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Supposa you are given shirts t0 "amall;" to se0 whathar or not they were worn by pecpl with Parkinson 9 Half ot the shirts were from Parkinson'8 patl...

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Supposa you are given shirts t0 "amall;" to se0 whathar or not they were worn by pecpl with Parkinson 9 Half ot the shirts were from Parkinson'8 patlents , and half were not Assuma throughout tnls actvlty that YOu cannot actually small Parkinsons Lo- YoU are Just quessing randomlyWhat is the probabilily that you wil guoss the first one right (express as @ percentage)?What is the probability that you wll guess the second one right (regardless 0l whelher or nol you got Ine first one

Supposa you are given shirts t0 "amall;" to se0 whathar or not they were worn by pecpl with Parkinson 9 Half ot the shirts were from Parkinson'8 patlents , and half were not Assuma throughout tnls actvlty that YOu cannot actually small Parkinsons Lo- YoU are Just quessing randomly What is the probabilily that you wil guoss the first one right (express as @ percentage)? What is the probability that you wll guess the second one right (regardless 0l whelher or nol you got Ine first one right)? Suppose 200 peaple do this same task How many will get the first shlrt rght? Among those who got the first shirt right hor many will additionally get tne Aicono snint nght? Out ol tne onginal 200, how many will gei both the first and Ihe second nght? What (5 tne probability of this happening (express [he probability as # percenL Among those who gol both the first and stcond night; how many will also Gel tne tnird ona righl? What Is the probabully that somuane will qat Ina Oirat (nree shirts ngnl? (express Porcenti Do you have an ideu Io Ina probubly Dettrg wbou: tho probablity getting 12 In Tol? Ihe {irat four righi? How



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Imprints Galore buys T-shirts (to be imprinted with an item of the customer's choice) from a manufacturer who guarantees that the shirts have been inspected and that no more than $1 \%$ are imperfect in any way. The shirts arrive in boxes of $12 .$ Let $x$ be the number of imperfect shirts found in any one box. a. List the probability distribution and draw the histogram of $x.$ b. What is the probability that any one box has no imperfect shirts? c. What is the probability that any one box has no more than one imperfect shirt? d. Find the mean and standard deviation of $x .$ e. What proportion of the distribution is between $\mu-\sigma$ and $\mu+\sigma ?$ f. What proportion of the distribution is between $\mu-2 \sigma$ and $\mu+2 \sigma ?$ g. How does this information relate to the empirical rule and Chebyshev's theorem? Explain. h. Use a computer to simulate Imprints Galore's buying 200 boxes of shirts and observing $x,$ the number of imperfect shirts per box of $12 .$ Describe how the information from the simulation compares to what was expected (answers to parts a-g describe the expected results). i. $\quad$ Repeat part h several times. Describe how these results compare with those of parts a-g and with part h.

In this question. To start off, we're given this relationship between lambda and P. Then in part A We are told that why is a binomial random variable based on parameters N. And lambda. Therefore why divided by N. Is an unbiased estimator for lambda. And we are asked to derive an unbiased estimator for P based on why we can rearrange the equation at the top of the sheet to give the following. This means that an estimator for P is given by the following. So that is our estimator for P. And now given and equals 80 And why equals 20. We want to find our estimate for P. So we just plug this into the formula for estimator And this comes out to 0.2. For part B. We want to show that our estimator is unbiased. So we really want to show that the expected value of our estimator is equal to P. So this is equal to the expected value of two. Y over em -0.3. That's just using this equation with why over and is equal to lambda. And then using the linearity of expectation. This can be re expressed as the following. So this is two times lambda And the expectation of .3 is .3. And this is equal to P. Since the expected value of our estimator is the parameter we're estimating for it is an unbiased estimator. And then for part C we are given a slightly different set up for the question which would result in this relationship between lambda and P Would now be 0.7 times P Plus 0.3 times 0.3. And now we are asked what our estimator for P would be the estimator for lambda remains Why over em since why is still a binomial random variable, the estimator for P is equal to the estimated Verlander -0.09, Divided by 0.7. And that's done simply by solving for P in this equation and then simply re expressing this, substituting why over. In for for the estimated for lambda, we get why over in -0.09 Over a 0.7. So this is now our estimator for P.

In this question. To start off, we're given this relationship between lambda and P. Then in part A We are told that why is a binomial random variable based on parameters N. And lambda. Therefore why divided by N. Is an unbiased estimator for lambda. And we are asked to derive an unbiased estimator for P based on why we can rearrange the equation at the top of the sheet to give the following. This means that an estimator for P is given by the following. So that is our estimator for P. And now given and equals 80 And why equals 20. We want to find our estimate for P. So we just plug this into the formula for estimator And this comes out to 0.2. For part B. We want to show that our estimator is unbiased. So we really want to show that the expected value of our estimator is equal to P. So this is equal to the expected value of two. Y over em -0.3. That's just using this equation with why over and is equal to lambda. And then using the linearity of expectation. This can be re expressed as the following. So this is two times lambda And the expectation of .3 is .3. And this is equal to P. Since the expected value of our estimator is the parameter we're estimating for it is an unbiased estimator. And then for part C we are given a slightly different set up for the question which would result in this relationship between lambda and P Would now be 0.7 times P Plus 0.3 times 0.3. And now we are asked what our estimator for P would be the estimator for lambda remains Why over em since why is still a binomial random variable, the estimator for P is equal to the estimated Verlander -0.09, Divided by 0.7. And that's done simply by solving for P in this equation and then simply re expressing this, substituting why over. In for for the estimated for lambda, we get why over in -0.09 Over a 0.7. So this is now our estimator for P.

In this question to start off, we are given this relationship between Land A and P, then in part a. We are told that why is a binomial random variable based on parameters N and Lambda? Therefore, why divided by n is an unbiased estimator for Lambda and we are asked to derive an unbiased estimator for P based on why we can rearrange the equation at the top of the sheet to give the following. This means that an estimator for P is given by the following. So that is our estimator for P and now given and equals 80. And why equals 20? We want to find our estimate for P. So we just plug this into the formula for estimator and this comes out to 0.2 for part B. We want to show that our estimator is unbiased. So we really want to show that the expected value of our estimator is equal to P. So this is equal to the expected value of two y over em minus 0.3. That's just using this equation with why over and is equal to Lambda and then using the linearity of expectation this can be re expressed as the following. So this is two times Lambda and the expectation of 0.3 is 0.3 and this is equal to P since the expected value of our estimator is the parameter we're estimating, for it is an unbiased estimator. And then for part C, we are given a slightly different set up for the question which would result in this relationship between Lambda and P would now be 0.7 times P plus 0.3 times 0.3. And now we are asked, What are estimator for P would be the estimator for Lambda remains. Why over em since why is still a binomial random variable? The estimator for P is equal to the estimated Verlander minus 0.9 divided by 0.7. And that's done simply by solving for P in this equation and then simply re expressing this substituting. Why over in for for the estimated for Lambda we get why over in minus 0.9 over a 0.7. So this is now our estimator for P

All right, So we're given an inventory of shirts. Some of them were shortly. Some are long. See, some are small, medium large sum are plaid printer stripe. We're giving the distribution in this table here and were asked a bunch of probability questions about this inventory. But before we do that, I'm gonna do something real quick just to make my life a little easier in solving this problem. I'm gonna write in a total call him in a total row, both of these tables as such. Now let's start with the short sleeves and small rope. In order, find the total for this row. Ijust add everything up in the road. So four plus two of six six plus five is 11. So this is 0.11 and we'll continue going down the line. So let's see. This is eight and seven, which is 15 15 and 12. It's 27 and this is that's 10 18. Now what these distributions mean these numbers are the distribution of small, medium and large short sleeve shirts, respectively. I'm going to do the same thing down here with the columns now. So plaid, we have four plus 8 12 Totals 5 15 2799 and seven At 16 eight and 12 2025 25. We're also gonna fill in this bottom right hand corner here. We're just gonna add up either the total rose or the total column. It doesn't matter. Basically, this will give us the distribution of all shorts T shirts. All right, We're just gonna add these three because 15 and 25 is a nice 40. We can add 16 to that rather easily. That's 56. This should be the same as thes numbers Here. You can check that for years. I'm gonna do the same thing down here. So this adds up to 0.8 This adds up to 0.22 This adds up to 0.14 It adds up to 0.170 point 09 0.18 The shadow add up 0.44 Yes, and something we should also check real quick is these two numbers they add up to one. And this should make sense because the percentage of short sleeve shirts puts the percentage Long sleeve shirt should be ah, 100% of the shirts. Which is this one, all right. With that in mind, let's start solving these problems. Our first problem. We're supposed to find the probability of selecting a medium long print shirt. It's rather simple. We can just look in the table long sleeves, medium print and that 0.5 all right, probability of a medium print in general. Well, we just add the probability of finding medium print shortly, plus a medium print long sleep. So that will be here. I'm just going to color code them waas 0.5 should give us 0.12 So those are answers apart and be part see were given were asked for the probability of selecting a short sharp Well, we can't id did that already, actually with peace total columns. So there you go 0.56 Same here with long shirts 0.44 So there are answers. Alternatively, you could add up everything inside here, but like I've established, that's already the same of is what we did here. We just added less terms. All right. Number four are part D. I should say we need to find the probability of just selecting a medium certain. General. Here's where these total columns will come in handy. All right, let's look at short sleeve. The total distribution of medium shortly shirts 0.27 As for long seeds, 0.22 So this illegal 0.27 0.22 which is 0.49 All right. As for the prince will do the same thing. We'll look in the print for the short sleeves. 0.16 the prince for long sleeves 0.9 sarah 0.16 plus 0.9 equal 0.25 All right, here we have some conditional probabilities content with male. We need to find the probability of a selectee and medium short given that it was short and plaid. Now, your reminder that probability is defined as the number of successful outcomes over the number of total outcomes in this case are successful. Outcomes are the medium shirts within the subset and our total outcomes are all the short sleeved plaid shirts. So that would be this column of the table right here now. Of those the mediums All right here. So in our numerator will have 0.8 in our denominator will have 0.15 This equals 0.533 rounding to three decimal places finally part off. So we need to find the probability of finding a short sleeve shirt, given that it was a medium and plaid shirt. Once again, successes over over total outcomes. So our successes, our short sleeve medium plaid shirts, which are right here. So that goes in the numerous up? Yeah. No, that's great. So that goes in the numerator. My fault. I thought I was circling something else. That's our total welcomes. We would just add our medium plaid shorts T shirts with our medium plaid long sleeve shirts. So we're gonna throw that nominator. Uh, hurray for color coding. So this equals 0.8 over 0.18 This is equal to this Reduces down to eight for ninth. So that 0.4443 decimal places and what we could use a similar role. Similar methodology down here. Well, since we only have two conditions outside of the medium and plaid shorts lever long sihf, you know that the probability of getting a long sleeve shirt, giving them medium. Platt is one minus. The probability of not getting a long sleep start given medium and Platt or the probability of finding a short sleeve shirt giving that it's medium in plaid. And since we've already found that very conveniently, you could just subtract that out 0.5 by six and there you go.


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