Question

BINOMIAL PROBABILITIES Big Box Store (BBS) has an annual rate of 4% of all sales being returned. In a recent sample of thirty randomly selected sales the number of returns was five. BBS is concerned about the event, and your advice is solicited. ( FOR 4 – 9, OPEN THE EXCEL EXAM TWO FILE. THE SPREADSHEET FOR THIS PROBLEM IS FOUND ON SHEET TWO. COMPLETE AND SAVE YOUR WORK IN EXCEL AND ENTER THE SOLUTIONS BELOW. (6) (2.5points) What is the probability that a random sample of 30 sales has 5 or fewer returns? Type the answer and any work below.   (7) (2.5 points) What is the probability that a random sample of 30 sales has less than four returns? Type the answer and any work below.

Answer 1

SInce no spreadsheet is given , thus we would compute the answers of 6 ,7 with only the information given

we will use binomial model as there are only two possible outcomes which in itself are mutually exclusive

n = 30 = total sample

p = probability of event occuring = 0.04 = probability of a sample returned

q =  probability of event not occuring = 0.96 = probability of a sample not returned

X= Number of samples returned from 30 samples

using binomial model

P(X=x) = nCx * p^x * q^(n-x)

P(X=0) = 30C0 * (0.04)^0 * (0.96)^(30 - 0) = 0.293858

P(X=1) = 30C1 * (0.04)^1 * (0.96)^(30 - 1) = 0.367322

P(X=2) = 30C2 * (0.04)^2 * (0.96)^(30 - 2) = 0.221954

P(X=3) = 30C3 * (0.04)^3 * (0.96)^(30 - 3) = 0.086304

P(X=4) = 30C4 * (0.04)^4 * (0.96)^(30 - 4) = 0.024273

P(X=5) = 30C5 * (0.04)^5 * (0.96)^(30 - 5) = 0. 005259

and so on , bu we require only these probabilities to compute the required probabilitues

P( random sample of 30 sales has 5 or fewer returns) = P(X<=5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

= 0.293858 + 0.367322 + 0.221954 + 0.086304 + 0.024273 + 0.005259

= 0.998970

So, there is 99.8970% probability that from 30 samples , 5 or less samples will be returned

P( random sample of 30 sales has less than four returns) = P(X<4 ) P(X=0) + P(X=1) + P(X=2) + P(X=3)

= 0.293858 + 0.367322 + 0.221954 + 0.086304

= 0.969407

So, there is 96.9407% probability that from 30 samples , less than 4 samples will be returned