Your firm purchased 10000 dvds the manufacturer assures you...

Your firm purchased 10,000 DVDs. The manufacturer assures you that only 12% of the 10,000 are defective. To test the claim that only 12% of the 10,000 are defective, you take a random sample of 100 DVDs and test each of the 100. Create a test such that you reject a good shipment about 5% of the time. (Get as close to 5% as you can). The correct answer will look like: reject the shipment if I find “Q” or more defects in the sample--- tell me what “Q” is 5. Using the test you created from problem 4, compute the probability of accepting a bad shipment if the supplier is sending us a shipment with a defective rate of 18%

Transcript

00:01 So in this question we're told that if nothing has gone wrong at most six chips in each lot would be defective.

00:08 So we have a lot of big n equals a thousand.

00:15 So if there's nothing wrong then we would have, let's call it x equals six defectives.

00:29 So what we want to do is test 100 chips from each lot and there's a trade -off between discarding a good lot and the cost of warranty claims and you want to have a chance for most two percent of discarding a lot given that the lot is good.

01:02 So let's say we discard if we find more than y defectives in our sample and we want the probability that, well let's let's call this, yeah let's call this y, and let's say that z is the number of defectives found.

01:44 We want the probability that z is greater than or equal to y given that nothing is wrong to be no more than two percent.

02:04 Well what we can say is that the probability that z equals little z is going to be the probability of choosing z defectives from our six in our lot of a thousand.

02:21 So it's going to be six choose z times 994 choose 100 minus z.

02:29 So this is choosing our defectives, this is choosing our non -defectives, and then divide that by the total number of choices which is a thousand choose a hundred.

02:45 So let's tabulate that because it only takes values from, well z can only be zero to six.

02:52 So z p z 0 1 2 3 4 5 6 and then i'll do a cumulative function as well because we want the cumulative function of y to be less than 0 .02.

03:06 So for zero we have 994 choose 100 divided by divided by a thousand choose 100 and this gives us a probability 0 .5306.

03:36 Then for one that's going to be six.

03:41 It's going to be six choose one times 994 choose 99 divided by a thousand choose 100.

03:49 So that's going to be 0 .3557.

03:59 But i mean really we don't need to tabulate this whole thing as we've already found that there's more than a 50 chance of getting zero.

04:09 So really the only thing you can say is that you should pick, oh no, you want the cumulative function actually to be greater than 99 so that the probability of selecting something greater than that is going to be no more than 2.

04:32 So let's keep going.

04:35 So 0 .5306 plus 0 .3557 gives us a cumulative function of 0 .8863.

04:48 Now for two we do six choose two times 994 choose 98 divided by a thousand choose 100 which is 0 .0982.

05:07 So let's add that on and we get 0 .9845.

05:15 So that means that if we take all of this, this is greater than 98 percent and that means that these three must be less than 2 percent...

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