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Problem 3 (ZLpoints Suppose the scores of a sample of your employees on mechanicul aptitude quiz have mound-shaped and symmetric distribution with mean X = 75 poin...

Question

Problem 3 (ZLpoints Suppose the scores of a sample of your employees on mechanicul aptitude quiz have mound-shaped and symmetric distribution with mean X = 75 points and standard deviation points. Illustrate your answers with graph. 31 Caleulate the percent of scores that are below 69. 3b Calculute the percent Of scores that are between 69 and 84. 39 Caleulate the S4th percentile of the data_

Problem 3 (ZLpoints Suppose the scores of a sample of your employees on mechanicul aptitude quiz have mound-shaped and symmetric distribution with mean X = 75 points and standard deviation points. Illustrate your answers with graph. 31 Caleulate the percent of scores that are below 69. 3b Calculute the percent Of scores that are between 69 and 84. 39 Caleulate the S4th percentile of the data_



Answers

Assume a normal distribution. The mean score on a national aptitude test is 498 with a standard deviation of 100 points. (a) What percentage of the population has scores between 400 and $500 ?$ (b) If we sample 300 test-takers at random, about how many should have scores above $700 ?$

So you Well, uh, you see that, huh? But, you know, we're national dissed scores. Um, the meanies, uh, the mean off reading us you mute or my 88 on the start. A standard deviation. She's written this 100 uh, scores. Yes, she s So we want to know. Uh, what is it? Probability off tests being a 405 over. But we have our best between for hundreds, but there's discrediting 400 five hundreds. So even though you don't mean that Sandra deviation, um, deviation is is what is it? Probably deal De cemento happens s so we can use the normal distribution. This were You're sits, It's ah for our samples. Behaves model is probably the very well. So the formula for their storm alleviation day pdf the probably be That's a distribution or normal. This delusion is like these men Demon six. These, uh she's Sigma. Stop. No deviation then. That is your mule. I mean, I'm a somebody vision again. Uh, so, uh, well, this is Deputy. Yes. When you are next. So there are radio. Some event to happen is you go to the inter go off the pdf between Well, they between J Range describing that event. So a number 405 5 100 is, But we want to describe. So if you use this thunder, they normal this tuition for that event with with our bodies for Sigma the Southern aviation on the meat, he's gonna be. So you go to these. Thank you, Will. Oh, I'm gonna write this exponential toe. He's here x with expert. I mean e too, that that exploded off. It's something all the meanies for 98 x for my its word. Bye bye. Two tens, 100 squared. And so? Well, they the problem is being given by these should not waste your time trying Thio. So be some politically, It's very hard. Uh, so, uh, just that for India a calculator and you get that number is simple to see you 0.3 for for So while this is, uh, probably do with one not event you got the one the reason being that up this whole time. So to put it in terms of our percentage of how likely that is to happen, we need to multiply this number they want. So the probably deal with that event is Ah, you're a 0.344 downs. 100. Um, but that much percent. So it's gonna be on Ah, about 3 34 point for. So the other question is, um, if you have a sample, some pool seven No, off free 100 people. These many people, so wasn't how many? With respect. You have a score. I should have a score Beer down. Sure scored bigger than discord over. So, uh, also fun thing with the first thing. First you mentioned the probably Oh, you're a scorpion, as the not score has to be about 700. So say that we got that body was equal to the number. Uh oh, you did that. It is you. So while the lobby times the number off the number of the sample he's gonna give you so you them's the number of the sample should give you about how many off those guys of those three hundreds That the sample should be a vote over 700. That is gonna be a number exceeding so uh huh. So for not over for I'm sorry. What is that? What is that? What is that? You mysterious you. We can do this interval so that you is gonna be dangerous girl from 700 Because that's the lowest discord and you won. And then well, you can take it up to infinity. Yeah, if you did she work. But like, if you have problems party plug in any anything numberless for the murders of computation are not trying, Uh, perhaps did. This should be good. His number should be for any cult leader. But your theory you're alone. Score sub team, Petey, even that is impossible. I mean, a score. Huzzah! Uh, fine. Eight. Does he remember, right? I mean, there's you can It's a reasonable to get a score on us. Taste off. Five thousands. Race that. So But you're like, this is the moral toll to make that model consistent, playing a large number, You know, you never the number. He's, uh he's seen saying with respect today reality of the problem. So we're again plugging in are mean or a standard deviation or those his course. Ana, this is the girl that you I want to do. What do you want to plug into your leader on? Uh, well, you amplify that and you find out that is, That number is about point. Oh, 21 Sort of the numbers of individuals that got about that score should be about for you. Oh, 21 thumbs. 300. So these number gives you 6.5. Also, uh, if you're counting people saying that 0.5 people needed is not an acceptable answer. You're gonna say that half a person did that. That doesn't make a reasonable sense. So you could take a while about six or seven. Six or seven people. So you're pushing. Made you have? Ah. When you get a number like these that is not need to hear Approximate begin the closest three into your values. But your answer will take, yeah.

For this question, we're told that scores for an exam are normally distributed and have a mean score of 525 and a standard deviation of 80. For Part A were asked what percentage of of testers score less than 350 on the exam. We're looking for the probability that X is less than 350 and to convert his head scores were using that is equal to X minus mu over Sigma. So this is equal to the probability that said is less than 350 minus 525 over 80 just equal to the probability that said it is less than minus 2.188 equals 0.144 So the probability of scoring less than 350 is 0.1 for four or, in other words, 0.144 of the students to take this exam score less than 350 for Part B, whereas what score is needed to make the top 12%. So we're looking for a score K, such that scoring higher than it is equal to 12% or a 0.12 probability of scoring higher than K is equal to 0.12 Another way to state this is to say the probability of Zed being less than K minus 525 over 80 is equal to 0.12 So you can look in the standard normal table or use a calculator or software to find the said value that has a cumulative area of 0.12 And that gives us value a zed score of 1.175 So we can say that K minus 525 over 80 is equal to 1.175 and therefore K is equal to 619 So you must score higher than 619 in order to be in the top. 12% of testers report see, were asked what the inter quartile range is. So the inter courthouse range is the range from the first quartile to the third, so the range is the difference of these court tiles. I remember the probability that of scoring less than the first quartile is equal to 0.25 So we can say that the probability of Zed being less than Q one minus 525 over 80 is equity 0.25 So we have Q one minus 525 over 80 is equal to 1.175 So this 1.175 is this ed score that has a cumulative area of 0.25 So therefore, yes is equal to you. This which were showing in this equation here. So then isolating Q one we get Q one is equal to 471 0.8 and then we can do the same thing for the beard quartile. But the probability that said is less than here three minus 525 over 80 is equal to 0.75 which tells us that Q three minus 525 over 80 is equal to zero point 67 four and then isolating que gives us 578 0.92 So those were the bounds on our intercourse. How range so to find the size of the range, we just subtract those two values where we subtract Q one from Q three and we get 107 0.84 So this is the answer now for Part D. Were asked, What grade do you need so that only one out of 500 score above it? The one out of 500 is equal to 0.2 So we want to find the probability of scoring greater than a value K being only equal to 0.2 So what value of K such that the probability of scoring higher than K is equal to 0.2 So we can say the probability of Zed being less than K minus 525 over 80 is equal to 0.2 This should be three zeros in here. Well, sorry. No, it's it's 200.0 to Okay, we're good to go. So we can say that K minus 525 over 80 is equal to you. 2.878 And that gives us K is equal to you. 755.2. So you must score higher than 755.2, such that only one out of 500 score that high

We'll start by drawing a picture for a. The important thing to know about the 80th percentile is that it is the value such but 80% of values are less than it. So in other words, the area to the left of songs that score is equal to 0.8. So in order to find this, we're going to use our Sandra normal table and we're going to look for the area that corresponds to 0.8. So we started the top. We see that these values are all way too small. So we keep going, Keep going. We scroll down here and we look for 0.8. So the closest values on here 20.8 are these two values right here. So the one on the left here corresponds with us. That score of 0.84 and this one corresponds with us. That's where 0.85 So we know where answer is gonna be somewhere between 0.84 and 0.85 So a good guess would be 0.845 And for B the middle 75%. It's going to be Sarah 0.75 in the middle and there's gonna be two zone scores that bound to this area. We can calculate the area in the tales, so we know that the total area needs to add upto one, which tells us that in either tail we're gonna have 12.5% or 0.1 to 5. So we're gonna find thes n score on our standard normal table. There corresponds with an area of 0.1 to 5. So we go back to our normal table, you scroll up and we're looking for 0.1 to 5. So we find that our answer is going to be somewhere between these two values. These are the closest to 0.125 on this table on this one on the left. Here corresponds with negative 2.24 and the one on the right corresponds with negative 2.25 So this value is somewhere between negative two point 24 and negative 2.25 So a good guess is going to be negative 2.245 for the value on the left and then this value on the right. We know that the normal distribution is symmetric. So this value was simply going to be at the positive version of this number. So is 2.245 on the right. So these are the two sat scores, such that the middle 75% is contained between them.

So in number 61 we're gonna find the 30th percentile. So since that's the bottom 30% it's one minus e to the negative. 0.75 X equals 35th percentile with his 0.3. So if we subtract one from both sides, we get negativity to the negative 0.75 X equals negative 0.7. Let me go multiplied by negative one to get rid of the negatives Should be an X there equals 0.7. So now we're gonna take the natural log the Ln of both sides. That leaves me with negative 0.75 X equals the natural log of 0.7. If we divide by negative 0.75 we have X equals the natural log of 0.7 a ver negative 0.75 And if you saw that, you get 0.47 six. So that would represent the 30th percentile


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