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	57	59	112	77	149	486
Midwest	31	79	68	120	63	150	511
South	53	85	93	158	106	204	699
West	41	65	67	78	45	65	361
Column Total	157	286	287	468	291	568	2057

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
  Your answer must be a fraction, integer, or exact decimal.

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

Answer 1

(a) has been loyal 10 to 14 years

ans ) 291/2057
(b) has been loyal 10 to 14 years, given that he or she is from the East

ans) 77/486
(c) has been loyal at least 10 years

ans) (291+568)/2057=859/2057

(d) has been loyal at least 10 years, given that he or she is from the West

ans) (45+65)/361=110/361
(e) is from the West, given that he or she has been loyal less than 1 year
ans) 41/157

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

(g) has been loyal 1 or more years, given that he or she is from the East

ans) (57+59+112+77+149)/486=444/486

(h) has been loyal 1 or more years, given that he or she is from the West

ans) (65+67+78+45+65)/361=320/361

(i) Are the events from the East and loyal 15 or more years independent?

P(East)=486/2057

P(loyal 15 or more years)=568/2057

P(East and Loyal 15 or more years)=149/2057=0.072 ......................(i)

P(East)*P(Loyal 15 or more years)=0.065 ..........................(ii)

Since i and ii are not equal hence not independent.

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