IB Mathematics Applications & Interpretation 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 $SS{R}_A=180$ and $SST=250$, while Model B has $SS{E}_B=10$ 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.