Question Type 4: Determining the R^2 value for different models on a set of data Exercises

In a regression through the origin, it can be shown that $SSR = r^2 \, SST$. Given that $r = 2x$, $SSR = x$, and $SST = 10$, find $x$.

Calculate the value of $SSR$ and the coefficient of determination, $R^2$, for this model.

A standard linear regression gives a sample correlation $r=0.75$. Compute $R^2$ and explain its interpretation.

A model explains $30\%$ of the total variation in $y$. State $R^2$ and compute the ratio $SSE/SST$.

Given that $\sum(x_i-\bar{x})(y_i-\bar{y}) = 45$, $\sum(x_i-\bar{x})^2 = 40$, and $\sum(y_i-\bar{y})^2 = 90$.

(a) Calculate the value of the Pearson correlation coefficient, $r$.

(b) Calculate the value of the coefficient of determination, $R^2$.

The sample Pearson correlation between $x$ and $y$ is $r = 0.8$ and the total sum of squares is $SST = 100$. Assuming a standard linear regression with an intercept, find the regression sum of squares $SSR$.

For a dataset, it is found that $SST = 200$ and $SSR = 160$. Compute $R^2$ and find the residual sum of squares, $SSE$.

Model A on a dataset has $SSR_A = 180$ and $SST = 250$, while Model B has $SSE_B = 10$ and $SST = 100$. Compute the coefficient of determination ($R^2$) for both models and state which is better.

If $\text{SSR} = 40$ and the total sum of squares $\text{SST} = 50$, find $R^2$.

Given the regression sum of squares $SSR = 45$ and the residual sum of squares $SSE = 15$, calculate the coefficient of determination $R^2$.

A regression yields $SSE = 25$ and $SST = 125$. Find $R^2$ and describe what it tells you about the model's fit.