A researcher collects a random sample of data across stores containing information on different cereals. Using this sample information, the researcher hopes to make some inferences about the cereal market.

The data is in the table below. Details of the variables are as follows,

ratings average ratings of a particular cereal (consumers assign higher ratings to their preferred cereal)

mfr Manufacturer of the cereal type. There are three manufacturers in the data, "N", "K" and "G".

shelf The presentation mode of a cereal in the store, where, 1 is when it is presented on promotion shelf, 2 is when it is presented on the regular shelf, and 3 is when it is presented on the shelf closest to the cashier.

caloriesamount of calories per one serving.

vitaminaverage vitamin (in milligrams) per one serving measured in milligrams.

(a)Based on a regression analysis, Manufacturer "N" has recently been claiming in its advertisements that its cereals are most preferred as compared to the corresponding cereals from manufacturers "K" and "G". Does your data show evidence of this claim?

(b)Manufacturer "G" is planning to come out with a new cereal with a calorie content and vitamin content equal to the average values observed in the data. The manufacturer is assured that the cereal will be presented across retailers on the shelf closest to the cashier. Using your regression results, predict the ratings that such a cereal is likely to get. Build a 90% confidence interval for your prediction.

(c) Every year the association for cereal manufacturers recognizes cereals that gross a consumer rating in excess of 50. Based on your data, what is the chance (probability) that this new cereal will get recognized by the "cereal manufacturers association"?

rating	mfr	calories	vitamins	shelf
68.4	N	70.0	25.0	3
34	K	120.0	0.0	3
59.4	K	70.0	25.0	3
93.7	K	50.0	25.0	3
34.4	K	110.0	25.0	3
29.5	G	110.0	25.0	1
33.2	K	110.0	25.0	2
37	G	130.0	25.0	3
49.1	K	90.0	25.0	1
53.3	N	90.0	25.0	3
18	K	120.0	25.0	2
50.8	G	110.0	25.0	1
19.8	G	120.0	25.0	2
40.4	G	110.0	25.0	3
22.7	G	110.0	25.0	2
41.4	K	110.0	25.0	1
45.9	K	100.0	25.0	1
35.8	K	110.0	25.0	2
22.4	G	110.0	25.0	2
40.4	K	110.0	25.0	3
64.5	N	100.0	0.0	2
46.9	K	110.0	25.0	3
36.2	G	100.0	25.0	3
44.3	K	100.0	25.0	3
32.2	K	110.0	25.0	2
31.4	K	110.0	25.0	1
58.3	K	100.0	25.0	2
40.9	N	120.0	25.0	3
41	K	120.0	25.0	3
28	N	110.0	25.0	2
35.3	N	100.0	25.0	1
23.8	G	110.0	25.0	2
52.1	N	100.0	25.0	3
53.4	N	110.0	25.0	3
45.8	N	120.0	25.0	3
21.9	K	120.0	25.0	2
31.1	G	110.0	25.0	1
28.7	N	110.0	25.0	1
36.5	K	110.0	100.0	3
36.5	K	140.0	100.0	3
39.2	G	110.0	25.0	2
45.3	K	100.0	25.0	2
26.7	G	110.0	25.0	2
54.9	G	100.0	25.0	2
37.1	K	150.0	25.0	3
34.1	K	150.0	25.0	3
30.3	K	160.0	25.0	3
40.1	G	100.0	25.0	1
29.9	K	120.0	25.0	2
40.7	K	140.0	25.0	3
59.6	K	90.0	25.0	3
30.5	G	130.0	25.0	3
37.8	N	120.0	25.0	3
41.5	K	100.0	100.0	3
60.8	K	50.0	0.0	3
63	K	50.0	0.0	3
49.5	K	100.0	25.0	3
50.8	K	100.0	0.0	1
39.3	K	120.0	25.0	2
39.7	G	100.0	25.0	3
55.3	K	90.0	25.0	3
42	K	110.0	25.0	1
40.6	K	110.0	25.0	1
68.2	N	80.0	0.0	1
74.5	N	90.0	0.0	1
72.8	N	90.0	0.0	1
31.2	K	110.0	25.0	2
53.1	K	110.0	25.0	1
59.4	N	90.0	25.0	2
38.8	G	110.0	100.0	3
28.6	G	140.0	100.0	3
46.7	G	100.0	100.0	3
39.1	G	110.0	25.0	3
27.8	G	110.0	25.0	2
49.8	K	100.0	25.0	1
51.6	G	100.0	25.0	1
36.2	G	110.0	25.0	1