Question: Tipping behaviour in restaurants. Can food servers increase their tips by complimenting the customers they are waiting on? To answer this question, researchers collected data on the customer tipping behaviour for a sample of 348 dining parties and reported their findings in the Journal of Applied Social Psychology (Vol. 40, 2010). Tip size (y, measured as a percentage of the total food bill) was modelled as a function of size of the dining party$\mathbf{\left(}{\mathbf{x}}_{\mathbf{1}}\mathbf{\right)}$and whether or not the server complimented the customers’ choice of menu items $\mathbf{\left(}{\mathbf{x}}_{\mathbf{2}}\mathbf{\right)}$. One theory states that the effect of the size of the dining party on tip size is independent of whether or not the server compliments the customers’ menu choices. A second theory hypothesizes that the effect of size of the dining party on tip size is greater when the server compliments the customers’ menu choices as opposed to when the server refrains from complimenting menu choices.

a. Write a model for E(y) as a function of ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$ that corresponds to Theory 1.

b. Write a model for E(y) as a function of ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$that corresponds to Theory 2.

c. The researchers summarized the results of their analysis with the following graph. Based on the graph, which of the two models would you expect to fit the data better? Explain.

Theory 1 suggests that the effect of the size of the dining party (denoted by ${\mathbf{x}}_{\mathbf{1}}$) on the tip size is independent of whether or not the server compliments the customer’s menu choices (denoted by ${\mathbf{x}}_{\mathbf{2}}$).

Therefore, the model under theory 1 would be $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$

Where,${\mathbf{x}}_{\mathbf{1}}$=size of the dining party

and ${\mathbf{x}}_{\mathbf{2}}$ = server complimenting the customer’s menu choices

Theory 2 suggests that the effect of size of the dining party (denoted by ${\mathbf{x}}_{\mathbf{1}}$ ) on the tip size is greater when the server compliments the customers’ menu choices (denoted by ${\mathbf{x}}_{\mathbf{2}}$) as opposed to when the server refrains from complimenting menu choices. Here the two variables have some dependency amongst them, hence, a new variable ${\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}$will be introduced in the model to represent this dependency.

Therefore, the model under theory 2 would berole="math" localid="1649842923237" $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+} {\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}$

where,${\mathbf{x}}_{\mathbf{1}}$=size of the dining party

and ${\mathbf{x}}_{\mathbf{2}}$ = server complimenting the customer’s menu choices

As can be seen in the graph, the tip percentage stays constant as the numbers in the party increase when the servers do not compliment customers’ menu choices. While the tip percentage declines when the server compliments the customers’ menu choices when numbers in the party increase.

To maintain a constant tipping percentage is a priority, therefore, model 1 would be preferred as a way to predict the tipping percentage.