted as [tex]\( r \)[/tex], measures the strength and direction of the linear relationship between these variables.
Upon calculating (considering the rounded values as given are correct), we find:
[tex]\[
r \approx 0.837
\][/tex]
Part B: Slope of the Linear Regression
Next, we need to determine the slope of the linear regression line, which is the line of best fit through our data points. The slope tells us how much the save percentage increases, on average, for each additional hour of practice.
After performing the linear regression, the slope of the line, rounding to the nearest thousandth, is found to be:
[tex]\[
\text{slope} \approx 0.008
\][/tex]
Explanation of the Slope in Context:
The slope of the linear regression line (0.008) represents the rate at which the save percentage increases per each additional hour of practice per week. In simpler terms:
- For every additional hour of practice per week, the save percentage improves by approximately 0.8%.
This means that there is a positive relationship between the number of hours practiced and the percentage of goals saved. More practice hours each week typically lead to a higher save percentage at the end of the season for these hockey goaltenders.
To summarize:
1. Correlation Coefficient: The correlation coefficient is 0.837, indicating a strong positive relationship between hours of practice and save percentage.
2. Slope: The slope of the linear regression line is 0.008, indicating that each additional hour of practice per week increases the save percentage by 0.8%.
