A company receives a very large shipment of components. A...

A company receives a very large shipment of components. A random sample of 16 of these components will be checked, and the shipment will be accepted if fewer than 2 of these components are defective. What is the probability of accepting a shipment containing each number of defectives? a. $5 \%$ b. $15 \%$ c. $25 \%$

Key Concepts

Binomial Distribution

The binomial distribution is a probability distribution that models the number of successes (or failures) in a fixed number of independent trials, each having the same probability of success. In quality control, it is applied to determine the likelihood of observing a certain number of defective items in a sample, assuming that each component has an independent chance of being defective.

Statistical Quality Control

Statistical quality control (SQC) involves the use of statistical methods and procedures to monitor and control a manufacturing process, ensuring that the product quality meets the required standards. Techniques like acceptance sampling and the use of the binomial distribution are central to SQC, enabling firms to make informed decisions about product acceptance based on probabilistic analysis.

Acceptance Sampling

Acceptance sampling is a statistical quality control process used to determine whether a batch of products should be accepted or rejected based on the quality of a randomly selected sample. This method is cost-effective, especially when testing every item is impractical, allowing decisions to be made about an entire shipment based on inspection of only a small portion.

Transcript

00:01 Now, in this problem, we are told that we have a large shipment of components.

00:03 A random sample of 16 is chosen, and we're going to be checked.

00:09 The shipment will only be accepted if fewer than two of the components are defective.

00:13 And we want to find the probability if on a, 5 % of them are defectives.

00:18 That means p is 0 .05.

00:19 Now, since this is a binomial, this means that p of x is 16 choose x times 0 .05 to the x times 0 .95 to the 16 minus x.

00:36 We want fewer than two, so this means that we need the probability of zero plus the probability of one.

00:44 The 16 choose 0 times 0 .05 to the 0 times 0 .95 to the 16 plus 16 choose 1 times 0 .05 to the 1 times 0 .95 to the 15.

01:03 And then we just plugged in 0 and 1 into that binomial distribution up above.

01:08 And now we're just going to evaluate this.

01:10 And just type these in straight on our calculator just like they look.

01:20 This tells us the probability of accepting 0 .8108.

01:29 0 .8108.

01:33 Now on b, we're going to change that to 15%.

01:36 And so 15 % of those that we have are defective.

01:40 And so for b, p is 0 .15.

01:44 So let's just go back up here and all we need to do is change these values.

01:48 Everything else about it will be the same.

01:50 So now it's 0 .15 .8.

01:51 0 .15 and 0 .85.

01:56 All we need to do is just change those values there, and then now we calculate the same thing.

02:00 And so now we just type this in on our calculator just like it looks...

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