In the Journal of Marketing Research (November 1996), Gupta studied the extent to which the purchase behavior of scanner panels is representative of overall brand preferences. A scanner panel is a sample of households whose purchase data are recorded when a magnetic identification card is presented at a store checkout. The table below gives peanut butter purchase data collected by the A. C. Nielson Company using a panel of 2,500 households in Sioux Falls, South Dakota. The data were collected over 102 weeks. The table also gives the market shares obtained by recording all peanut butter purchases at the same stores during the same period.

a. Show that it is appropriate to carry out a chi-square test.
b. Test to determine whether the purchase behavior of the panel of 2,500 households is consistent with the purchase behavior of the population of all peanut butter purchasers. Assume here that purchase decisions by panel members are reasonably independent, and set α = .05.

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Number of Purchases by Household Panel Market Shares 20.10% 10.10 5.42 16.01 28.56 12.33 7.48 Goodness of Fit Test %of Brand Jif Jif Jif Peter Pan 10 Skippy Skippy Skippy Total Size 18 oz. 28 40 3,165 1,892 726 ,079 6,206 1,627 1,420 19,115 3165 1892 726 4079 6206 1627 1420 19115 obs expected O-E O- E/E chi 677.115 38.615 310.033 8.689 746.756 3842.115 1930.615 1036.033 31 3060.312 101 5459.244 2356.880729.880 1429.802 119.331 0.772 92.777 339.092 102.147 226.029 0.067 880.216 13.56 0.09 10.54 38.52 11.60 25.68 0.01 100.00 18 28 40 -9.802 0.000 19115.000 880.22 chisquare 6 df 0.0000 p-value