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Question Let Z denote standard normal variablei. (5pts) Find P(-0.44 < Z < 2.68).ii. (6pts) Determine the value of Z which satisfies P(Z Z Zo) = 0.5285.(5pts)...

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Question Let Z denote standard normal variablei. (5pts) Find P(-0.44 < Z < 2.68).ii. (6pts) Determine the value of Z which satisfies P(Z Z Zo) = 0.5285.(5pts) Suppose that 10 of the probability for certain distribution that is N(0,o?) is below 60 and that 5% is above 90_ What are the values of / and 0?

Question Let Z denote standard normal variable i. (5pts) Find P(-0.44 < Z < 2.68). ii. (6pts) Determine the value of Z which satisfies P(Z Z Zo) = 0.5285. (5pts) Suppose that 10 of the probability for certain distribution that is N(0,o?) is below 60 and that 5% is above 90_ What are the values of / and 0?



Answers

Let $Z$ be the standard normal variable. Find the values of $z$ if $z$ satisfies: a. $P(Z<z)=.8907$ b. $P(Z<z)=.2090$

Okay. Welcome to enumerate Today. We are given the problems for standard Normal believers. Standard norman said this problems normal one. Because standard Normal is having mean at zero ingredients set up. What? So this is how a standard graph would look like. And I will first explain you quickly what is the relation of the stable and this standard normal distribution. Imagine let us take any any point. Okay, So yes, this one we can take 0.1401 So if I write 0.1401 what would that mean? That would mean probability that the normal variable is greater than some particular value. Did not. Okay. And what is that? Did not value That can be obtained based on the position of the number. Okay, So if you see first is here 1.0 and then we have 0.8. So 1.0 and zero point funny 0.8 Thanks. 0.0. So the total of this too, that is 1.8 should be the value. So where is 1.0? If this is one somewhere over here we can think now. What is the implication of it? It says then more than this value. Okay, something roughly this is the probability upset to be more than this would be 0.1401 This is the understanding. Okay, so you might take a screenshot of the current picture and keep it handy for understanding and future as of now the screen? Well, yeah, we are given as the first problem. What what would be the probability? Uh huh You don't not is equals to 0.5. So you just learned that you have to find this value in this table saying well we have to find this value in the stable if you see the first value itself, is that value That means what it should be. Uh huh this value and that's right right. The Rwanda quantum is this value. And so this should be the viability of that greater than that not. It would be probability zero greater than 0.0 That means they have not is equals to 0.0 It is this now think the zero when Z is equal to zero, that is our point of symmetry. So of course this side will be 0.5 as well as this side will be 0.5 because the total area is plus of this which is one. So now you can understand if you know this side this one side we can also find the other side from that given probabilities. They will go to that for our next problem and that's why I introduced it. So Lucas. Yes, so for this part we are given probability said less than did not such that. It is 0.8 643 No promote table. We can get the values when it is greater than not less than so. How can we altered thing? What just I showed you if there is any normal distribution like this? Okay. And says that notice this. So if this area is that greater than that? Not What is this idea? This will be zero less than did not. Right. So therefore probabilities of the total this area and this area will be one correct probability of less than said not plus zed greater than that. Not that will always yield up to one. Therefore if we if we have this and if I send this to this side then priority of the greater than sign we will get. Therefore, probability of Z greater than said not he's equals two one minus zero point 8643 which is 0.1 357 probability that greater than that. Not That means what we have to find. Where is one 10.1257 is Ankara. Okay, so let us try to find where is the 0.1357 If we look here in corresponding to 1.1, you can see that 1357 values are correct. That means what it will be probability zed greater than 1.1 and in second this month place it is zero. So our did not here. He is one point why for the next again we'll have the problem statement written over here. It is probability minus that. Not less than that. Less than that not. Is equal to 0.90 Now we will you quickly get another good help for our reference so that we can visualize it better. Like kink of the stew being does it not? Okay, let us Britain the two is ours it not. So if this is our miners it not and if this is our plus they've got we are basically interested the area between that. Right? So let me highlight that with some blue tops. Oh we are interested in the same. Now we know the total area is one. If the total area is one then what about that other areas? Okay, I can highlight them with black dots and red box. Okay now you see this black dots and red dots. Those who are having seen areas. So if I want to have the total area okay that is equivalent to saying one minus black about 100 bucks these two. And they're basically saying so I can say two times the red dots red box area clear. So with this understanding we will now proceed Okay, so let us change no. If I'm having this I can right over here one minus this less than area. So probability said less than minus did not minus the probability that great weapons that that not eating. They said greater than that not and which is given to the 0.9. Now we know these two areas are same so we can replace by the same value. This we're not taking this because this table does not give us the less than values. It gives the better than value. So very basic. So we will have minus I'm sorry. We will have 0.1 is equals to two times probability zed greater than that. Not that means we will have 0.5 is equal to probability said greater than said. Not now. How will we read them? Uh Said not from the table. We have to search for 0.5 in the table. So let us look over here and we will find 0.5 lying between these two. 0.5 is less than this value under the second. The smell being four and between second and smell being fight. So what can we do? We can write this to be probability of said greater than 1.6. We are confirmed till there and then it's between 0.4 and 0.5. So what's the mean of 0.4 and 0.5? So it is zero point to the point 05 20. P 0.9 So divided by two and you get 0.45 Right? So the first decimal places six here. Right? So we will get the 2nd and 3rd decimal place to be this. It is a constant probability that later than zip. Not. So that notice this. So what's for our third problem? Did not These equals to 1.645 Now we will move to the next problem which is exact same like this. So you may not have to worry at all. So I believe quickly you can get the normal tips. Yeah, I think this much is enough. So our problem says it is probability minus said not less than zero, less than plus it not is equal to 0.99 Now again we can think that 0.99 So this can be the bracket where the entire area 0.99 is happening. So we have this area is being left out so that we can write us one minus probability zed less than did not miners did not minus probability dead greater than said not. Which is basically the same because the areas are the same over there. So that means one minus two times probability zed greater than that not is equal to 0.99 That means we will have probability said greater than said not is because to one minus 0.99 divided by two, which is zero point double 05 Right? So we have to find this probability from the table when we are having greater than sign and that left hand side, right. So let us look first look 1005 See here again. Same case 0.5 lies between these two. These two probably. So that means 2.5 and then seven and eight. We have to get the midpoint of it. That means we can write probability zero greater than said not is equals two. Probabilities said greater than 2.5. Were confirmed to him there. Now we don't know if it is 0.7 or zero point it well, it's none of them. It's basically the midpoint of both of them. So how much would that be? It will be 0.0. Uh No, it would be 0.15 Uh huh. Divided by two. So it would be 0.75 Like half of it. Just like how we got last time. So we have 75 That means in this problem that not is 2.575 I hope this was sufficient for your understanding. Let me know if you have any questions by

To find easy value. We can use the inverse normal function because you want to obtain the raw value. Given a probability or per cental, the inverse normal function utilizes the area and autograph from negative infinity to a Z value. So we want to make sure to find the area to the left off the the value that we are interested in. Given his ease at the 75th percentile, we can translate that to an area off 0.75 under the normal distribution cough. We can then use inverse non function on a graphic calculator to obtain the result off 0.6745 Given that Z's at the 80th percentile, we can translate that to an area of 0.8. Under the normal distribution cough. We can then use the inverse non function to obtain the result off 0.8416 Lastly, given Aziz at the 92nd percentile, we can translate that to an area of 0.92 Under the normal distribution cough. We can then use the inverse non function to obtain the result off 1.4051

All right. And this problem we wish to use a normal distribution to be able to find the following the scores. A through D. This question is challenging understanding of how to match the score in a normal distribution to the associated area under the curve. To solve its first review relevant material for normal distributions before proceeding so as detail remember that's the scores on the probabilities. So an example the probably the greater than zero equals peanut implies that the area and purple peanut is the area to the right of arsenal scored. As an example, the probabilities is greater than 0.5 because the area on either side of these normal distribution is equal symmetric or one half. To solve this problem, we need to rely on two properties normal curves. First, the symmetry of the normal curve as well as the fact that the total area under the normal curve is one. So with this logic, we only need to solve proceeding through we see that A through D all can make use of symmetry us all. So first for part A we can write this as quickly as the probability less than that gives, you know, is one minus 10.95. Over to that is this 0.25. Area each Tales the corresponding Xena plus or minus 1.96. We apply the exact same principle to solve B through D. So it be the probability of the tail is one minus point number two equals 20.5 giving zero equals plus or minus 2.33 and see the area in the tales 0.170 is plus or minus 0.96 and finally, indie the area, and the tails is 0.135, giving zero plus or minus 3.0.

Uh huh. In this problem we wish to use the normal distribution table to find the following Z scores given for a through this problem is challenging our understanding of how to understand the relationship between the Z score in the area under a normal curve or C under normal distribution to solve. Before we proceed to find these Z scores directly, we're going to relate or rather review relative information for normal distributions. So as the people have the scores on the probabilities as we weren't as an example that probably these great additions and you know it's peanut or peanut is the area in purple and not as much as black as an example is the standard normal distribution has a mean zero, probably these great and 0.5 or half the area under the normal. So to solve this problem we need to remember the symmetry of the normal case. That is probably the lessons do not is probably greater than negative Z not similarly, we have to remember that the total area is one. Well, this is the reason we only need to solve so the probability lessons, you know, equals 0.9 gives. It probably is is less than negative United's 0.1. Thus Z 91.28 probably easy lessons, you know, it's .5 is equivalent to the is he not equal zero. This is from the identity we identified above coincidentally see probably the greater than zero equals 00.1 is probably the lessons are not equal 0.9 this time again, Xena is 1.28 because A and E are equivalent Fergie. The probably the greater than 0.9 is now negative 1.28 by symmetry. Finally, and I've heard either probably easy Between negative 1.24 and 19.8 is probably the lessons do not mind is probably less than negative 1.2, So I went to the scene gives 1.33.


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