Given the following bivariate dataset: $(1,2)$, $(2,3)$, $(3,5)$, $(4,4)$, $(5,6)$, what is the coefficient of determination $R^2$ for the best-fit linear model?

• A. $1.00$
• B. $0.75$
• C. $0.50$
• D. $0.92$

Answer: (D)

Verified step by step guidance

1. Calculate the means of the x-values and y-values. Use the formulas: \(\bar{x} = \frac{1+2+3+4+5}{5}\) and \(\bar{y} = \frac{2+3+5+4+6}{5}\).
2. Compute the slope \(b\) of the best-fit line using the formula: \(b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\), where the summations run over all data points.
3. Calculate the intercept \(a\) of the best-fit line using: \(a = \bar{y} - b \bar{x}\).
4. Using the slope \(b\) and intercept \(a\), find the predicted \(y\) values (\(\hat{y}_i\)) for each \(x_i\) with \(\hat{y}_i = a + b x_i\).
5. Compute the coefficient of determination \(R^2\) using the formula: \(R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}\). This measures the proportion of variance in \(y\) explained by the linear model.