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Doiy random sample ot I onservalols2.81(a) Frid 92e3 coniidence intenval II <(olmd90% contidence inic val for / 0 <(C! Fing 95* confdence Intnvall f0l / Su &...

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Doiy random sample ot I onservalols2.81(a) Frid 92e3 coniidence intenval II <(olmd90% contidence inic val for / 0 <(C! Fing 95* confdence Intnvall f0l / Su <

Doiy random sample ot I onservalols 2.81 (a) Frid 92e3 coniidence intenval II < (olmd 90% contidence inic val for / 0 < (C! Fing 95* confdence Intnvall f0l / Su <



Answers

Use Figure 12.2 "Cumulative Normal Probability" to find the probability indicated. a. $P(-1.99<Z<-1.03)$ b. $P(-0.87<Z<1.58)$ c. $P(0.33<Z<0.96)$

Okay for these problems. Were using the Z. Table to look up. Uh Probabilities. Alright, let her a wants to know the probability that you get a Z value less than Drative. Okay so all you do is you take the negative 1.17 and look up look it up on the table negative 1.17 gives us an answer of 12.10%. And that is all this is asking. All right. The only consideration is make sure it says less. Then if it says greater than we need to subtract it from one, We need to attract the probability it gives us from one. Okay, part B is asking about less than negative 0.05. So on our table, 0.05 has a 48 0.1 Percent chance. Let her see less than .66. 3.66 Is a 74.5. chance. All right now D. E. And f. You notice these are a little different. This is the probability that Z is greater than this time. Greater than negative 2.43 The only tricky part about this is whatever answer we get has to be subtracted from one. So if you look up negative 2.43 in your table, you'll see in a number of .0075 people want. All right. So to get your final answer, you must track. So it's 99.25%. Mhm. Alright. Similarly, for E N F E is probability getting az greater than negative one. So, greater than so, 1-. Then look up -1 on the table and you'll see .5178. Excuse Me? five points .1587. That's the number one. So that gives us 84 13%. All right. And then finally, letter F f is Z values greater than 2.19 so one minus. So look up 2.19 And you should see 98 Points 5 7.9857, Which comes out to 1.43%. Mhm.

For this question of a sample size of 24 and they have to find key, you know, degrees of freedom, Missy. Quote and manage when Israelis 20 for minus one. And this is 23 and we had elevated C distribution. And so the truth, Our area here, this one going to use this diagram to solve the question 50 here the total. Okay. From the question, I have negative 2.69 See, value. And that is that lies to the left of you. Now, for the distribution of total area to the right of zero is 0.5. The total area to the left off zero is 0.5. And the question gives us a value off 0.965 between two t values and because their value is more than 50% partof realized to the left and impartial eyes to the right. So this down would be my key. Well, you look up the value off. Praise your 69 from the tea distribution table. They're from the table. Cooper is your 69 inside are far level off 0.25 and interviews off them off 23. So I'm going to find a negative off this value that gets you negative. 2.69 Therefore, 0.25 is the area to the left off negative 2.69 and then are faced the area to the right off key. The total area here, this one. So when I add the areas up, I should go one. And this case, I'm going to do a letter. Bits off budget right here. So how far is it? Quote one minus zoo points 99 And the value is zoo point as you want. So well, you look up this value from the sea distribution table, the values 2.5 years you and so this value is the value off key. Now let's get to the next question. Parts be for parts B. The total every opportunity to value is 0.95 and I have a T value of 2.807 The area to the right of you is your 0.5. The area to the left of zero is 0.5. I'm going to look up a value from let's see distribution table from the distribution table offer. Level off. Super is your five with degrees of freedom off 23 gives me two points easier with seven. Therefore the area to the right of two points easier. 70.5 The question tells us that the area between point is your seven and then another value, which is key is 0.95 I noticed that when we added two values the Allessandro print for a free just way they lie to the right of you. So you're looking for the total area to the right off key and that would be our Alfa. So 0.95 plus 0.5 would be quote Alfa That issue coaches your prints when were to look at this value from the tea distribution table and that is equal to 1.3 a 19 Therefore, this value is the value off key to the next question at sea. I have zero here and then I have a key to the right key to the left. I mean, negative key to the left of you read to the right off zero is 0.5, and then the area to the left off zero is your brain's five. The question tells me that area between key and Negative Key is 0.9. Therefore, our farm to the left in the offer to the right in this case. So when I add the two office and zero point and zero points now, I should get one, have to off apply 0.98 is equal to one, and then I do a letter bits off algebra here and that issue quote 0.5 So when you look up this value from the city suspicion table, I have a value off 1.714

So in this question we are given a normally distributed random variable X week, mean 112 and Standard Deviation 15. And whereas to find the probabilities of X, so here to find the properties of X, we're basically going to transform X into our standard normal variable, see using the formula x minus mu over sigma. So we're going to apply that to each of the values of X listed here. So for a we basically have instead of X between 100 125 we have Z between -170 and point 87. So if we look at our table that's basically probably see less than .87 minus. The probability is the less than minus pointed, which will give us The property in this interval which is .807, 8 -2119, Which is .5959 B. We have the probability we're going to do the conversion right here of Z between -1.4 and minus .33, we're going to get .37 For 7 -1808. It's going to give us .2899. See we have the probability here, we're going to convert that to our normal variables, e so we have .4 and three. That's going to give us .9993 -16 554 from the table, which will give us .34 394 parts. See, so we have partly B and C.

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


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