Question

A chain of EX1150 jewelry stores conducted an experiment to investigate the effect of price markup 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. 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 \(3 .\)

Short answer

Explain your reasoning and provide practical implications. Answer: To determine if there is an interaction between markup and location in terms of their effect on the demand for diamonds, we need to perform a two-way ANOVA test. The null hypothesis states that there is no interaction between the two factors, while the alternative hypothesis states that there is an interaction between markup and location. If the calculated p-value from the ANOVA test is less than the significance level (\(\alpha = 0.05\)), we will reject the null hypothesis in favor of the alternative one, which indicates an interaction between markup and location. If the p-value is less than \(\alpha\), the practical implication is that the choice of price markup for the jewelry stores might affect the demand differently depending on the location (small-town stores or suburban mall stores).

Step-by-step solution

Hypothesis Testing for Interaction Between Factors

To test whether there is an interaction between markup and location, we need to perform a two-way ANOVA test. The null hypothesis states that there is no interaction between the two factors, while the alternative hypothesis is that there is an interaction between markup and location. Using the given data, perform a two-way ANOVA test at a significance level of \(\alpha = 0.05\). If the calculated p-value is less than \(\alpha\), we will reject the null hypothesis in favor of the alternative one, which indicates an interaction between markup and location.

Practical Implications

After performing the two-way ANOVA test and obtaining the p-value, compare it with the significance level (\(\alpha = 0.05\)) to determine if the data provides enough evidence to conclude there is an interaction between markup and location. If the p-value is less than \(\alpha\), the practical implication is that the choice of price markup for the jewelry stores might affect the demand differently depending on the location (small-town stores or suburban mall stores).

Drawing a Line Graph

To visualize the results of this experiment, create a line graph with percentage markups on the x-axis and percentage gain/loss in sales on the y-axis. Draw two lines representing the small-town stores and suburban mall stores. This graphical representation will help in gaining insights about the relationship between the markups and sales for different store locations.

Summarizing the Results

Analyze the line graph to describe the relationship between markup and sales change for each store location. Observe if one type of location responds differently to markup changes as compared to the other type of location. This step is crucial to understand if a general pattern emerges from the data, which can help in making decisions about price markups for the jewelry stores.

Confidence Interval for Difference in Mean Sales Change

To find a 95% confidence interval for the difference in mean change in sales between small-town stores and suburban mall stores when using price markup 3, use the t-distribution and the required formula. The confidence interval will provide an estimate of the difference between the mean sales change for the two types of locations at the chosen price markup, giving an idea of how the markup may affect them differently.

Key concepts

Interaction Effects in Two-Way ANOVA

Understanding interaction effects is crucial when analyzing how two or more factors influence an outcome. In the case of the jewelry stores, we are interested in how the price markup and location together impact sales. An interaction effect occurs when the effect of one factor depends on the level of another factor. This means that the impact of price markup might be different in small-town stores compared to suburban mall stores.
In a two-way ANOVA, we specifically test if there is a statistically significant interaction between the factors. By doing this, we can see if the relationship between markup and sales varies across locations. If the interaction is significant, it suggests that strategies should consider both factors together rather than independently.
The null hypothesis in this context is that there is no interaction between markup and location. If our p-value from the ANOVA is less than the designated significance level (0.05), we reject this hypothesis, indicating there's indeed an interaction.

Confidence Intervals Explained

Confidence intervals provide a range within which we expect the true difference in means to lie, with a certain level of confidence. For instance, in our study, we are looking at the 95% confidence interval for the difference in mean sales change between small-town stores and suburban malls for a specific markup level.
This interval helps us understand the magnitude of the difference in sales change, giving us a clearer picture of how sales might vary depending on the location. Using the t-distribution, we calculate this interval to ensure our estimates reflect the variability in our data.
A narrower confidence interval means more precision in our estimate, while a wider interval suggests more variability or uncertainty. Therefore, understanding these intervals helps managers make informed decisions about pricing strategies across different store locations.

Hypothesis Testing in ANOVA

Hypothesis testing is a key statistical tool used to determine if there is enough evidence to support a specific claim about a population parameter. In the context of our two-way ANOVA, we perform hypothesis testing to evaluate if there's an interaction between location and price markup on sales.
The process begins by establishing a null hypothesis, which asserts no interaction between the factors. Then, an alternative hypothesis suggests that an interaction does exist.
We calculate a p-value during the ANOVA test to see if we can reject the null hypothesis at the significance level \( \alpha = 0.05 \). A small p-value indicates strong evidence against the null, hence supporting the idea of an interaction. This test is fundamental to understanding complex relationships in data and helps in making strategic decisions.

Price Markup Impact Analysis

The impact of price markup on sales is a critical focus for the jewelry stores. By evaluating the percentage gain or loss in sales with different markup levels, businesses can identify the optimal pricing strategy.
The analysis involves comparing how sales change with each markup percentage in both small-town and suburban mall stores. Understanding these patterns can indicate whether a higher or lower markup yields better sales outcomes in different locations.
Price markup strategies might need adjustments based on location characteristics. The observed differences can help tailor marketing and pricing strategies effectively, ensuring each store maximizes its potential sales.

Location Effect Analysis

Analyzing the effect of location on sales helps in identifying how geographical factors influence consumer behavior. For the jewelry chain, this means understanding if small-town or suburban mall locations have distinct sales patterns with different price markups.
The analysis delves into how location-specific factors such as foot traffic, competition, and local economic conditions impact demand. By doing so, stores can identify if certain locations are more sensitive to price changes than others.
This information is crucial for strategic decision-making, allowing the jewelry chain to allocate resources effectively, optimize inventory, and plan location-specific marketing campaigns that align with local consumer preferences.