Consider the partially completed ANOVA table showing the results of a regression analysis shown.

$\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the t-value we obtain when doing a t-test using a hypothesized slope of 1.0?

Question

Consider the partially completed ANOVA table showing the results of a regression analysis shown.
​
 $\begin{array}{|lcrcl|}
\hline \text { Source of } & {\text { Sum of }} & {\text { Mean }} & \
\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \
\text { Regression } & 305.1 & 1 & ? & ? \
\text { Residual } & 1247.3 & 24 & ? & \
\text { Total } & 1552.4 & 25 & & \
\hline
\end{array}$
-If the slope is -1.64, the residual mean square is 3.6, the sum of squares for X is 2.5, and the sample size is 16, what is the t-value we obtain when doing a t-test using a hypothesized slope of 1.0?

Quizplus · Accepted Answer

The t-value for testing a hypothesized slope (β1) can be calculated using the formula: $t = \frac{\hat{\beta}_1 - \beta_{1,0}}{SE(\hat{\beta}_1)}$, where $\hat{\beta}_1$ is the estimated slope, $\beta_{1,0}$ is the hypothesized slope, and $SE(\hat{\beta}_1)$ is the standard error of the estimated slope. Given $\hat{\beta}_1 = -1.64$, $\beta_{1,0} = 1.0$, and the residual mean square (MSE) is 3.6, we can find $SE(\hat{\beta}_1)$ using the formula $SE(\hat{\beta}_1) = \sqrt{\frac{MSE}{SSX}}$, where SSX is the sum of squares for X. Given SSX = 2.5, $SE(\hat{\beta}_1) = \sqrt{\frac{3.6}{2.5}} = \sqrt{1.44} = 1.2$. Thus, $t = \frac{-1.64 - 1.0}{1.2} = \frac{-2.64}{1.2} = -2.2$.

Consider the Partially Completed ANOVA Table Showing the Results of a Regression