Final answer:
A linear regression equation is used to find a line that best fits a set of data and is calculated using the least-squares method. An r² value of .72 indicates 72% of the variability is explained by the independent variable. Without the specific data points, we cannot compute the regression equation to estimate the homework grade for a test score of 68.
Step-by-step explanation:
To find the linear regression equation that best fits a given set of data, one must calculate the least-squares regression line. This involves finding the line that minimizes the sum of the squares of the vertical distances (residuals) between the observed values and the line's predicted values.
The coefficient of determination, denoted as r², measures the proportion of the variability in one variable that is predictable from the other variable. An r² value of .72 indicates that 72% of the variability in the dependent variable can be explained by the independent variable. The coefficient of determination is always between 0 and 1 and cannot be negative because it represents the squared value of the correlation coefficient.
Using the given data, one would typically use a calculator or statistical software to input the (x, y) pairs and calculate the line of best fit. In the absence of such tools, the median-median line approach could be used as an alternative to calculate a less precise fit. Once the regression equation is obtained, you can use it to estimate unknown values by substituting the known value into the equation. For example, if the equation is ï = a + bx and you want to predict the homework grade (x) for a test grade (y) of 68, you would solve the equation for x by substituting 68 for y.
To answer the student's question, we would need the actual data points to calculate the regression equation. Without this data, we cannot compute the necessary equation to estimate the homework grade for a student with a test grade of 68.