Question 19.1.a: THE EFFECT OF SHELF HEIGHT ON CEREAL SALES Does it matter wh......
THE EFFECT OF SHELF HEIGHT ON CEREAL SALES
Does it matter which shelf a popular brand is placed on? It certainly might, because we tend to purchase items that are easiest to see. To test this, suppose that Midway is a large chain of supermarket stores with many stores in many locations. Midway selects 125 of these stores for an experiment. Specifically, it selects these particular 125 stores to be as alike as possible, so that store size, amount of customer traffic, types of customers, and other characteristics are as similar across stores as possible. Each store stocks cereal in a similar location in the store on five-shelf displays. In the experiment, 25 randomly selected stores place a particular popular brand of cereal—we’ll call it Brand X—on the lowest shelf for a month. Another randomly selected 25 stores place Brand X on the next-to- lowest shelf, another 25 place it on the middle shelf, another 25 place it on the next-to- highest shelf, and the final 25 place it on the highest shelf. Then the number of boxes of Brand X sold is recorded at each of the stores for the last two weeks of the experiment. (The first two weeks allow customers to get used to the shelving arrangement.) The resulting data are in the file Cereal Sales.xlsx, as shown in Figure 19.3 (with some rows hidden). Does shelf height appear to make a difference in sales?
Objective To use one-way ANOVA to see whether shelf height makes any difference in mean sales of Brand X, and if so, to discover which shelf heights outperform the others.
First, the sample sizes are equal—this is a balanced design. This is not absolutely necessary in an experiment of this type, but since Midway is able to specify which stores use which shelving heights, it makes sense to use a balanced design. Second, this is a designed experiment, not an observational study. Midway deliberately chose the 125 stores in the experiment to be alike in as many ways as possible. This helps to ensure that any differences in sales across the five groups can be attributed to differences in shelf heights and not to other extraneous factors. Of course, it is virtually impossible to control for all other factors in an experiment such as this—the 125 stores are certainly not identical in all of their characteristics—but Midway has tried its best to keep them similar. Also, it has randomly assigned the stores to treatment levels (shelf heights), rather than arbitrarily assigning them. By using a random assignment, Midway avoids any possible bias it might have unconsciously introduced with a nonrandom assignment.
To analyze the data, select One-Way ANOVA from the StatTools Statistical Inference group, and fill in the resulting dialog box as shown in Figure 19.4. In particular, click the Format button and make sure the Unstacked option is checked (because there is a separate sale column for each of the shelf heights), and select all five variables for analysis. (Note that the Stacked option would be appropriate if there were two columns of length 125 each, one with the shelf height and the other with sales.) Finally, make sure only the Tukey option for confidence intervals is checked. (We will discuss these confidence interval options in Section 19-4.)
The one-way ANOVA output is shown in Figure 19.5. The summary statistics at the top indicate that the next-to-highest shelf height has the largest average sales, 426.3, almost 100 boxes larger than the lowest shelf height, which has the smallest average sales. This information is confirmed by the side-by-side box plots in Figure 19.6. (Although these box plots are not created as part of the ANOVA output, they are always a useful addition.) The sample standard deviations vary from about 61 to 85 over the five treatment levels. Although these tend to indicate unequal variances, the equal-variance assumption is almost never satisfied exactly in any study, and this much discrepancy in the standard deviations is nothing to worry about—it certainly does not invalidate the analysis.
It appears from the summary statistics and the box plots that mean sales differ for different shelf heights, but are the differences significant? The test of equal means answers this question. It appears in rows 26−28 of the output. The values in this ANOVA table are based on Equations (19.1)–(19.3). (The only part we didn’t discuss is the Total variation in row 28. It is based on the total variation of all observations around the grand mean in cell B10 and is used mainly as a check of the calculations. Note that SSB and SSW in cells B26 and B27 add up to the total sum of squares in cell B28. Similarly, the degrees of freedom add up in column C.) The F-ratio in cell E26 is 4.58, the ratio of the mean squares in cells D26 and D27. Its corresponding p-value is 0.0018, nearly zero. This leaves practically no doubt that the five population means are not all equal. Shelf height evidently does make a significant difference in sales.
Measure of Between Variation
\text{MSB}=\frac{\sum\limits_{j=1}^{J}{n_j(\bar{Y}_j-\bar{\bar{Y}})^2} }{J-1} (19.1)
Measure of Within Variation
\text{MSW}=\frac{\sum\limits_{j=1}^{J}(n_j-1)s_j^2}{n-J} (19.2)
F-ratio for ANOVA Test
\text{F-ratio}=\frac{\text{MSB}}{\text{MSW}} (19.3)
The 95% confidence intervals for ANOVA in rows 32–41 indicate which shelf heights differ significantly from which others. Any difference whose confidence interval does not include 0 is boldfaced. In this example, there is only one such difference, the one between the next-to-highest height and the lowest height. Not surprisingly, these are the treatment levels with the largest and smallest average sales. None of the other differences are significant. For example, even though the difference between the next-to-highest and next-to-lowest heights is 47.6, the corresponding confidence interval extends from a negative number to a positive number. Therefore, we cannot declare this difference to be statistically significant.
The main conclusion from this example is that shelf height definitely appears to make a difference in mean sales, at least for the population of stores similar to the ones in the study. Customers tend to purchase fewer boxes of cereal when they are placed on the bottom shelf, and they tend to purchase more when they are placed on the next-to-highest shelf—presumably right around eye level.
