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Conduct a two-way ANOVA (Analysis of Variance) test using the given data to check for an interaction between markup and location. Use the following steps: 1. Compute the Grand Mean: $$ \bar{Y}_{..} = \frac{\sum_{i=1}^{\text{rows}} \sum_{j=1}^{\text{columns}} Y_{ij}}{\text{rows} \times \text{columns}} $$ 2. Compute the Sum of Squares due to Location (SS_Location): $$ \text{SS}_\text{Location} = \text{rows} \times \sum_{i=1}^{\text{rows}} (\bar{Y}_{i.}-\bar{Y}_{..})^2 $$ 3. Compute the Sum of Squares due to Markup (SS_Markup): $$ \text{SS}_\text{Markup} = \text{columns} \times \sum_{j=1}^{\text{columns}} (\bar{Y}_{.j}-\bar{Y}_{..})^2 $$ 4. Compute the Sum of Squares due to Interaction (SS_Interaction): $$ \text{SS}_\text{Interaction} = \sum_{i=1}^{\text{rows}} \sum_{j=1}^{\text{columns}} (\bar{Y}_{ij}-\bar{Y}_{i.}-\bar{Y}_{.j}+\bar{Y}_{..})^2 $$ 5. Compute the Sum of Squares due to Total (SS_Total): $$ \text{SS}_\text{Total} = \sum_{i=1}^{\text{rows}} \sum_{j=1}^{\text{columns}} (Y_{ij}-\bar{Y}_{..})^2 $$ 6. Compute the Sum of Squares due to Error (SS_Error): $$ \text{SS}_\text{Error} = \text{SS}_\text{Total}-\text{SS}_\text{Location}-\text{SS}_\text{Markup}-\text{SS}_\text{Interaction} $$ 7. Compute the Mean Square due to Location (MS_Location), Markup (MS_Markup), Interaction (MS_Interaction), and Error (MS_Error): $$ \text{MS}_\text{Location} = \frac{\text{SS}_\text{Location}}{\text{rows}-1} \\ \text{MS}_\text{Markup} = \frac{\text{SS}_\text{Markup}}{\text{columns}-1} \\ \text{MS}_\text{Interaction} = \frac{\text{SS}_\text{Interaction}}{(\text{rows}-1)(\text{columns}-1)} \\ \text{MS}_\text{Error} = \frac{\text{SS}_\text{Error}}{\text{rows}\times\text{columns}} $$ 8. Finally, compute the F-statistic for Interaction: $$ F_{\text{Interaction}} = \frac{\text{MS}_\text{Interaction}}{\text{MS}_\text{Error}} $$ 9. Compare the computed F-statistic with the F-critical value at \(\alpha=0.05.\) If the F-statistic is greater than the F-critical value, reject the null hypothesis, which means there is a significant interaction between markup and location.

Discuss the practical implications of the test in part (a). If there is a significant interaction between the factors of markup and location, it indicates that the effect of changes in price markup depends on the location of the store, and vice versa. Business owners could use this information to determine the optimal pricing strategy and store locations to maximize profits.

Provide a line graph that compares the mean change in sales for stores in small towns versus suburban malls, given the three different price markups. This visualization will help us better understand the interaction effect and its potential influence on business decisions. To draw the line graph: 1. Use the mean change in sales for each category and plot them on a graph. 2. Create a line that connects the points for each location. 3. Compare the lines to see if there is a noticeable difference in their patterns. If the lines are parallel or nearly parallel, it suggests no significant interaction. If the lines show different patterns, it may indicate an interaction effect.

To find a \(95\%\) confidence interval for the difference in mean change in sales for small towns and suburban malls when using price markup 3, use the following formula: $$ \text{CI} = (\bar{Y}_{1.3}-\bar{Y}_{2.3}) \pm t_{\alpha/2, df} \times \sqrt{\frac{\text{MS}_\text{Error}}{\text{rows}}} $$ Where: - \(\bar{Y}_{1.3}\) is the mean change in sales for small towns at price markup 3 - \(\bar{Y}_{2.3}\) is the mean change in sales for suburban malls at price markup 3 - \(t_{\alpha/2, df}\) is the t-critical value for a \(95\%\) confidence interval and the degrees of freedom (df) of MS_Error - \(\text{MS}_\text{Error}\) is the mean square due to error - \(\text{rows}\) is the number of stores in each category By calculating the confidence interval, we will be able to determine if there is a significant difference in the mean change in sales for stores in small towns versus suburban malls when using price markup 3.