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A company is considering two investment projects whose present values are described as follows:Project 1: there are three possible NPW outcomes: $2,000, $4,000,and ...

Question

A company is considering two investment projects whose present values are described as follows:Project 1: there are three possible NPW outcomes: $2,000, $4,000,and S8000 with associated probabilities: 0.4, 0.35, 0.25 respectively:Project 2: NPW(1O%) 10 X + 8 XY + 3 XZ,where X, Y; and Z are statistically independent discrete random variables with the following distributions:Variable XVariable YVariable ZEventProbability 0.7EventProbability 0.25EventProbability520S1181.50.6S400.35220.7553.750.4[a]

A company is considering two investment projects whose present values are described as follows: Project 1: there are three possible NPW outcomes: $2,000, $4,000,and S8000 with associated probabilities: 0.4, 0.35, 0.25 respectively: Project 2: NPW(1O%) 10 X + 8 XY + 3 XZ, where X, Y; and Z are statistically independent discrete random variables with the following distributions: Variable X Variable Y Variable Z Event Probability 0.7 Event Probability 0.25 Event Probability 520 S11 81.5 0.6 S40 0.3 522 0.75 53.75 0.4 [a] Compute the mean and variance of the NPW for project [b] Plot the distribution of the NPW for project 1. X-axis is NPW value and Y-axis is probability: [c] If all NPW values from Project are independent from NPW values from Project 2. Find the probability that Project 1 is better than Project 2_



Answers

(a) identify the expected distribution and state $H_{0}$ and $H_{a}$, (b) find the critical value and identify the rejection region, $(c)$ find the chi-square test statistic, $(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. The pie chart shows the distribution of the opinions of U.S. parents on whether a college education is worth the expense. An economist claims that the distribution of the opinions of U.S. teenagers is different from the distribution for U.S. parents. To test this claim, you randomly select 200 U.S. teenagers and ask each whether a college education is worth the expense. The table shows the results. At $\alpha=0.05,$ test the economist's claim.
(FIGURE CAN'T COPY) (TABLE CAN'T COPY)

Okay. So for this example we are asked to find basically the expected values of V. W. X. And Y. Which is certificate of deposit and so on. So we're just gonna take the probability that the strong occurs which is going to be 40% times the outcome of that and then added to the other one. So the probability that we make $6000 of the certificate of deposit. Uh Well that's actually just gonna be 6000 since we're gonna make $6000 in either case. Mhm. The probability that we make 15,000 is strong which is 0.4. So we're gonna write 0.4 times 15,000. And then we're gonna add, it's the probability that we get weak which is 5000, mm. Okay. And then we're going to repeat this process for the other two. And here are the results that we get for X and Y. So now be determine the expected value of each variable. Well, that's exactly what it is. Find the probability distribution of each random variable. Uh Well, there we go. And which investment is the best expected payoff? That would be the w so the office complex has the best and the worst one is going to be the land speculation and this. I would, if I were to invest in this, I would probably choose the green one because it's highly valued. So I go into the office complex.

Its first. We have to find whether the board, the contracts or independent So Indra ability the two events are independent of the instance of one event is not affected, probably off the other event. So in this case, getting contract number two depends on the first contract. So both the card racks or not independent. Next, we have to find a probably to get the boat. The contracts show the probability to get off his contract. Probably that you get those what's going back. It's given us 0.8. Oh, and probably that you get this second good black after getting the first contract. The 0.2, if we would be modification rule. You have problems. See, Did you get bored of contracts? Little 0.8 Because your point girl, that is a critical your 0.16 16%. Next, we have to find probably that you don't all get both the contracts. So using compliment room, we have probably that you do not good contract one is equal to one minus. Probably that you get the contract. One. It is one minus. There a 0.8 that is G quick Original point. Good. No probability that you do not get contract to without getting contract. One that is equal to one miners. Probably. They did not get contractor. Yeah, so it is equal to one minus. Probably that you get contacto without contract one. Did you get going? Classical to the Probably that you can't contract to without getting contract. One is given a zero point three. Okay, that is equal to 0.7. So probably they did not get contract one and door using my application route. We have zero point group. Probably the June contract. One don't. Probably They did not get contract without contract. One that is General Greek. Seven is equal to 0.14 2014% age. Next, we need toe. Assume X. To be the number off contracts you get. We have to find the probably mortal forex tonight, XB. Then I'm off contracts you get. You have to find the probability motor on the so X takes a value to one not getting any contracts or getting one contract Getting more. The contracts in the respective probabilities? Yes. You want getting in equal any contact? 0.14 and getting bored. The contracts I found that it is 16% age this 0.16 and getting anyone off the contact is nothing but one minus 10.14 minus 0.16 is one minus 10.3 just 0.7. Next, we need to find the expected value. Understand the deviation expected value it affects. It's given us X in tow. Your fix that is 0.0 in 2.14 Place one in 2.7 plus doing toe little 0.16 zero place 2.7 bliss 0.32 That is equal to one point in orderto I understand the deviation. You have to find the first on a division. First, we'll find a radiance. Sigma squared, that is summation X minus knew the whole square in to pee off eggs that is equal to zero minus one point. You do the whole square digital 0.1 full anise one minus 1.2 was quite in tow. Little 0.7 list Tu minus one point They do. They were square in tow. One point does it'll 0.16 That is your point 14 56 place. Several point. There was little zero to be place zero point 1537 There is a quarto. Consider a point to 996 So signifies equality zero point 54 74

Okay. In this question, we have a fighter that shows the distribution of how much married US male adults, just their spouses to manage their finances. Now there is a financial services company that claims that the distribution of how much the married females trust their spouses to manage their finances is the same as the picture. Now we're skeptical as always. So what we do is in order to test the claim, we randomly select 400 married US females and ask each of them how much they trust their spouses to manage their finances. So we have the results of our survey with us. The first column is the response column has always What we do is we create a table to solve these kinds of questions. So we have the response column with us, right? So they completely dressed the first. The first category is completely dressed. What? Yeah, yeah, completely. Trust the next one is trust with certain aspects. Yeah, so let me just write this as they trust them. They trust them. They don't completely trust them. They don't blindly trust them, but they just trust them. The fourth, the third one is that they do not trust them. So this is they don't trust. And in the end, we have the not sure category, not short category. Then we have the observed value. These are the frequencies. Or let me just write these as the observed values. Okay, For the completely trust we have 2 43 2, 43. After that, we have trust with certain aspects. This is 108 then do not trust is 36 and then we have studied. All right now we have a by charged with us, which is the distribution of how much the married male US adults trust and the financial services company claims that the distribution is the same for males as well as females. So the probabilities, the probabilities for these categories will be the same as given to us in the pie chart. So completely trust at 65.6. So let me just write this as 0.656 Trust with certain aspects is 27.8. So this is 0.278 Do not trust is 5.7. So this is 0.57 I'm not sure few people who are not sure this is 0.9% of this is zero point 009 Right? 0.90 point 009 Right? Uh, yeah. So this is your 0.9 Okay, Now, what are going to be our now? An alternative hypothesis. The null hypothesis will be that the distribution mhm. The distribution for us married male adults mean? Mm. Yeah. Adults fits the distribution. Yeah, it's the distribution for us married female adults. All right, what will be the alternative hypothesis? The alternative hypothesis will be that the distribution for us married male adults does not fit the distribution. Four. Yeah. US married female adults. All right, Now what we are going to do over here is the chi square. Goodness of fit test. What is the first time? The first step in this analysis is to find the expected values for all the categories. And what is the formula for that? The formula for that is sample size that is in multiplied by the probability for each category. The probability for each category that is P I. All right, Let us look at this formula in action. This is the column for expected values. This will be the column for expected values. Okay, now, what is a sample taste? It is 400 females, 400 married US females. So what will be the expected value For the first category? It is going to be 400 multiplied by zero point 656 or this is to 62.4 to 62.4. For the next category, it will be 400 multiplied by 0.278 This is going to be 111.2. After that, we have 400 multiplied by 0.57 which is 22.8, 22.8 and then it will be 0.9 multiplied by 400. So this should be three 0.6. These are the expected values. Now. The next step in this analysis is finding the chi square statistic. How do we do that now? There is a formula for this. What are you going to do is in the next column you're going to fill up the values for each category. For each category, you're going to find the difference of observing the expected values. Mhm You square them. You divide this value by the expected value and this particular value this expression we find for all the categories and in the end we sum them up. So this gives us the overall price quest at the stake for our problem. Let us go here. Let us look at this formula in action. So for the first category, what is the difference between observed and expected values? It is too. 62.4 minus 2. 43. This is 19.4. I square this. So this becomes 30. 76.36 and I divide this by the expected value to 62.4. So this is 1.434 1.434 For the next one, we have 111.2, minus 108. The square this and divided by 111.2. So this 0.92 0.92 Then we have 36 minus 22.8. We square this this is 1 74 and divide this by 22.8. This is 7.642 7, 70.642 Then we have the difference between 13 and 3.6. We square this and divide this by the expected value. This is 24.54 24.54 Again, we have solved enough questions to see that these the addition of this anti column is very high. It goes into 30 or 32 or something right 32 or 33 something so we can, without any analysis, say that the distributions are different. We will reject Donald Hypothesis, but still let's verify this by doing the proper analysis by using the proper method I some all of these up. So this is 24.54 plus 7.642 plus 0.92 plus 1.434 This is 33.71 So my overall chi square statistic is 33.71 Now, if I want to reject my null hypothesis or not, how do I decide this? I decided this with by the help of two methods. Okay, there are two methods to do this. The first one is the P value method And the second one is the critical funding method. Now, in order to do both of these methods, I need the degrees of freedom and the degrees of freedom is given by the formula. Yeah, number of categories. Number of categories. Yeah, minus one. All right. How many categories do I have? If I look at this, there are just four categories here. So this is going to be four minus one or this will be three. So now I have my chi square value. I have my degrees of freedom. Now, the first matter that I'm using is the P value matter. What is the alpha for this question? The alpha is 0.1. The alpha is 0.1. Now I will find the p value. And if my P values less than Alpha, I will reject final hypothesis or else I will fail to reject it. Now, in order to find the p value, I can use either the Chi square table. But the thing over here is that if I use the chi square table, I won't get the exact value. I will get an approximate range which will still be good enough to reach an answer. But if you use a statistical software like SPS s or R or python or Excel, you will get the exact value. So this is what I'm doing. I'm using an online tool, a way that I have. My chi square statistic is 33 point. This is 33.71 and degrees of freedom is so this is 33.71 and my degrees of freedom is three. And my alpha is 0.1. I hit calculate, and the P value that I get is much less than 0.1 So let me just write this zero from a P value is approximately 4 to 0. My p value is approximately equal. Yeah, my P value is approximately equal to zero now, Since my P value is less than alpha, this will suggest that I will reject. I will reject my final hypothesis H north. Right now this was a P value method. How do I find this? Using the critical value method, I need to find the critical chi Square statistic, the critical value. So what will be my critical value? My degrees of freedom is three and my alpha is 0.1. So I just put in these values My advice. One my degrees of freedom is three and I hit calculate and I get 6.251 My critical value turns out to be 6.251 So what exactly does this mean? If this is my graph for the class quested to stick, let's say that this is my critical value 6.25 and 6.251 to the right of this, I have the rejection region. This is going to be the rejection region. Okay, so if I get a chi square statistic that that is to the right of 6.251 that is, it is greater than 6.251 It will fall here in the rejection region and I will reject minor hypothesis. I can see that my value is 33.71 which falls way towards the right, which is much bigger than 6.251 Hence, I will reject minor hypothesis. See the same result that we got from the P value method. Now, if we go up and look at the non hypothesis, the hypothesis was that the distribution for us married male adults fix the distribution for us married female adults. We are rejecting this. So how do I frame my answer? In the end, my answer will be that ad 10% significance level at 10% significance level. I have enough statistical evidence. Statistical? Yeah. Evidence to suggest that the distribution yes, of men married us adults, US adults, mhm of male married US adults. Yeah, and the distribution of female married us Adults are different, are different. Yeah, are different. Which means if you look at the question, what was the claim? The claim was the financial services communicate the distribution of how much US married women illustrates their spouses. The man is the finance is the same. The claim of the financial services company. What we are effectively saying is that the claim? Yeah. Of the financial services company of the financial services company. Mhm. It doesn't seem to be right. Yeah. Doesn't seen yeah to be right. Okay, so let me just go with this what we had first of all, we had a nail in the alternative hypothesis. The null hypothesis was that the distribution for US male adults about how much they trust their female counterparts to manage their finances is the same for the US married females, farmers that trust their male counterparts. And the alternative hypothesis was that the distribution for us married males in the US married females was different. We had a sample of 400 a females. We had the probabilities, the pie chart was given to us, and based on the probabilities and the sample size, we calculated the expected values. The formula was sample size multiplied by the probability for each category. This gave us the expected values for this category, and in the end, we wanted to find the guy Specialist IX. So for all the categories we applied this formula, will you Oh, my inner city hall square upon e, we sum them all up, and we found that our chi square statistic is 33.71 Now, we could have just said over here that yes, this, uh, we have project and a hypothesis, since this guy's for value is very high. But we use this two methods to find this to verify our answer. The first one was a p value matter. We found that with the degrees of freedom three, our P value was very close to zero. Hence, we reject around a hypothesis and even using the critical value matter for Alpha of 0.1, we found that the critical value 6.25 and our value of 33 falls in the rejection reason. So, yes, Beyonce's we're matching now. The main thing was to frame the correct answer. In the end, the conclusion. The conclusion was that at 10% significance level, we had enough statistical evidence to suggest that the distribution of male married US adults and the distribution of the female married US adults were different. Which means that the claim of the financial services company doesn't seem to be right there. Claim is not correct. And this is how we go about doing this question, Yeah.

Now, In this case, there is an organization that claims the number of prospective Homebuyers one that next house to be larger or smaller or the same size as the current house is not going to form the distribution. Okay, now, in order to disclaim, what we do is we randomly select 800 prospective Homebuyers and ask them what size they want their next house to be. So we have 800 prospective home bias. We complete our survey, and we have the results with us, we have the table. As always, The first thing is draw the table and note down the observations. So the first problem is response. First category is larger. The second one is small size, and the third one is smaller. All right, what are the observed values that we have? The frequencies that we saw in our survey Philosopher it is 25 for small size is 2 24. And for smaller, it is to 91. All right, now, what will be the null and the alternative hypothesis? The null hypothesis will be that the number of prospective home buyers who want to buy the next house to be larger, smaller the same side of the current houses. Not even a formulation. Okay, so the distribution. Yeah. Then distribution off opinions of prospective Homebuyers. Mm. Opinions of prospective Homebuyers of opinions of prospective Homebuyers is uniform. Yeah, is uniformly distributed. Okay. Okay. What would be the alternative hypothesis? Each a. The older hypothesis would be that the distribution? Yeah. Mhm, uh, opinions. Or we can write this as choices. Also, the choices of prospective Homebuyers, of course. Prospective Homebuyers. Yeah. Is not. Your family distributed is not uniformly distributed. Well, right. We have the Nile and the alternative hypothesis. Now, what is the distribution? This point is very important. Now, since we are saying that it is uniformly distributed, we can say that the probabilities for all the categories the probabilities, just a moment. The probabilities for all the categories. There are three different categories. They should have the same probability. So they should be won by three. For all of them. One by 31 by 31 by three. All right Now, what is our sample size 800. This adds up to 800. Okay. Yeah. Now we have the probabilities. We have the absorbed values. We have the sample size. Now the next step is always finding the expected value for other categories. So e I will be given us the sample size. Yeah, the sample says multiplied by the probabilities for each category were deployed by the probability for each category P I. So let's look at this formula in action. A sample sizes 800 the probability for every categories one by three. So the expected value will also be the same. So this is 800 divided by three. This is 2. 66.66 six so administrate. These are the expected values the expected values. This will be to 66.6 seven. Let me write it like this. So this will be same for all three of them. So this is to 66.67 Okay, The next thing is calculating the chi square statistic. Now, how exactly do we do that? In order to calculate the chi square statistic for every category we apply this formula, we find the difference between the observer and the expected values. We square the difference and we divide this value by the expected value. In the end. After doing this for all the categories. We sum them all up and we end up getting our chi square statistics. All right, let us look at this formula in action for the first can be The difference between observed and expected values is 2 85 minus 2, 66.67 We square this or 18.33 square and we divide this by 2. 66.67 This is 1.2599 Let me just write this as 1.26 for the next one. The difference is between 2. 66.67 and 2. 24. We square this this is 42.67 and we divide this by 2. 66.67 So this is 6.8276 So this is 6.8276 or eight minutes. Right. This is 6.8 two. Similarly, now the differences between 2 91 and to 66.7. Now we have to square this. So this is 24.3 square and direct this by 2. 66 point 67 This is 2.214 2.21 four. All right, now what do we do? We add, all of these are so this is 1.26 plus 6.83 plus 2.214 This is 10.304 This is 10.30 for this addition. Turns out to be 10.304 Now, this is our chi square statistic. Now, another calculation that we need to do is for the degrees of freedom. The reason freedom is given by the formula. Number of categories, number of categories, minus one. How many categories do we have in this question? We just have three categories. So this is going to be three minus one or I can write this as my degrees of freedom. Now I have my chi square statistic. I have my degrees of freedom. Now I want to decide whether I wanted to reject minor hypothesis or not. In order to do that, there are two methods. The first one we are going to look at is the P value method. What is the alpha for this question? If I look at the question I have and 50.5 Alpha is 0.5 Now, in this weather, what I do is I find the p value. And if my p values less than Alpha, I reject one of my prosthesis. In order to find the p value, I can either use a chi square table. But if I use a high score table, I will not get an exact value. I'll just get arranged. So, in order to get the exact value, what I will do is I will use a statistical software. I have an online tool. You can use anything you like. Excel s, P. S s R. Python, anything. I'm using an online tool over here. I'm just putting in the value. So 10.304 And to my guys go to the sticker standpoint, 304 10.30 force. And maybe he's a freedom is too significant level of 0.5 And I had to calculate my p values 0.578 minutes. Right. This, uh, just a moment. So, yeah. So my p value is 0.0 578 So let me just write this is 60.6 Now, if I look at my alpha and my P value, I can see that my P values less than Alpha. Right hence. This implies that what I'm going to do is I am going to reject final hypothesis. It's not. This is a P value method. What is the critical value method over here? I'll just find the critical value of my chi square statistic. My critical value of the Chi Square statistic. What is my Alpha Alpha 0.0 fight? What is my degrees of freedom? My degrees of freedom is too. So I am just going to put in these values in the calculator and this is going to give me my chi square value and it is 5.991 My critical value is 5.991 If I look at the graph of the distribution, let's say that this is the draft for my by square distribution. This is the critical value. This is the critical value 5.991 Now, any value that I get to the right of this lights in this shaded region, which is the rejection region This is the rejection region. Mhm the rejection region. So any value that is greater than 5.991 or that lies in the rejection region and rejected what is my chi square statistic? It is 10.304 which means it lies over here somewhere. So this is in the rejection region greater than 5.991 So I will reject minor hypothesis. This is exactly what I've got here. So what is going to be my conclusion? After all of this analysis that we went through, what will be our conclusion conclusion will be. What I will write is that act 5% level of significance at 5% level of significance. I have enough statistical evidence to suggest evidence to suggest that what was the wording of another hypothesis? That the distribution of opinions of prospective Homebuyers is not uniformly distributed? Okay, that mhm the distribution, Yeah, of opinions or choices. Opinions of prospective Homebuyers of cross. Yeah, directive home buyers Home buyers is not uniformly distributed is not uniformly distributed. Mhm distributed. Okay, or what I can say is what was the claim movie? The organization? The organization claims that the number of prospective who were smaller and so is not enough for ministry. So we can say that the claim of the organization what I'm essentially saying is that the claim of the organization of the organization seems to be correct. Seems to be correct. Mm. And this is how we go about doing this. Question what we have done. Well, the first thing that we did was just stayed the night at the alternative hypothesis. And the important thing to catch in this question was to understand that the probability will be won by three for all the categories and following that. The calculations were very simple. We just applied the formulas to find the expected values, and we found the guy's question distinct. Another very important formula was the degrees of freedom formula. You have to remember this. We found a degree of freedom to be, too, and we use two methods. The P value with her. We found that a P value was close to zero, which was less than alpha. Hence we rejected Arnold hypothesis. And even with the critical value matter, we got the same answer, right? Uh, Chi Square statistic for fell to the right of the critical value, which is in the rejection region away. So we rejected it and this was the most important. This is how you will frame your conclusion. The answer over here will be that at 5% level of significance, we have enough statistical evidence to suggest that the distribution of opinions or choices of prospective Homebuyers is not uniformly distributed, which means that the claim of the organization seems to be correct, and this is how we go about doing this question.


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NOTE: PLEASE SHOW ALL YQUR WORK IN ORDER TQ RECEIVE FULL CREDITLet A = [27 3] (a) Find A; (b) Using part (a), solve the following system of simultaneous equations by using matrices Zx +y = 5 ~Tx + 3y = 4
NOTE: PLEASE SHOW ALL YQUR WORK IN ORDER TQ RECEIVE FULL CREDIT Let A = [27 3] (a) Find A; (b) Using part (a), solve the following system of simultaneous equations by using matrices Zx +y = 5 ~Tx + 3y = 4...
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Question (15 marks) Use the Residue Theorem to evaluate the integral(22 + 1)2Clearly specify the contour you are using and provide careful discussion of all required esti- mates
Question (15 marks) Use the Residue Theorem to evaluate the integral (22 + 1)2 Clearly specify the contour you are using and provide careful discussion of all required esti- mates...

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