Use the value of the linear correlation coefficient $r$ to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.
$r=0.941$, where $x=$ distance in miles and $y=$ fare in dollars
What is the value of the coefficient of determination?
The coefficient of determination is
(Round to four decimal places as needed.)
Answer
Final Answer: The coefficient of determination is \(\boxed{0.8855}\). This means that approximately 88.55% of the total variation in fare can be explained by the linear relationship between distance and fare.
Step 1 ：Given the value of the linear correlation coefficient \(r=0.941\), we are asked to find the coefficient of determination. The coefficient of determination, denoted as \(r^2\), is a measure of how well the regression line represents the data. If the coefficient of determination is closer to 1, the better the regression line (or in simpler terms, the line of best fit) fits the data.
Step 2 ：The formula for the coefficient of determination is simply the square of the correlation coefficient: \(r^2\)
Step 3 ：Substitute the given value of \(r\) into the formula, we get \(r^2 = (0.941)^2 = 0.885481\)
Step 4 ：Final Answer: The coefficient of determination is \(\boxed{0.8855}\). This means that approximately 88.55% of the total variation in fare can be explained by the linear relationship between distance and fare.
