The data from exercise I follow. (xi|1 2 3 4 5)/(yi|3 7 5 11 14) The estimated r

step 1 In this exercise, we are given the data set shown here, and we are also told that the estimated

The data from exercise I follow. (xi|1 2 3 4 ...

The data from exercise I follow. $$ \frac{x_{i}|1 \quad 2 \quad 3 \quad 4 \quad 5}{y_{i}|3 \quad 7 \quad 5 \quad 11 \quad 14} $$ The estimated regression equation for these data is $\hat{y}=.20+2.60 x$ $$ \begin{array}{l}{\text { a. Compute SSE, SST, and SSR using equations }(12.8),(12.9), \text { and }(12.10) .} \\ {\text { b. Compute the coefficient of determination } r^{2} \text { . Comment on the goodness of fit. }} \\ {\text { c. Compute the sample correlation coefficient. }}\end{array} $$

The data from exercise I follow.
$$
\frac{x_{i}|1 \quad 2 \quad 3 \quad 4 \quad 5}{y_{i}|3 \quad 7 \quad 5 \quad 11 \quad 14}
$$
The estimated regression equation for these data is $\hat{y}=.20+2.60 x$
$$
\begin{array}{l}{\text { a. Compute SSE, SST, and SSR using equations }(12.8),(12.9), \text { and }(12.10) .} \\ {\text { b. Compute the coefficient of determination } r^{2} \text { . Comment on the goodness of fit. }} \\ {\text { c. Compute the sample correlation coefficient. }}\end{array}
$$

Key Concepts

Sample Correlation Coefficient

The sample correlation coefficient quantifies the strength and direction of a linear relationship between two variables. It is a standardized measure that ranges from -1 to 1, where values close to 1 or -1 indicate a strong positive or negative relationship, respectively, and values close to 0 indicate a weak linear relationship. This coefficient is essential in understanding how closely changes in one variable are associated with changes in another.

Error Sum of Squares (SSE)

SSE is a measure of the discrepancy between the observed data values and the values predicted by the regression model. It quantifies the variation that is not explained by the model, and is computed by summing the squared differences between the actual and predicted values. A lower SSE indicates a model that better fits the data.

Total Sum of Squares (SST)

SST measures the total variability in the observed data. It is calculated by summing the squared differences between each observed value and the overall mean of the data. SST serves as a baseline to assess how much of the total variability can be explained by the regression model.

Regression Sum of Squares (SSR)

SSR represents the portion of the total variability in the dependent variable that is explained by the regression model. It is obtained by summing the squared differences between the predicted values and the overall mean. A higher SSR implies that the model accounts for a larger proportion of the variation in the data.

Coefficient of Determination (r²)

The coefficient of determination, r², is a statistical measure that indicates the proportion of the variance in the dependent variable which is predictable from the independent variable. It is calculated as the ratio of SSR to SST, demonstrating how well the regression model fits the data. An r² value closer to 1 suggests that a large proportion of the variance is explained by the model.

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