Question

Booked Solid, a small independent bookstore in Bradford is trying to decide whether to discontinue selling magazines. The owner suspects that only 7% of the customers buy a magazine and thinks that she might be able to use the display space to sell something more profitable. Before making a final decision, she decides that for one day he'll keep track of the number of customers and whether or not they buy a magazine.

1. What is the probability that exactly 5 of the first 15 customers buy magazines? Round your answer to 4 decimal places.
2. 
   What is the probability that at least 5 of her first 50 customers buy magazines? (10 points)
   3. She had 280 customers that day. Assuming this day was typical for her store, what would be the mean and standard deviation of the number of customers who buy magazines each day? (10 points)
   4. Surprised by the high number of customers who purchased magazines that day, the owner decided that her percentage estimate of customers who still buy magazines must have been too low. How many magazine sales would it have taken to convince you? Justify your answer.

Answer 1

What is the probability that exactly 5 of the first 15 customers buy magazines? Round your answer to 4 decimal places.

What is the probability that at least 5 of her first 50 customers buy magazines? (10 points)

P(X>5) = 1-P(X<=4)  = 1 - (P(X=0) + P(X=1)+ P(X=2)+P(X=3)+ P(X=4))

P(X=0)

P(X=1)

P(X=2)

P(X=3)

P(X=4)

P(X>5) = 1-P(X<=4) = 1-(0.0999 + 0.1843 + 0.2219 + 0.1963)=0.2975

She had 280 customers that day. Assuming this day was typical for her store, what would be the mean and standard deviation of the number of customers who buy magazines each day? (10 points)

4. Surprised by the high number of customers who purchased magazines that day, the owner decided that her percentage estimate of customers who still buy magazines must have been too low. How many magazine sales would it have taken to convince you? Justify your answer.

This a binomial distribution with p = 0.07, q = 0.93

Hence np = 19 and nq= 271, which are both greater than 10, hence we can approximate to normal distribution

According to normal distribution, it would be unusual to see sales more or less than 2 standard deviation from the mean

2 standard deviation above the mean
20 + 2(4)= 28

2 standard deviation below the mean
20-2(4) = 12

Hence if more than 28 customers bought the magazin the the estimate was probably too low.