Compute the coefficient of determination. Round your answer to four decimal places.
-The regression equation for the data below is $\hat { y } = 3 x$.
$\begin{array}{r|rrrr}
\mathrm{x} & 2 & 4 & 5 & 6 \
\hline \mathrm{y} & 7 & 11 & 13 & 20
\end{array}$
A)$0.8873$
B) $0.7265$
C) $0.9420$
D) $0.4839$

Question

Quizplus · Accepted Answer

The coefficient of determination, denoted as $R^2$, measures the proportion of the variance in the dependent variable that is predictable from the independent variable. To find $R^2$, we first need to calculate the total sum of squares (SST), the regression sum of squares (SSR), and the sum of squares of residuals (SSE). However, given the regression equation and the data, we can simplify the process by focusing on the formula for $R^2 = 1 - \frac{SSE}{SST}$.Given the regression equation $\hat { y } = 3x$, we can calculate the predicted values ($\hat{y}$) for each $x$:- For $x = 2$, $\hat{y} = 3(2) = 6$- For $x = 4$, $\hat{y} = 3(4) = 12$- For $x = 5$, $\hat{y} = 3(5) = 15$- For $x = 6$, $\hat{y} = 3(6) = 18$The actual $y$ values are given, so we can calculate the mean of $y$ ($\bar{y}$) to find SST and SSE:- $\bar{y} = \frac{7 + 11 + 13 + 20}{4} = \frac{51}{4} = 12.75$SST (total variance in $y$) is calculated using the actual $y$ values:- $SST = (7 - 12.75)^2 + (11 - 12.75)^2 + (13 - 12.75)^2 + (20 - 12.75)^2$SSE (variance in $y$ not explained by the model) is calculated using the residuals (difference between actual and predicted $y$):- $SSE = (7 - 6)^2 + (11 - 12)^2 + (13 - 15)^2 + (20 - 18)^2$After calculating SST and SSE, $R^2$ can be found using $R^2 = 1 - \frac{SSE}{SST}$. The calculations for SST and SSE are not directly shown here, but when performed correctly and substituted into the formula for $R^2$, the result is closest to the value given in option A, $0.8873$, after rounding to four decimal places. This indicates that approximately 88.73% of the variance in $y$ is explained by the regression model $\hat { y } = 3x$.

Compute the Coefficient of Determination y^=3x\hat { y } = 3 xy^​=3x

Compute the Coefficient of Determination $\hat { y } = 3 x$