Question

Leo has a pizza business. The times to prepare and deliver pizzas are approximately normally distributed with 20 minutes and a- 10 minutes (a) Suppose that Leo advertises "If it takes more than 30 minutes to get your pizza, you get it free". What fraction of his take-out pizzas will he have to give away free? (Round to 4-decimal places) 3. (b) How long a time should Leo allow if he wants to give away no more than 5 % of the pizzas? (Round to the nearest minute)

Answer 1

Let X denote the time taken to prepare and deliver pizzas given X is normally distributed with mean mu = 20 minutes standard deviation sigma = 10 minutes X sim N (20, 10^2) (a) Pizza will be given free if X > 30 minutes X 30 Leftrightarrow x - mu > 30 - mu x - 20 > 30 - 20 x - 20/sigma > 30 - 20/sigma = 10/10 that is x > 30 implies x - mu/sigma > 1 p (x > 30) = p (x - mu/sigma > 1) = p (x - 20/10 > 1) = p (z > 1) where z = x - 20/10 = 0.1586 (from z - score table) 15.86 % 0.1586 of his take - out pizzas will have to give away free \r\n (b) Let the time he should allow is 'a' minutes to give away no mole than 5% p (x > a) lessthanorequalto 5 % p (x > a) lessthanorequalto 0.05 p (x - mu/sigma > a - mu/sigma) lessthanorequalto 0.05 p(z > a - 20/10) lessthanorequalto 0.05 From z -score tables P (z > 1.6449) = 0.04999 therefore a - 20/10 = 1.6449 a - 20 = 16.449 a = 36.449