Sure, let's go through this step-by-step.
1. Understand the correlation coefficient (r):
- The given correlation coefficient [tex]\( r \)[/tex] is -0.96.
- This indicates a strong negative linear relationship between the weight of a car (x) and the gas mileage (y).
2. Calculate the coefficient of determination ([tex]\( r^2 \)[/tex]):
- The coefficient of determination is calculated by squaring the correlation coefficient.
- [tex]\( r^2 = (-0.96)^2 \)[/tex].
3. Compute the actual value:
- Squaring -0.96 gives 0.9216.
4. Round to three decimal places:
- When rounding 0.9216 to three decimal places, we get 0.922.
5. Interpret the coefficient of determination:
- The coefficient of determination tells us the proportion of the variance in the dependent variable (gas mileage in this case) that is predictable from the independent variable (weight of the car).
- [tex]\( r^2 = 0.922 \)[/tex] means that 0.922 (or 92.2%) of the variation in gas mileage (y) is explained by the variation in the weight of the car (x).
6. Forming the interpretation in context:
- Approximately 92.2% of the variation in miles per gallon (mpg) is explained by the weight of the car.
So, the coefficient of determination is 0.922. And, approximately 92.2% of the variation in miles per gallon is explained by the weight of the car.
