ANOVA, Regression and Correlation

1. A chain of electronic stores is planning to expand the number of stores that it operates. Management believes that there is a relationship between the number of households in a store's market area and its level sales volume. Management wants to verify its belief that there is a relationship between the two factors before making further capital investments. To test this hypothesis, management has gathered data on the most recent month's sales volume (in thousands of dollars) for a random sample of 15 of its outlets. It then collected data on the number of households located in each store's market area (in thousands). The consultant has suggested that it is likely that all possible combinations in the bivariate relationship between the two factors (sales volume and number of households) are not normally distributed. These data are presented below.

Store Number of Households Sales Volume
1 161 157.27
2 99 93.28
3 135 136.81
4 129 123.79
5 164 153.51
6 221 241.74
7 179 201.54
8 204 206.71
9 214 229.78
10 101 135.22
11 231 224.71
12 206 195.29
13 248 242.16
14 107 115.21
15 205 197.82

2. A marketing research firm has been hired to develop a forecasting model of sales for a client's products in various sales districts. The sales department of the firm wants to use the model to demonstrate to top management the consequences of increasing the firm's level of promotional activities. The consultants have been provided with the following data for the past quarter.

Sales Advertising Expenditures Sales
District (in thousands $) (in thousands $)
1 1.814 55.420
2 1.112 45.819
3 0.718 16.940
4 1.421 35.818
5 3.085 85.090
6 2.119 62.025
7 0.525 17.918
8 1.108 32.845
9 1.621 41.180
10 2.645 62.910
11 2.927 87.013
12 0.847 22.150
13 0.621 16.660
14 0.981 27.050
15 1.394 39.121

Compute forecasting model from the data. What is the equation? Test the significance of the model at the 0.05 level of significance. What proportion of the variation in sales can be attributed to variation in advertising expenditures?

3. A local bakery which sells baked good to local grocery chain stores in the local metropolitan market wants to study the effect of the shelf display heights used by the stores on monthly sales volumes. The display heights (or location) have three levels: bottom (B), middle (M), and top (T). For each shelf height, six stores with equal sales potential are randomly selected, and each supermarket displays the baked product using its assigned shelf height for a month. At the end of the month, sales of the baked product at the 18 participating stores are recorded. The bakery uses a one-way ANOVA to test its hypothesis of no treatment effect. The results are:

ANOVA
Source df SS MS 'F'
Height 2 2273.88
Error 15 92.40
Total 17 2366.28

Complete the ANOVA table. Is the 'F' statistic significant at the 0.05 level? What are your conclusions?

4. NAOCO has three types of gasoline (A, B, and C) under development. It wants to compare the effects of the types of gasoline on gasoline mileage. NAOCO randomly selected 5 automobiles from a pool of a thousand Honda Accords and assigned to gas type A. Another 5 vehicles were randomly selected from the pool and assigned to gas type B, and finally, another 5 cars were randomly selected from the pool and assigned to gas type C. Each test vehicles was driven using the assigned gas type for an appropriate distance under normal driving conditions and the mileage per gallon was recorded. The results are found below:

Gas type A Gas type B Gas type C
34.0 35.3 33.3
35.0 36.5 34.0
34.3 36.4 34.7
35.5 37.0 33.0
35.8 37.6 34.9

Test the hypothesis of no treatment effects at the 0.05 level of significance.