Question

Are customers more loyal in the East or in the West? The following table is based on information from a recent study. The columns represent length of customer loyalty (in years) at a primary supermarket. The rows represent regions of the United States. Less Than 1 Year 1-2 Years 3-4 Years 5-9 Years 10-14 Years 15 or More Years Row Total East 32 49 59 112 77 143 472 Midwest 31 67 68 120 63 171 520 South 53 96 93 158 106 208 714 West 41 35 67 78 45 67 333 Column Total 157 247 287 468 291 589 2039 What is the probability that a customer chosen at random has the following characteristics? (Enter your answers as fractions.) (a) has been loyal 10 to 14 years (b) has been loyal 10 to 14 years, given that he or she is from the East (c) has been loyal at least 10 years (d) has been loyal at least 10 years, given that he or she is from the West (e) is from the West, given that he or she has been loyal less than 1 year (f) is from the South, given that he or she has been loyal less than 1 year (g) has been loyal 1 or more years, given that he or she is from the East (h) has been loyal 1 or more years, given that he or she is from the West (i) Are the events from the East and loyal 15 or more years independent? Explain. Yes. P(loyal 15 or more years) = P(loyal 15 or more years | East). No. These events cannot occur together. No. P(loyal 15 or more years) ≠ P(loyal 15 or more years | East). Yes. These events can occur together

Answer 1

	<1	1-2 yr	3-4 yr	5-9 yr	10-14 yr	15 or more	total
east	32	49	59	112	77	143	472
midwest	31	67	68	120	63	171	520
south	53	96	93	158	106	208	714
west	41	35	67	78	45	67	333
total	157	247	287	468	291	589	2039

a)

P( has been loyal 10 to 14 years )=291/2039

b)P( has been loyal 10 to 14 years, given that he or she is from the East )=77/472

c)

P( has been loyal at least 10 years )=880/2039

d)

P( has been loyal at least 10 years, given that he or she is from the West )=112/333

e)

P( is from the West, given that he or she has been loyal less than 1 year )=41/157

f)

P( is from the South, given that he or she has been loyal less than 1 year )=53/157

g)

P( has been loyal 1 or more years, given that he or she is from the East )=440/472

h)

P( has been loyal 1 or more years, given that he or she is from the West )=292/333

i)

No. P(loyal 15 or more years) ≠ P(loyal 15 or more years | East).