IB Mathematics Applications & Interpretation Question Type 5: Connecting Pearson's Correlation Coefficient and Determination for Linear Model with or Without Parameters Exercises

In a linear model with an intercept, the total sum of squares ($SST$) is $20$ and the error sum of squares ($SSE$) is $12$. Calculate the Pearson correlation coefficient, $r$, assuming a positive relationship between the variables.

For a set of bivariate data $(x,y)$ where the means of both variables are zero (i.e., $\bar{x}=0$ and $\bar{y}=0$), a linear regression through the origin is performed.

The sum of squares due to regression ($SSR$) is 8 and the total sum of squares ($SST=\sum y^2$) is 10.

Assuming the slope of the regression line is positive, calculate the Pearson correlation coefficient, $r$, between $x$ and $y$.

A no-intercept linear regression yields the following summary statistics:

$$\sum x_i{y}_i=3,\sum x_i^2=2,\sum y_i^2=8$$

Calculate the regression sum of squares (SSR), the coefficient of determination ($R^2$), the correlation coefficient ($r$), and the sum of squared errors (SSE).

In a simple linear regression model with an intercept, the Pearson correlation coefficient is $r=-0.8$ and the total sum of squares is $\text{SST}=100$.

Find the value of the coefficient of determination $R^2$, the regression sum of squares $\text{SSR}$, and the sum of squared errors $\text{SSE}$.

A simple linear regression model yields $SSR=5$ and $SSE=5$. Given that the relationship between $x$ and $y$ is positive, calculate the Pearson correlation coefficient, $r$.

Suppose in a model with intercept, $r^2=0.64$ and $\text{SST}=50$. Find $\text{SSR}$ and $\text{SSE}$.

In a linear model with intercept, you have $r=-3x$, $\text{SSR}=2x$, and $\text{SST}=18$. Compute $x$ and $R^2$.

Given a linear regression model with an intercept, the Pearson correlation coefficient satisfies $r=2x$, the regression sum of squares (SSR) is $x$, and the total sum of squares (SST) is 10. Find the value of $x$.

In a model with an intercept, $r=-0.6$ and the error sum of squares is $SSE=40$. Compute the total sum of squares ($SST$) and the regression sum of squares ($SSR$).

For a simple linear regression model, the estimated slope is $b=-1.5$, the regression sum of squares is $\text{SSR}=45$, and the total sum of squares is $\text{SST}=100$. Find the Pearson correlation coefficient $r$.

In a linear regression model with an intercept, the regression sum of squares is $SSR=30$ and the Pearson correlation coefficient is $r=0.5477$.

Find the total sum of squares ($SST$) and the error sum of squares ($SSE$).

For a linear regression model, the correlation coefficient is $r=2x$, the sum of squares due to regression is $\text{SSR}=x$, and the total sum of squares is $\text{SST}=10$. Given that $x>0$, find the value of $x$.