Questions: A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 59 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 7000 aspirin tablets actually has a 5% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 59 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 7000 aspirin tablets actually has a 5% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Transcript text: Points: 0 of 4 Save A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 59 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 7000 aspirin tablets actually has a $5 \%$ rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Solution

Solution Steps

Step 1: Define the problem

We are given a shipment with a total of $N = 7000$ items, from which we randomly select $n = 59$ items. The shipment is accepted if at most $x = 1$ of these items are defective. The defect rate of the shipment is $p = 0.05$.

Step 2: Calculate the probability of acceptance

The probability that a shipment is accepted is calculated using the formula: \[P(\text{accept shipment}) = \sum_{k=0}^{x} \binom{n}{k} p^k q^{n-k}\] where $q = 1 - p = 0.95$.

Step 3: Perform the calculation

Substituting the given values into the formula, we calculate the probability of acceptance.