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	72	59	112	77	144	496
Midwest	31	60	68	120	63	189	531
South	53	98	93	158	106	178	686
West	41	44	67	78	45	69	344
Column Total	157	274	287	468	291	580	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


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

Answer 1

Solution:-

Given data:-

(a)

The probability that a customer has been loyal 10 to 14 years = (291)/(2057)

(b)

The probability that a customer has been loyal 10 to 14 years, given that he or she is from the East
=(77)/(496)

Required probability = (77) / (496)

(c)

The probability that a customer has been loyal at least 10 years = (291+580) / (2057)

Required probability = (871) / (2057)

(d)

The probability that a customer has been loyal at least 10 years, given that he or she is from the West = (45 + 69 ) / (344)

Required probability = (114) / (344)

(e)

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

Required probability = (41) / (157)

(f)

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

Required probability = (53) / (157)

(g)

The probability that a customer has been loyal 1 or more years, given that he or she is from the East = (72+59+112+77+144) / (496)

Required probability = (464) / (496)

(h)

The probability that a customer has been loyal 1 or more years, given that he or she is from the West = (303) / (344)

Required probability = (303) / (344)

(i)

Actually probability loyal 15 or more years > probability 15 or more years | East

So,the answer is No

These events cannot occur together.