A chain of jewelry stores conducted an experiment to...

A chain of jewelry stores conducted an experiment to investigate the effect of price and location on the demand for its diamonds. Six small-town stores were selected for the study, as well as six stores located in large suburban malls. Two stores in each of these locations were assigned to each of three item percentage markups. The percentage gain (or loss) in sales for each store was recorded at the end of 1 month. The data are shown in the accompanying table. $$ \begin{array}{lrrr} {\text { Markup }} \\ \hline \text { Location } & 1 & 2 & 3 \\ \hline \text { Small towns } & 10 & -3 & -10 \\ & 4 & 7 & -24 \\ \hline \text { Suburban malls } & 14 & 8 & -4 \\ & 18 & 3 & 3 \end{array} $$ a. Do the data provide sufficient evidence to indicate an interaction between markup and location? Test using $\alpha=.05 .$ b. What are the practical implications of your test in part a? c. Draw a line graph similar to Figure 11.11 to help visualize the results of this experiment. Summarize the results. d. Find a $95 \%$ confidence interval for the difference in mean change in sales for stores in small towns versus those in suburban malls if the stores are using price markup $3 .$

A chain of jewelry stores conducted an experiment to investigate the effect of price and location on the demand for its diamonds. Six small-town stores were selected for the study, as well as six stores located in large suburban malls. Two stores in each of these locations were assigned to each of three item percentage markups. The percentage gain (or loss) in sales for each store was recorded at the end of 1 month. The data are shown in the accompanying table.
$$
\begin{array}{lrrr}
{\text { Markup }} \\
\hline \text { Location } & 1 & 2 & 3 \\
\hline \text { Small towns } & 10 & -3 & -10 \\
& 4 & 7 & -24 \\
\hline \text { Suburban malls } & 14 & 8 & -4 \\
& 18 & 3 & 3
\end{array}
$$
a. Do the data provide sufficient evidence to indicate an interaction between markup and location? Test using $\alpha=.05 .$
b. What are the practical implications of your test in part a?
c. Draw a line graph similar to Figure 11.11 to help visualize the results of this experiment. Summarize the results.
d. Find a $95 \%$ confidence interval for the difference in mean change in sales for stores in small towns versus those in suburban malls if the stores are using price markup $3 .$

Key Concepts

Factorial Design

A factorial design involves studying the effects of two or more factors by varying them simultaneously. This design allows the experimenter to efficiently investigate not only the individual (main) effects of each factor but also how the factors interact with one another.

Interaction Effect

Interaction effect occurs when the effect of one independent variable on the dependent variable differs depending on the level of another independent variable. This is crucial to understand because it indicates that the factors do not simply add their effects but may modify each other's influence on the outcome.

Two-Way ANOVA

Two-way ANOVA is a statistical technique used to analyze the effects of two categorical independent variables on a continuous dependent variable. It helps in assessing both the main effects of each factor and any potential interaction effect between them.

Confidence Interval for Mean Difference

A confidence interval for a mean difference provides a range of values within which the true difference between the group means is expected to lie with a specified level of confidence. It quantifies the precision of the estimated difference and is useful in assessing practical significance.

Graphical Analysis of Interactions

Graphical methods, such as line graphs or interaction plots, are used to visualize how the response variable changes across the levels of different factors. These plots help in identifying interactions by highlighting any non-parallel lines that indicate that the effect of one factor depends on the level of another.

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