1. Are customers more loyal in the east or in the west? The following table is based on information from Trends in the United States, published by the Food Marketing Institute, Washington, D.C. A poll was conducted among a random sample of 2008 supermarket customers, and the poll results are summarized in this table. The rows represent length of customer loyalty (in years) at a primary supermarket. The columns represent regions of the United States. East Midwest South West Row total Less than 1 year 32 31 53 41 157 1 - 2 years 54 68 92 56 270 3 - 4 years 59 68 93 67 287 5 -9 years 112 120 158 78 468 10 - 14 years 77 63 106 45 291 15 or more years 118 173 158 86 535 Column total 452 523 660 373 2008 According to the information presented in this table, what is the probability that a supermarket customer chosen at random a) has been loyal no more than 2 years? b) has been loyal at least 1 year, given that he or she is from the South? Show all the necessary work to obtain your answer. c) has been loyal 5 or more years or is from the Midwest? Show all the necessary work to obtain your answer. d) Are the events "from the West" and "loyal less than 1 year" independent? Use a probabilistic argument to support your answer. (Hint: Remember that if events A and B are independent, then P (B) = P(BA))