The following regression equation was computed from a sample...

The following regression equation was computed from a sample of 20 observations: $$ \hat{y}=15-5 x $$ SSE was found to be 100 and SS total was 400 . a. Determine the standard error of estimate. b. Determine the coefficient of determination. c. Determine the correlation coefficient. (Caution: Watch the sign!)

The following regression equation was computed from a sample of 20 observations:
$$
\hat{y}=15-5 x
$$
SSE was found to be 100 and SS total was 400 .
a. Determine the standard error of estimate.
b. Determine the coefficient of determination.
c. Determine the correlation coefficient. (Caution: Watch the sign!)

Key Concepts

Correlation Coefficient

In simple linear regression, the correlation coefficient (r) measures the strength and direction of the linear relationship between the two variables involved. It is related to the coefficient of determination by the equation r = ?R² if the slope is positive (or r = -?R² if the slope is negative), thereby retaining the sign of the relationship.

SSE and SST

SSE (Sum of Squared Errors) quantifies the total deviation of the observed data points from the regression line, reflecting the unexplained variation by the model. SST (Total Sum of Squares) represents the overall variance present in the data, which serves as a benchmark to assess the proportion of variance explained by the regression model through the coefficient of determination.

Standard Error of Estimate

The standard error of estimate measures the typical distance between the observed values and the predicted values obtained from a regression model. It is computed as the square root of the mean squared error (MSE), where the MSE is the sum of squared errors (SSE) divided by its degrees of freedom. This statistic provides an assessment of the accuracy of the regression predictions.

Coefficient of Determination

The coefficient of determination, often denoted as R², indicates the proportion of the variation in the dependent variable that is explained by the independent variable(s) in the regression model. It is calculated as 1 minus the ratio of SSE to the total sum of squares (SST). A higher R² value represents a better fit of the model to the data.

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