You generate a scatter plot using Excel. You then have Excel plot the trend line and report the equation and the $r^{2}$ value. The regression equation is reported as and the $r^{2}=0.1936$. \[ y=19.55 x+61.35 \] What is the correlation coefficient for this data set? \[ r= \]
Solution
Step 1 ：The regression equation is reported as \(y=19.55x+61.35\) and the \(r^{2}\) value is 0.1936.
Step 2 ：The correlation coefficient, often denoted by r, is a measure of the strength and direction of a linear relationship between two variables.
Step 3 ：The \(r^{2}\) value, also known as the coefficient of determination, is the square of the correlation coefficient.
Step 4 ：To find the correlation coefficient, we need to take the square root of the \(r^{2}\) value.
Step 5 ：The sign of the correlation coefficient is the same as the sign of the slope of the regression line.
Step 6 ：In this case, the slope of the regression line is positive, so the correlation coefficient should be positive.
Step 7 ：Therefore, the correlation coefficient r is \(\sqrt{0.1936}\) which equals to 0.44.
Step 8 ：Final Answer: The correlation coefficient for this data set is \(\boxed{0.44}\).
