Super-marketing. A Data Investigation. A note to teachers:

Transcription

1 Super-marketing A Data Investigation A note to teachers: This is a simple data investigation requiring interpretation of data, completion of stem and leaf plots, generation of box plots and analysis of data. If you are at a beginning level with STAT mode it will be useful to work through the Self-Guided Stat document, at More data investigations are available at: and also at: NOTE: If you desire to modify this activity and therefore desire the original word document you may request it by ing casio.edusupport@shriro.com.au 1

2 Super-marketing A company that manages an Australia-wide chain of supermarkets decides to investigate the following three approaches to advertising goods for sale in its supermarkets: Approach A: No advertising at all Approach B: Advertisements in the local newspaper Approach C: Advertisements in the local newspaper and advertising material distributed to letterboxes The company decides to use forty-five of its supermarkets. These supermarkets are of similar size and have a history of similar gross monthly turnovers. Three simple random samples of fifteen supermarkets are chosen from the forty-five supermarkets. Each sample has Approach A, B or C randomly allocated to it, and the supermarkets in the respective samples test the approach for one month. The table below shows the gross turnover ($'000) of each supermarket for the month in which the approaches were tested. The data are in rank order. Use this data in answering the following questions. 1) Using the stems provided, complete the individual stem plots which compare the distributions of gross monthly turnovers for the three approaches to advertising. Stem and Leaf Plots

3 2) Complete the summary statistics table below. Minimum Score st Quartile 264 Median rd Quartile Maximum Score Standard Deviation (1dp) 17.8 Mean (1dp) Sum of all scores 3976 Number of Scores ) Is there any correlation between the mean and the sum of all scores for each approach? If possible use a Stat Graph to help explain your answer. 4) Use your Graphic Calculator to draw three box plots comparing the monthly turnovers for the three approaches. Leave outliers OFF. Then copy the box plots and sketch them in the space provided below. Include a scale below each box plot. 3

4 5) Using the 'shapes' of the stem plots in Q1 and the shapes of the box plots in Q3 compare the distributions resulting from Approaches A and C in regard to skewness. Through your answer explain the concept of skewness. 6)a) Has the cost of advertising been taken into consideration in these results? If not, what would you need to do with the advertising costs in regard to the results of the three approaches? b) Let us assume the median cost of advertising for the 15 supermarkets of Approach B was $ and for Approach C was $ Recalculate the medians taking into account these advertising costs. Approach A: Median revenue (no advertising costs) = Approach B: Median revenue minus advertising costs = Approach C: Median revenue minus advertising costs = 4

5 c) If you were the Managing Director of the company managing this chain of supermarkets what would be your conclusions from this investigation, taking into consideration advertising costs? Would you say the results are conclusive? Which (if any) of the advertising strategies appear to be the more effective? Justify your answers referring to the available information. 5

6 Super-marketing SOLUTIONS 1) Stem and Leaf Plots ) Complete the summary statistics table below. Minimum Score st Quartile Median rd Quartile Maximum Score Standard Deviation (1dp) Mean (1dp) Sum of all scores Number of Scores ) By entering the 3 means in a list and the 3 sums-of-scores in another list, drawing a scatter plot and generating a linear regression line we see that the correlating coefficient is (close enough to) 1. The reason for this is obvious. Each approach has 15 scores. The highest sum divided by 15 will give the highest mean, the lowest sum divided by 15 will give the lowest mean, etc. 4) Draw boxplots to compare the gross monthly turnovers for the three approaches. Or, the three graphs together: 6

7 (NOTE: A suitable scale needs to be included on your paper-graphs) 5) The distribution resulting from Approach A is fairly evenly distributed around the mean (265.1). This is reflected by the symmetry of the box plot. The even distribution of Approach A data can be seen also in the stem and leaf plot where approximately the same number of scores exist either side of and in similar (but opposite) positions to the mean. Approach C however, has a distribution negatively skewed. Although, considering the stem and leaf plot for Approach C, there are also approximately the same number of scores either side of the mean (285.3) the scores below the mean extend over a greater range than those scores above the mean (min score is 33 below the mean, max score is 24 above). Therefore Approach C is considered to be negatively skewed. The negative skew is displayed in the box plot by the fact that the length of box and whisker to the left of the median line is far greater then to the right ie. the scores are negatively skewed. 6)a) The cost of advertising has not been taken into account. The figures are gross monthly turnovers. If we knew the advertising costs we would subtract the local newspaper costs from Approach B gross turnovers and subtract local newspaper and letterbox drop costs from Approach C gross turnovers. b) Approach A: Median revenue (no advertising costs) = $ Approach B: Median revenue minus advertising costs = $ $ = $ Approach C: Median revenue minus advertising costs = $ $ = $ c) NOTE: A variety of conclusions may be drawn. Prior to taking into account the advertising costs it would appear that Approach C is the most successful approach (greatest total revenue, highest median and mean revenues; Approach C had 6 supermarkets which gained revenue for the month in the $29 000s compared to 1 for Approach A and 2 for Approach B) However, if we assume the effects of advertising to be accurately represented in this data then we see that once the advertising costs are considered it appears the net benefit from advertising is small. This is represented by the net median revenue figures. Advertising in the local paper created a net median increase in revenue of $3 300 for the month ( ) and the letter box drops netted a median revenue increase of $200 ( ) A more detailed way of investigating this follows. In Approach B the cost of local paper advertising was $ and this generated $ of extra revenue (median B minus median A). In Approach C the cost of letter box advertising was $3 800 and this generated $4 000 of extra revenue (median C minus median B). Another way of putting this: #Each dollar of local paper advertising generated 15.9c additional median revenue after the advertising cost was recovered, and ##Each dollar of letter box advertising generated 5.3c additional median revenue after the advertising cost was recovered. Clearly, if these figures can be relied upon, then local paper advertising was more fruitful (3 times more) than the letter box drops. It could be argued the local paper advertising should be continued. However, it may be wise to run this trial again, perhaps over a longer period as it could be argued the results are not conclusive. Calculation for #: 24000/20700x100 = 115.9% = 15.9% median revenue in addition to the advertising cost (15.9 cents in the dollar) Calculation for ##: 4000/3800x100 = 105.3% = 5.3% median revenue in addition to the advertising cost (5.3 cents in the dollar) 7