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	31	59	112	77	148	459
Midwest	31	71	68	120	63	177	530
South	53	69	93	158	106	161	640
West	41	75	67	78	45	76	382
Column Total	157	246	287	468	291	562	2011

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.

No. P(loyal 15 or more years) ≠ P(loyal 15 or more years | East).Yes. These events can occur together.    No. These events cannot occur together.Yes. P(loyal 15 or more years) = P(loyal 15 or more years | East).

Answer 1

a) P(has been loyal 10 - 14 years) = 291/2011 = 0.1447

b) P(10 - 14 year | East) = 77/459 = 0.1678

c) P(loyal at least 10 years) = (77 + 148)/2011 = 225/2011 = 0.1119

d) P(loyal at least 10 years | west) = (45 + 76)/382 = 121/382 = 0.3168

e) P(West | less than 1 year) = 41/157 = 0.2611

f) P(south | less than 1 year) = 53/157 = 0.3378

g) P(loyal 1 or more years | East) = (459 - 32)/459 = 0.9303

h) P(loyal 1 or more years | west) = (382 - 41)/382 = 0.8927

i) P(loyal 15 or more years) = 562/2011 = 0.2795

P(royal 15 or more years | East ) = 148/459 = 0.3224

Since P(loyal 15 or more years) P(royal 15 or more years | East ) so these events are not independent.

No, P(loyal 15 or more years) P(royal 15 or more years | East ). These events can occur together.