A fast food franchise is test marketing three new menu items. To find out if they have the same popularity, 18 franchisee restaurants are randomly chosen for participation in the study. Six of the restaurants are randomly chosen to test market each of the new menu items. Below are the sales figures, in number of orders per day, of the three new menu items in these restaurants after a week of test marketing. Item Sales Total 1 22 42 44 52 45 37 242 2 52 33 8 47 43 32 215 3 16 24 19 18 34 39 150 The sum of squares of the observations is ∑ 3 i=1 ∑ 6 j=1 y 2 i j = 23,415. (a) Write down a suitable model for these data and any necessary assumptions, explaining your notation. (b) Compute the analysis of variance table and test factor item. (c) Using an appropriate constraint, write down in matrix form the multiple regression model that is equivalent to the analysis of variance model for the data. (d) By using the model in part (c), find the least squares estimates of the parameters.
A fast food franchise is test marketing three new menu items. To find out if
they have the same popularity, 18 franchisee restaurants are randomly chosen for participation in the
study. Six of the restaurants are randomly chosen to test market each of the new menu items. Below are
the sales figures, in number of orders per day, of the three new menu items in these restaurants after a
week of test marketing.
Item Sales Total
1 22 42 44 52 45 37 242
2 52 33 8 47 43 32 215
3 16 24 19 18 34 39 150
The sum of squares of the observations is ∑
3
i=1 ∑
6
j=1
y
2
i j = 23,415.
(a) Write down a suitable model for these data and any necessary assumptions, explaining your
notation.
(b) Compute the analysis of variance table and test factor item.
(c) Using an appropriate constraint, write down in matrix form the multiple regression model that is
equivalent to the analysis of variance model for the data.
(d) By using the model in part (c), find the least squares estimates of the parameters.
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