A shipment of 10 items has two defective and eight...

A shipment of 10 items has two defective and eight nondefective items. In the inspection of the shipment, a sample of items will be selected and tested. If a defective item is found, the shipment of 10 items will be rejected. a. If a sample of three items is selected, what is the probability that the shipment will be rejected? b. If a sample of four items is selected, what is the probability that the shipment will be rejected? c. If a sample of five items is selected, what is the probability that the shipment will be rejected? d. If management would like a .90 probability of rejecting a shipment with two defective and eight nondefective items, how large a sample would you recommend?

A shipment of 10 items has two defective and eight nondefective items. In the inspection of the shipment, a sample of items will be selected and tested. If a defective item is found, the shipment of 10 items will be rejected.
a. If a sample of three items is selected, what is the probability that the shipment will be rejected?
b. If a sample of four items is selected, what is the probability that the shipment will be rejected?
c. If a sample of five items is selected, what is the probability that the shipment will be rejected?
d. If management would like a .90 probability of rejecting a shipment with two defective and eight nondefective items, how large a sample would you recommend?

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution is used to determine the probability of a given number of successes (such as defective items) in a sample drawn without replacement from a finite population. It is crucial in problems involving quality control and acceptance sampling where the composition of the entire lot is known, and the sample is taken without putting selected items back.

Sampling Without Replacement

This concept refers to the process of selecting items from a set where each selected item is not returned to the population before the next draw. It impacts the probabilities because the chance of drawing a defective item changes as items are removed from the shipment, making the draws dependent on each other.

Combinatorial Analysis

Combinatorial analysis involves using combinations to count the number of ways in which items can be selected from a set. In the context of sampling, it is used to calculate the number of possible samples that contain a certain number of defectives versus the total number of samples, forming the basis for probability calculations.

Quality Control and Acceptance Sampling

In quality control, acceptance sampling is a statistical method used to decide whether to accept or reject a shipment based on a sample of items. The approach typically involves defining criteria (such as the presence of a defective item) that determine if a lot meets the required quality standards, thereby minimizing both inspection costs and risks.

Sample Size Determination

Sample size determination is the process of calculating the number of items to test in order to achieve a desired probability of detecting defects if they are present. This concept is essential in designing effective sampling plans that balance risk and cost, ensuring that a sufficient portion of a shipment is inspected to make informed decisions.

Transcript

00:01 Well, this problem is told that we have a shipment of 10 items with two defective and eight non -defective.

00:07 So you have 10 total items.

00:09 Two of them are defective.

00:10 That means your p value is 0 .2.

00:15 This is 20%.

00:18 Now notice that we have a binomial distribution here because each item is chosen independently, and there are only two options that's either going to be defective or in effect.

00:27 And so let's look here on part a.

00:30 On part a, we were told that we have a sample of three items.

00:35 We want the probability that the shipment will be rejected, and it's rejected if a defective item is fat.

00:41 Now here, n is 3.

00:45 And so we want the probability that one of the three, again, we are taking a sample of 3.

00:53 We want the probability that at least 1 is defective.

00:59 Now, probability of at least 1 is able to 1 minus the probability of 0.

01:04 And so since this is a binomial distribution, this is 1 minus 3 -20 times 0 .2 to the 0 .0 .8 to the 3rd.

01:19 And so then we evaluate this, and this is 1 minus 0 .8 to the third, and that's 0 .488.

01:29 And so the probability it's rejected is 0 .488.

01:33 That's with that sample of 3.

01:37 Now on b, we want a sample of four items to be selected.

01:43 We want to know the probability that it is rejected.

01:48 And so it means with a sample of four, that means we want the probability that at least one of the four.

01:55 I'm sorry, so it should still be x as random one.

01:57 We want at least one of the four, but n is four now.

02:02 This is still the same as one minus the probability that x is zero.

02:07 And since it is four, this is now one minus...

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