This case study considers data on sales volume, price, and advertisting display activity for packages of Borden sliced cheese, available as “cheese.csv” here. For each of 88 stores (store) in different US cities, we have repeated observations of the weekly sales volume (vol, in terms of packages sold), unit price (price), and whether the product was advertised with an in-store display during that week (disp = 1 for display). Altogether there are 5,555 observations in the data set. We want to understand whether the displays are effective at increasing demand,

1. To start, examine the visual evidence for the effect in two different stores using the following code:

   boxplot(vol ~ disp, data=subset(cheese, store == "HOUSTON - KROGER CO"))
   boxplot(vol ~ disp, data=subset(cheese, store == "ORLANDO,FL - FOOD LION"))
   

   Based on these plots, do you think an additive effect makes sense here? (That is, does running the display seem to increase the predicted sales by adding some fixed amount?) If not, what type of effect seems to make more sense? Can you think of why this might be the case?

2. Ignoring price and store for now, do the in-store displays appear to have an effect on sales volume? Using an appropriate simple linear regression model, provide an estimate, confidence interval, and an appropriate hypothesis test.

3. Different stores have different overall sales volumes, and different preferences for running display ads, making store an important potential confounder of the effect of display ads. Using a regression model, estimate the expected percentage change in sales volume when a display is present, controlling for the store, and provide a 95\% confidence interval.

4. You suspect that this relationship is further confounded by pricing strategies (for example, some stores may have be aggressive in general, leading them to both set lower prices and run displays). Propose a model that allows you to adjust for both price and store differences in assessing the effect of in-store displays on sales volume.

   (Remember back to our milk demand example: a typical model for price elasticity of demand is of the form $\hat{y}_i = \gamma x_i^{\beta_1}$ where $\hat{y}_i$ is expected sales, $x_i$ is price, $\gamma$ is a constant, and $\beta_1$ is the elasticity. We talked about how to fit this model with least squares; adapt that strategy to this problem where we also have the store and presence of display as predictors of demand.)

   Fit this model and provide the estimated expected percentage change in sales volume when a display is present, controlling for store and price. Were you right to suspect confounding?

5. You suspect that the shape of the demand curve – specifically the price elasticity (the partial relationship between price and sales volume) – might change when an in-store display is present. Construct and fit an appropriate model to explore this possibility. Are your suspicions supported by the data?

6. What should Kroger’s in Dallas/Ft. Worth charge for cheese in display weeks? Should their price change when they’re not running a display ad? Assume that the wholesale cost of cheese is $1.50 per unit.

   (Hint: Think back to our milk demand case study; what do you need to change in that code to answer this problem?)