Let's carefully examine the given data and solve each part step by step.
### Given Data
- Hours spent practicing per week [tex]\( x \)[/tex]: 5, 8, 10, 6, 4, 10, 13, 8
- Save percentage (written as a decimal) [tex]\( y \)[/tex]: 0.875, 0.892, 0.931, 0.883, 0.846, 0.918, 0.92, 0.927
### Part A: Find the correlation coefficient for the linear regression for these data, rounded to the nearest thousandth.
The correlation coefficient, often denoted as [tex]\( r \)[/tex], is a measure that determines the strength and direction of the linear relationship between the two variables.
For the given data, the correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r \approx 0.837 \][/tex]
This coefficient indicates a strong positive linear relationship between the number of hours spent practicing and the save percentage.
### Part B: Identify the slope of the linear regression to the nearest thousandth, and explain what it represents in this context.
The slope of the linear regression line is a measure of how much the dependent variable (in this case, the save percentage) is expected to increase (or decrease) as the independent variable (the hours of practice) increases by one unit.
For the given data, the slope of the linear regression line, rounded to the nearest thousandth, is:
[tex]\[ \text{slope} \approx 0.008 \][/tex]
#### Explanation of the Slope:
The slope of 0.008 means that for each additional hour spent practicing per week, the percentage of goals saved is expected to increase by 0.008 (or 0.8%).
In summary:
- Part A: The correlation coefficient [tex]\( r \)[/tex] is approximately 0.837.
- Part B: The slope of the linear regression is approximately 0.008, indicating that each additional hour of practice per week is associated with an increase of 0.8% in the save percentage.
