The following data glves the number of hours 5 students spent studying and their corresponding grades on their midterm exams. \begin{tabular}{|c|c|c|c|c|c|} \hline Hours Spent Studying & 0 & 1 & 2 & 4 & 5 \\ \hline Midterm Grades & 66 & 75 & 84 & 90 & 96 \\ \hline \end{tabular} Calculate the coefficient of determination, $R^{2}$. Round your answer to three decimal places. Answer How to enter your answer (opens in new window) Tables Keyboard Shor

Step 1 ：The given data represents the number of hours 5 students spent studying and their corresponding grades on their midterm exams. The hours spent studying are [0, 1, 2, 4, 5] and the corresponding grades are [66, 75, 84, 90, 96].

Step 2 ：We are asked to calculate the coefficient of determination, often denoted as \(R^{2}\). This is a statistical measure that shows the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. In this case, the independent variable is the hours spent studying and the dependent variable is the midterm grades.

Step 3 ：To calculate \(R^{2}\), we first need to calculate the correlation coefficient (r) between the two variables. The formula for r is: \(r = \frac{\Sigma[(x_{i} - \bar{x})(y_{i} - \bar{y})]}{\sqrt{[\Sigma(x_{i} - \bar{x})^{2} * \Sigma(y_{i} - \bar{y})^{2}]}}\) where \(x_{i}\) and \(y_{i}\) are the individual sample points indexed with i, \(\bar{x}\) is the mean of the x values, and \(\bar{y}\) is the mean of the y values.

Step 4 ：Calculating the mean of hours and grades, we get \(\bar{x} = 2.4\) and \(\bar{y} = 82.2\).

Step 5 ：Substituting these values into the formula, we get the numerator as 96.6 and the denominator as 98.91086896797542.

Step 6 ：Dividing the numerator by the denominator, we get the correlation coefficient \(r = 0.9766368550586324\).

Step 7 ：Finally, we calculate \(R^{2}\) by squaring the correlation coefficient. This gives us \(R^{2} = 0.954\).

Step 8 ：Final Answer: The coefficient of determination, \(R^{2}\), is \(\boxed{0.954}\).