Question

Your mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders recieved in the past week for an audit. The audit reveals that 86 of these orders were shipped on time.

(a) If the company really ships 90% of its orders on time, what is the probability that 86 or fewer in an SRS of 100 orders are shipped on time?

(b) A critic says "Aha! You claim 90%, but in your sample the on-time percent is only 86%. So the 90% claim is wrong." Explain in simple language why your probability calcualtion in part (a) shows that the result of the sample does not refute the 90% claim.

Answer 1

(a) Here as sample size is less than 5% of population so we will count the distribution is binomial.

Here n = 100

sample proportion = 86/100 = 0.86

standard error of sample mean = sqrt [0.86 * 0.14/100] = 0.0347

Test statistic

Z = (0.86 - 0.90)/0.0347 = -1.153

p - value = Pr(Z < -1.153) = 0.1245

so here the probability that 86 or fewer in an SRS of 100 orders are shipped is 0.1245

(b) Here as we know that we are just using a sample to check the authenticity of mail order company's claim. Here, as the we get the 86 shipped orders on time that have a probability of around 12% which is not unusual so here we cannot refute the 90% claim.