When purchasing bulk orders of​ batteries, a toy manufacturer uses this acceptance sampling​ plan:

Question

Randomly select and test 42 batteries and determine whether each is within specifications. The entire shipment is accepted if at most 3 batteries do not meet specifications. A shipment contains 7000 ​batteries, and 1​% of them do not meet specifications. What is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?

**May someone please walk me through this

Answer 1

There are 7000 batteries in the shipment, and 1 % of them do not meet the specifications. Thus there are 70 batteries that do not meet the specification, and 7000 – 70 = 6930 batteries that do meet specifications.

The whole shipment will be accepted if, out of 42 randomly selected batteries, less than 4 batteries do not meet specifications. Put another way, the shipment will be accepted if more than 38 batteries (39, 40, 41, or 42 batteries) meet specifications.

There are two main ways to approach this problem.

The first method is to get an exact answer by using the hypergeometric distribution. For this problem, the probability distribution of the number of randomly selected batteries that meet specifications can be described by what is known as the hypergeometric distribution.

The hypergeometric distribution has 3 unique parameters. These are: N = the size of the population, K = # of success states in population, and n = # of draws (sample size). Let’s say that for this problem, we’ll define a battery as a “success” if it meets specifications.

In this case, N = 7000, K = 6930, and n =  42. It is possible to plug these into a hypergeometric calculator to get the exact answer. However, it involves large factorials (i.e. very big numbers) so most calculators will probably run into an error. So this first method isn't practical, although if you do find a way to calculate the answer using the hypergeometric distribution it will be exactly right.  

The second way gives an approximate number, but it should be a very good approximation since the population size and # of successes are large compared to the size of the sample. In this case, we can treat the population (the 7000 batteries) as being so large that for all practical purposes it can be thought of as infinite. So we can treat the probability distribution of the number of randomly selected batteries that meet specifications as a binomial distribution.

The binomial distribution has 2 unique parameters. They are: N = # of draws (sample size), and p = probability of success.

In this case, N = 42 and p = 0.99 (since 99% of the batteries do meet specifications). We can use a binomial calculator with these parameters to calculate the probability that there are 39, 40, 41, or 42 batteries that meet specifications. This yields a probability equal to 0.9991735, or approximately 99.92 % chance that the shipment will pass. So the vast majority of shipments will be accepted.