91Ó°ÊÓ

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 $\mathbf{y}\mathbf{=}{\mathbf{β}}_{\mathbf{0}}\mathbf{+}{\mathbf{β}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{β}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{Î\mu }$

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" $\mathbf{y}\mathbf{=}{\mathbf{β}}_{\mathbf{0}}\mathbf{+}{\mathbf{β}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{β}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+} {\mathbf{β}}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}$

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.