Using a binomial distribution to evaluate toy manufacturing quality control

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to use a binomial distribution to evaluate toy manufacturing quality control.  Here's our problem statement: When purchasing bulk orders of batteries, a toy manufacturer uses this acceptance sampling plan: Randomly select and test 35 batteries and determine whether each is within specifications.  The entire shipment is accepted if at most three batteries do not meet specifications . A shipment contains 3000 batteries, and 2% of them do not meet specifications.  What is the probability that this whole shipment will be accepted?  Will almost all shipments be accepted, or will many be rejected?

Part 1

OK, the first part of this problem is asking us to calculate the probability that the whole shipment will be accepted.  And to do that, we're going to use the binomial distribution calculator in StatCrunch.  So first we need to pull up StatCrunch, and I can do that here.  I'm going to pop this window out, and then I'm going to resize it so we can see a little bit better everything that's going on here.  Then inside StatCrunch, I'll go to Stat --> Calculators --> Binomial.

Here in my binomial calculator, I need to add in parameters of my distribution.  The sample size is 35 batteries.  Why am I using the 35 and not the 3000?  Well, because 3000 is the population.  3000 is the entire shipment.  We're just taking a portion of that population.  That's what a sample is --- a portion of the population.  So the 35 batteries is our sample size and not the 3000.  Probability of success?  Well, we're going to define success as not meeting specifications, and we do that because it just works out better that way.  I know it sounds funky that, you know, not meeting specifications is going to be a success, but it just makes the problem easier.  The percentage is 2% of the population aren't meeting specifications, so that's the probability of success.

Then we have to look to see that the entire shipment will be accepted if that most three batteries do not meet specifications.  So we can have no batteries, or one battery, or two battery, or three batteries, and that would mean that we are accepting the shipment.  So here we're actually calculating the probability based on one number, but we need four different numbers: 0, 1, 2, and 3.  So I'm going to come up here and press the Between option on my calculator so I can put in everything between zero and three.  And there we would get our probability, 0.9948918.  We were asked to round to four decimal places.  So that comes out to be 0.9949.  Nice work!

Part 2

And now the second part has a few different fields for us to fill in.  The first is asking for an acceptance rate.  And we have that acceptance rate right here.  We just calculated it.  It's in decimal form.  We need to convert it to percent form.  And we do that by moving the decimal place over two places.  So that becomes 99.49.  And then the rejection rate is just a complement of the acceptance rate.  So if we subtract that from 100 and I can do that with my handy dandy calculator here, just subtract that out from 100.  And that gives me my rejection rate, which is awfully low. So we've got some good stuff going on here because it was such a low rejection rate.  It's going to be that almost all the shipments are going to be accepted.  Fantastic!

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