Question

Question: Forecasting daily admission of a water park (cont’d). Refer to Exercise 12.165. The owners of the water adventure park are advised that the prediction model could probably be improved if interaction terms were added. In particular, it is thought that the rate at which mean attendance increases as predicted high temperature increases will be greater on weekends than on weekdays.

The following model is therefore proposed:

$\mathit{E}\left(y\right)\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{ }{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{3}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{3}}$

The same 30 days of data used in Exercise 12.165 are again used to obtain the least squares model,$\hat{\mathbf{y}}\mathbf{=}\mathbf{250}\mathbf{-}\mathbf{700}{\mathit{x}}_{\mathbf{1}}\mathbf{+}\mathbf{100}{\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathbf{5}{\mathit{x}}_{\mathbf{3}}\mathbf{+}\mathbf{15}{\mathit{x}}_{\mathbf{1}}{\mathit{x}}_{\mathbf{3}}$ with $\mathit{s}{\mathit{\beta }}_{\mathbf{4}}\mathbf{=}\mathbf{3}\mathbf{,}{\mathit{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{96}$.

a. Graph the predicted day’s attendance, y, against the day’s predicted high temperature,, for a sunny weekday and for a sunny weekend day. Plot both on the same graph forbetweenand. Note the increase in slope for the weekend day. Interpret this.

b. Do the data indicate that the interaction term is a useful addition to the model? Use$\mathbf{\alpha }\mathbf{=}\mathbf{.}\mathbf{05}$.

c. Use this model to predict the attendance for a sunny weekday with a predicted high temperature of$\mathbf{95}\mathbf{°}\mathbf{F}$.

d. Suppose the 90%prediction interval for part c is (800, 850). Compare this result with the prediction interval for the model without interaction in Exercise 12.165, part e. Do the relative widths of the confidence intervals support or refute your conclusion about the utility of the interaction term (part b)?

e. The owners, noting that the coefficient${\hat{\mathbf{\beta }}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{700}$, conclude the model is ridiculous because it seems to imply that the mean attendance will be 700 less on weekends than on weekdays. Explain why this is not the case.

Short answer

Answer

a. Graph of the predicted days attendance.

b. Thevalue for the model is 0.96 indicating that 96% of the variation in the data is explained by the model meaning that the model is a good fit for the data. This means that the interaction term is useful in explaining model.

c. The predicted attendance on a sunny weekday at a temperature of$95°F$is 825.

d. The 90% prediction interval for daily attendance is (800, 850) indicating that the future values of the dependent variables will fall between the interval. From part c, the value of 825 is also falling into the prediction interval.

e. The coefficient of x₁ is -700 indicating that there is an inverse relation between attendance and days of the week. The number 700 indicates that for every 1 unit change in attendance, the no of days’ changes by 700.

Step-by-step solution

Given Information

The least square regression equation is:

$\hat{y}=250-700x_1+100x_2+5x_3+15x_1{x}_3$

Graph

a.

The question involves interpretingvalues which represents the fraction of the sample variation of the y-values (measured by) that is explained by the least squares prediction equation.

The graph can be drawn by taking individual values of y and , for second line y and to understand the effect of individual x variables on the dependent variable y.

Interaction term

b.

The$R^2$ value for the model is 0.96 indicating that 96% of the variation in the data is explained by the model meaning that the model is a good fit for the data. This means that the interaction term is useful in explaining model.

Prediction value

c.

The regression equation is $\hat{y}=250-700x_1+100x_2+5x_3+15x_1{x}_3$.The prediction value of daily attendance on sunny weekday at 95⁰F can be calculated when , $x_1=1,x_2=1and{x}_3=95$.

$$\begin{gathered} \hat{y}=250-700\left(0\right)+100\left(0\right)+5\left(95\right)+15\left(0\right)\left(1\right) \\ \hat{y}=825 \end{gathered}$$

Therefore, the predicted attendance on a sunny weekday at a temperature of $95°F$is 825.

Prediction interval 

d.

The 90% prediction interval for daily attendance is (800, 850) indicating that the future values of the dependent variables will fall between the interval. From part c, the value of 825 is also falling into the prediction interval.

Implication of coefficient of x1

e.

The coefficient of is -700 indicating that there is an inverse relation between attendance and days of the week. The number 700 indicates that for every 1 unit change in attendance, the no of days’ decreases by 700.