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:

$\sum x_i y_i = 3, \quad \sum x_i^2 = 2, \quad \sum y_i^2 = 8$

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 $x$.