Problem 1 A typical convenience store customer buys gasoline with probability 0.

Question

Problem 1

A typical convenience store customer buys gasoline with probability 0.50 and groceries with probability 0.8. The events A={buying groceries} and B={buying gasoline} are independent.

a) What is the probability that a customer buys gasoline but does not buy groceries?

b) What is the probability that in a random sample of 4 customers at least one will buy both gasoline and groceries?

c) Consider a random sample of 20 customers. What is the probability that in this sample the number of customers who buy both gasoline and groceries is larger than 2 and smaller than 8?

Problem 2

A drug manufacturer claims that a new drug for lowering blood pressure is effective in 80% of patients. Consider a random sample of 20 patients. Use the information provided by a manufacturer to compute

a) the probability that the drug will be effective in 15 to 18 patients. b) the probability that the drug will be effective in at least 8 patients? c) the expected number of patients and the standard deviation of the

number of patients for which the drug will not be effective.
d) When the drug was tested it lowered blood pressure in 7 out of 20 patients. In view of this, what do you think about manufacturer’s claim concerning the rate of the effectiveness of the drug? (To answer this question

use an appropriate probability calculation)

Problem 3

The management of the supermarket knows from the past experience that expenditures per customer are normally distributed with mean = $95 and a standard deviation of = $21.

a) What is the probability that a randomly selected customer will spend more than $105?

b) The management plans to give a bonus to the customers whose expen- ditures are in the top 5% of the distribution of expenditures per customer. What is the least expenditure that qualifies a customer for a bonus?

c) Consider a random sample of 4 customers. What is the probability that at least one customer’s expenditures will be above the 80th percentile of the expenditures per customer distribution?

d) Consider a random sample of 100 customers. What is the expected number customers and standard deviation of the number of customers whose expenditures will be above the 80th percentile of the distribution of expen- ditures per customer?

Explanation / Answer

Problem 1

A typical convenience store customer buys gasoline with probability 0.50 and groceries with probability 0.8. The events A={buying groceries} and B={buying gasoline} are independent.

a) What is the probability that a customer buys gasoline but does not buy groceries?

Answer : Pr(Buys Gasoline but not groceries) = P(A) * P(B') = 0.50 * (1 - 0.80) = 0.10

b) What is the probability that in a random sample of 4 customers at least one will buy both gasoline and groceries?

Answer : Pr(Both gasoline and groceries) = P(A) * P(B) = 0.50 * 0.80 = 0.40

Now there are 4 people and probability that any one of them will buy both things is 0.40 . That would be calculated by calculating that noone will buy both of the things.

so by binomial distribution

Pr(X >=1; 4 ; 0.40) = 1 - Pr(X = 0; 4 ; 0.40) = 1- (0.60)4 = 1 - 0.1296 = 0.8704

(c) Consider a random sample of 20 customers. What is the probability that in this sample the number of customers who buy both gasoline and groceries is larger than 2 and smaller than 8?

Answer : Now here n = 20 and Pr(Both gasoline and groceries) = 0.40

so, Pr(2 < X < 8 ; 20 ; 0.40) = BIN(2 < X < 8 ; 20 ; 0.40) = BIN (X <8 ; 20; 0.40) - BIN(2 <=2 ; 20; 0.40)

= 0.4159 - 0.0036 = 0.4123

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