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[1Opts] The IRS classifies income into 3 ranges: low (35% of the population), medium (45%), and high (20%_ and historically audits tax returns from these groups wit...

Question

[1Opts] The IRS classifies income into 3 ranges: low (35% of the population), medium (45%), and high (20%_ and historically audits tax returns from these groups with rates 0. 75%/, 1.75%/, and 3%, respectively_ What is the probability that; given a tax return has been audited, it comes from the high income group?

[1Opts] The IRS classifies income into 3 ranges: low (35% of the population), medium (45%), and high (20%_ and historically audits tax returns from these groups with rates 0. 75%/, 1.75%/, and 3%, respectively_ What is the probability that; given a tax return has been audited, it comes from the high income group?



Answers

Solve each problem. Tax Time If the probability of a tax return not being audited by the IRS is 0.91 , then what is the probability of a tax return being audited?

All right. 72. We have you come and were given under 25 25 to 49. 50 to 99 100 for 99 and greater than or equal to 500. And our probabilities, our 0.431 for under 25 0.248 for 25 to 49. Wait to 15 or 50 to 99 0.1 for 104. 99.0 06 So, in part A, we're going to look at, um we're randomly selecting someone. No, we don't want to know the probability that they're making 50,000 or more. So I'm gonna add up my 0.215 plus 0.1 plus 0.6 in that equals 0.3 21 part B. Um, given they make over, find the probability given they make over 50,000 at based. What's that probability? They make at least 100,000. So my formula is at least 100,000 and at least 50,000. Well, that's just gonna be at least the hunt. Good 1000 which is gonna be 0.106 over the probability of at least 50,000 which is 0.3 21 No, when I divide that I get 0.3302

Here we have data on the sources of federal revenue already expressed in percentages. The probability that a particular source chosen at random would be in the category of individual or corporate taxes. Individual is 50% of the data and corporate taxes is 10% of the data. So we're simply going to had those together. That would be 60% probability.

Alright in this problem, we're talking about the fact that Delaware has the highest per capital annual income, and we know that the average of that annual income is 51,803. We know that the standard deviation is $4850 and we're talking about the entire population of Delaware in this problem. We're also talking about the fact that we're going to take a random sample of 34 residents. So from that population, we're going to pull a sample and the sample size is 34. And then we're going to talk about the average of those 34 residents. So the average of the sample means and we've got to talk about the standard deviation of the sample means. And there's two questions that we're going to answer in this problem. The first one is what is the probability that the mean of those 34 residents that were chosen is greater than 50,000, and the second one is going to be less than 48,000. So let's work on the fact that we're doing the mean being greater than 50,000, so we're going to need to construct are bell shaped curve. Um, again, it's the bell shaped curve is because of our sample size being large enough, it's greater than 30 so the sample distribution will be normally distributed. The average of our sample means, based on the central limit theorem, will be equivalent to the average of the population, or, in this case, 51,803. And the standard deviation will be equivalent to the standard deviation of the population divided by the Santa Square root of the sample size or, in other words, 4850 divided by the square root of 34. So in order to solve our problem, we're going to put the average in the centre Rod Bell, and we want the fact that the average has to be greater than 50,000, which would fall about here on our bell, and we want to be greater. So we're going to need the standard Z score. So we're going to calculate the Z score for 50,000 and the formula for the sea scores e equals X bar minus mu sub x bar all over Cygnus of expert. So in this instance, the sea score would be just move that over a little bit. Disease score would be 50,000 minus 51 803 all over 4850. Divide by the square root of 34 which calculates out to be approximately negative 2.17 So negative 2.17 is up there. So when we're asking for the probability that the average is greater than 50,000, it's no different than saying What's the probability that the Z score is greater than negative 2.17? And that is the same thing as saying one, minus the probability that the Z score is less than negative 2.17 And then if you go to your standard normal table and look up negative 2.17 you will find that the area to the left is 0.1 50 When you subtract that from one, we get 10.9 eight five zero. So in summary of the first part of this problem, the probability that the 34 Delaware residents have an average income or a mean income greater than 50,000 is 0.98 50 Now let's go to part two of this, we're going to have to use the same information. I am going to screwed it up a little bit here. We're going to need the Z score associated with 48,000. So the second question is, what's the probability that the average is less than 48,000? We're going to calculate the Z score using that same formula we're gonna do 48,000 minus 51,803 over 400,850 divided by the square root of 34. And if you use your calculator, you will calculate that out to be approximately negative 4.57 So when we say that we're looking for the probability that the average is less than 48,000, since 48,000 has a Z score of negative 4.57 We could also write this problem as what's the probability that the Z score is less than negative 4.57 Now, if you look at your standard normal table, you're going to see that there is no negative 4.57 It's off the chart, and as you look as we get closer and closer to that value. The numbers are getting closer and closer to zero. So we're going to say that the probability that Z is less than negative 4.57 is close 20 So it's a highly unlikely event. So to summarize Part two of this question, the probability that X bar were the average of the 34 residents that were selected in a sample having a salary or income sorry of less than $48,000 is close 20

This problem we are told that 75% of customers to a certain department store or from a high income group. We want to find the probability That fewer than four of the next nine do not belong to this income group. Yeah. And so since 75% are part of it, that means peace .75. Since we're choosing nine people, that means that are in value is nine. And some mines in general, the probability effects is nine choose acts From his .75 to the Act Times .25 To the nine -X. Just using our formula for binomial distribution. Now we want fewer than four. So we want to find the probability that X is less than four. And so this means that acts zero, one, two or three. And we're gonna find each of those probabilities and then just add them to get. The probability of zero Is 9- zero. Times .25 to the 9th probability of one. There's nine choose 1 Times .75 to the one I was .25 to the 8th. The probability of two is 9 choose to 1.75 to the second, 1.25 to the 7th. And then the probability of three There's nine choose 3, 1.75 to the third Times .25 to the sixth. Again, all these just came from plugging them all in to the formula. Up here for the binomial distribution. And then now you can just type all these in on a calculator in order to evaluate them. Okay? And when we do this should give us 0.0 999 five.


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