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[s]At onliue store claims that 80% of all orders ure delivered within days: The qqunlity coutrol deputtment would like to test this claim by randomly selecting 200 ...

Question

[s]At onliue store claims that 80% of all orders ure delivered within days: The qqunlity coutrol deputtment would like to test this claim by randomly selecting 200 recemt Orders. Out oftlc 2u0 ordlers; 156 of tlcm wet delivered withun "duvs Using (.12I cun YOu (Olcludlc tlmt if Alwre i5 sullticient evidlence to Kuppcnt thc &lm? Use the [-value %pprouch.

[s] At onliue store claims that 80% of all orders ure delivered within days: The qqunlity coutrol deputtment would like to test this claim by randomly selecting 200 recemt Orders. Out oftlc 2u0 ordlers; 156 of tlcm wet delivered withun "duvs Using (.12I cun YOu (Olcludlc tlmt if Alwre i5 sullticient evidlence to Kuppcnt thc &lm? Use the [-value %pprouch.



Answers

It is suspected that some of the totes containing chemicals purchased from a supplier exceed the moisture content target. Samples from 30 totes are to be tested for moisture content. Assume that the totes are independent. Determine the proportion of totes from the supplier that must exceed the moisture content target so that the probability is 0.90 that at least 1 tote in the sample of 30 fails the test.

So in this question, were given information about the proportion of orders that arrive on time from a catalog company. So they took a sample and found that 88% of the orders people in their sample received arrived on time, with a margin of error of 6%. So this and this is a 95% confidence interval. So that means that our confidence interval 88% minus 6% is 82% at the low end and 88% plus 6% 94% at the top end. And this is our confidence interval about what we believe the true proportion of catalog orders received by customers to be based on this sample proportion of 88%. So if we interpret what that means, we can be 95 percent confident that the true proportion of customers who received I think I spoke that wrong looks better. Orders on time is between 82% Andi, 94%. So that's what that means. We're estimating about the parameter, the true proportion, using our sample to do that. So this is our best guess, and if we were to repeat this procedure lots and lots and lots of times. We believe that 95% of those would capture the true proportion. Not that they would all be the same thing is this. So were given some other scenarios and were asked to say if that interpretation is correct or incorrect on the 1st 1 is that between 82% and 94% of all orders arrive on time and this is an incorrect statement. It's partly incorrect. Well is incorrect because they use the expression they think they will arrive implying some kind of guarantee. Um and that is too much certainty. Do you wanna be careful with that in statistics and interpreting any kind of statistics? But confidence intervals a swell part B asks. About 95% of all samples will show that 88% of the orders arrive on time and again, we have another incorrect statement. Um, this is a confidence interval. Based on this sample, each sample will produce its own sample proportion. So Oops, I don't know what I was spelling there, but wasn't the right word. So each sample will produce its own. He hat Not the same P hat we don't expect the same proportion. We don't expect the same confidence interval for each individual sample. Part C says that 95% of all random samples of customers will show that 82% to 94% of the order survive on time. On again, we have another incorrect statement. Um, each sample will have its own confidence interval and the interval that were that What we get is making some kind of inference about the parameter, not about the sample. So we're not trying to predict what other samples are gonna be. We're trying, toe. Um oh, jeez. Misspelling here is about the population we're trying to infer about the population, not other samples. Okay, Um, and then part D, we are 95% sure that between 82% and 94% of the orders placed from the sampled customers will arrive on time and again, we have another incorrect statement. We know what P hat Waas. We don't have to infer about it in this sample. He hat was 88% and we're not making some kind of inference about the sample. Were making an inference about the population and then finally, e 95% of the days between 82% and 94% of the orders will arrive on time and again we have another incorrect statement. And that's because it's not 95% of the days we're talking about this inference again. It's kind of the same thing as C this inferences about the population in other words, all days, not 95% of the days. So the only correct interpretation that we have here is this. 1st 1 up here tells us that we could be 95% confident that the true proportion ISS somewhere in this interval of 82% to 94% That's based on our sample, um, proportion of 88%. I got a different sample. My interval will be slightly different, and I would hope that that would still capture whatever the true proportion is

So we know that we have aspirin and we know that we're taking a sample random sample of 40 of the shipments. And we know that the probability of defect Is about three according to them. And we know that there are 5000. Uh they're gonna sample 40 of the 5000 and they will accept a shipment. So they'll accept a shipment if less than or equal to one, so zero or none is uh defective. And so we need to find what the probability is of zero And the probability of one being defective. And so we know that this is a binomial setting approximately. So, And we know there's the probability of a defect, there is the probability of not effective and in the first case we're going to have 40 right here And we're gonna choose none of them to be defective. None. And all 40 of them will be good. And then we want to find the likelihood one being defective And that means 39 of them will be good and we have 40, choose one. And again this is just one and this is just 40. And so we can find that probability. I'm actually going to use my feature on my calculator of the binomial syria And I'm going to type in 40 and the .3 and then I'm going to type in that. I want one or fewer and so and that is underneath the distribution button second and distribution. And we can find that there at the scroll quite a ways down Or you can do all this separately and the 40 trials. The probability of success is .03. My ex value is one And paste. It depends on your software on your couch later. But we find out that this comes out to be 66, about 66%. So the likelihood of accepting a shipment is 66%. If they have a three chance that each one is a defective and the question is you're going to accept About 66%. But that means you're going to reject Mhm 34%. That's a very high rejection rate. So we would suggest that they actually try to do something to decrease this number of defective and do something with their quality control because that is a very high defective rate, verify, high rate of rejecting the aspirin.

A person. Probability is when we're determining a probability in a defined space, a period of time or volume. And this particular problem is talking about finding the probability of receiving two orders in a defined number of advertisements going out. And that would be a defined space. 100 advertisements. So let's write down what we know. We know that there's five orders that are received per 500 solicitations or advertisements. We know that it is a plus on probability, and the formula for Hassan probability is the probability of X given an average or Lambda equals E to the negative. A Lambda Times Lambda raised to the X power all over ex factorial and Lambda in this case is the average number of orders in 100 advertisements rather than 500. So we're going to set up a proportion to solve this. We know that there are five orders generated when we send out 500 ads or solicitations and we've got to figure out our lambda when we only send out 100 ads and if we do are cross product, we must apply. We would end up with 500 lambda equals 500 or a Lambda equals one. So we know that we average one order for every 100 solicitations were adds that go out. Now we're trying to find the probability of receiving at least two orders. So if we define what at least two orders means, well, that could mean that we have two orders or three orders or four or five, and it's an infinite list. So it's hard for us to do the probabilities of all of those. So in order to solve this, we're going to have to think in terms of a probability distribution chart and in a probability distribution chart, there are cursed certain characteristics. And when you add up, the probabilities associate it with every single outcome possible, the sum of those probabilities will always equal one. So when we are soliciting for orders, the values of X in this instance could be zero. Could be one could be two or three. We're more than that. So we're focused on the 2345 So we're looking for this value here, So in order to find that value, what we're going to have to do is we're gonna first have to find the probability of zero orders and the probability of one order. And then in order to find what we need right here, we're going to do one, subtract the probability of zero and subtract the probability of one which means that we're going to have to do use this formula two different times. So when we use the formula first time we're going to do the probability of zero, given that there is one order per 100 solicitations. So then it would be e to the negative one power multiplied by one to the zero power all over zero factorial. And when we do that, we get in a value of about point three six seven, eight eight and then we have to calculate the probability of one. So we're going to have e to the negative one power times one to the first power over one factorial which, lo and behold, gives us the same value 0.367 eight, eight. So when we substitute those values in here and in here, we're then going to have that the probability of two or more orders is equal to one minus two of these 20.36 788 values because there were two of them and your probability turns out to be approximately 0.264


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