Chapter 2.5, Problem 96E

(a)

To determine

To find: The four residuals for the data provided on the counts over different times.

Answer to Problem 96E

Solution: The residuals for the four counts are obtained as $\underset{\_}{49.9 }$, $\underset{\_}{-61.7}$, $\underset{\_}{-26.3}$, and $\underset{\_}{38.1}$.

Explanation of Solution

Calculation: The regression equation for the provided data set is as follows:

$\text{Count}=\text{602}\text{.8}-\left(74.7\times \text{Time}\right)$

Here, the response variable $\left(\text{represented as }y\right)$ is ‘count,’ and the explanatory variable $\left(\text{represented as }″x″\right)$ is ‘time.’ The estimated counts are obtained by substituting the respective values of time (1, 3, 5 and 7) in the regression equation.

For time $x_1=1$, the value of count (represented as ${\widehat{y}}_1$ ) is calculated as follows:

$\begin{array}{c}\text{Count}=\text{602}\text{.8}-\left(74.7\times \text{Time}\right)\\ =\text{602}\text{.8}-\left(74.7\times \text{1}\right)\\ {\widehat{y}}_1=528.1\end{array}$

For time $x_2=3$, the value of count (represented as ${\widehat{y}}_2$ ) is calculated as follows:

$\begin{array}{c}\text{Count}=\text{602}\text{.8}-\left(74.7\times \text{Time}\right)\\ =\text{602}\text{.8}-\left(74.7\times 3\right)\\ {\widehat{y}}_2=378.7\end{array}$

For time $x_3=5$, the value of count (represented as ${\widehat{y}}_3$ ) is calculated as follows:

$\begin{array}{c}\text{count}=\text{602}\text{.8}-\left(74.7\times \text{time}\right)\\ =\text{602}\text{.8}-\left(74.7\times 5\right)\\ {\widehat{y}}_3=229.3\end{array}$

For time $x_4=7$, the value of count (represented as ${\widehat{y}}_4$ ) is calculated as follows:

$\begin{array}{c}\text{Count}=\text{602}\text{.8}-\left(74.7\times \text{Time}\right)\\ =\text{602}\text{.8}-\left(74.7\times 7\right)\\ {\widehat{y}}_4=79.9\end{array}$

The observed $y$ values have been provided in the data and are written below:

$\begin{array}{l}{y}_1=578\\ \\ y_2=317\\ \\ y_3=203\\ \\ y_4=118\end{array}$

The residuals (represented as $e_i$ ) are the difference between the observed $y$ values and predicted $y$ values. The residuals for the above four observed and predicted values are calculated as follows:

For $y_1$, the residual $e_1$ is calculated as follows:

$\begin{array}{c}{e}_1=\text{Observed }{y}_1-\text{Predicted}{y}_1\\ =y_1-{\widehat{y}}_1\\ =578-528.1\\ =49.9\end{array}$

For $y_2$, the residual $e_2$ is calculated as follows:

$\begin{array}{c}{e}_2=\text{Observed }{y}_2-\text{Predicted}{y}_2\\ =y_2-{\widehat{y}}_2\\ =317-378.7\\ =-61.7\end{array}$

For $y_3$, the residual $e_3$ is calculated as follows:

$\begin{array}{c}{e}_3=\text{Observed }{y}_3-\text{Predicted}{y}_3\\ =y_3-{\widehat{y}}_3\\ =203-229.3\\ =-26.3\end{array}$

For $y_4$, the residual $e_4$ is calculated as follows:

$\begin{array}{c}{e}_4=\text{Observed }{y}_4-\text{Predicted}{y}_4\\ =y_4-{\widehat{y}}_4\\ =118-79.9\\ =38.1\end{array}$

Hence, the four residuals obtained are

$e_1=49.9$

$e_2=-61.7$

$e_3=-26.3$

$e_4=38.1$

(b)

To determine

To graph: The model for residual versus time plot.

Explanation of Solution

Graph: The residual versus time plot is obtained using the Minitab software by following the steps below:

Step 1: Enter the data in the Minitab worksheet.

Step 2: Go to Stat and select Regression and then select Regression again.

Step 3: Enter the response variable as ‘Count’ and Predictors as ‘Time.’

Step 4: Under Graphs option, choose Residuals versus order.

Step 5: Fill in Residual versus the variables as ‘Time.’

Step 6: Click on Ok.

The resultant graph is obtained as follows:

(c)

To determine

To explain: Interpretation of the obtained residual plot.

Answer to Problem 96E

Solution: Since there is no pattern observed in the residuals versus time plot, it can be concluded that the provided model is appropriate to estimate the count variable.

Explanation of Solution

The residual plot has no specific pattern. The plot seems to be scattered and nonlinear. The two residuals, $e_1$ as $49.9$ and $e_4$ as $38.1$, are positive, and the two residuals $e_2$ as $-61.7$ and $e_3$ as $-26.3$ are negative. This implies that the residuals are scattered above as well as below the x- axis. The uniform scattering of the residuals represents a good sign. Thus, it can be said that the provided regression model is suitable to predict the count variable.