Question

damaged during A shipping company guarantees customers that no more than 2% of their shipments are transportation. On a typical day, the company transports 2000 shipments. 4. a. What is the average and standard deviation of the damaged shipments? b. Calculate the exact probability that less than the average number of shipments are damaged. Use any computer software to find it. c. Calculate the probability in part (b) using normal approximation. d. What is the % error between (b) and (c)? e. Calculate the exact probability that 60 or more shipments are damaged. f Calculate the probability in part (e) using normal approximation. g. What is the % error between (e) and (f)?

i need complete Answer,all parts

Answer 1

Let's write the given information

n = sample size = 2000

p = probability of damage = 0.02

a) $\mu$ = average = n * p = 2000 * 0.02 = 40

$\sigma$ = standard deviation = square root of n*p*(1-p) =  $\sqrt{2000*0.02*.98}=6.261$

b) Here average = 40 so we need to find P( X < 40) = P(X <= 39)

let's used excel:

P( X <= 39) = "=BINOMDIST(39,2000,0.02,1)" = 0.4783

c) Here we need to make continuity correction

P( X < 40) = P( X < 40 - 0.5) = P( X < 39.5)

By using excel:

P( X < 39.5) = "=NORMDIST(39.5,40,A3,1)" = 0.4682

d) the difference is = 0.4783 - 0.4682 = 1.01%

e) P( X >= 60) = 1 - P(X < = 59) = "=1-BINOMDIST(59,2000,0.02,1)" = 0.001692

f) P( X >=60) = 1- P(X < 60) = 1- P(X < 59.5) (By making continuity correction)

= "=1-NORMDIST(59.5,40,A3,1)" = 0.000921

g ) the difference = 0.001692 - 0.000921 = 0.000771 = 0.077%