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Question 1 (5 points) ISAT Math scores are Normal with a mean: /l; of 500 and standard deviation 0,of 91 If one student is selected at random: what is the probabli...

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Question 1 (5 points) ISAT Math scores are Normal with a mean: /l; of 500 and standard deviation 0,of 91 If one student is selected at random: what is the probablity their SAT Math scorel is above 650Your Answer:

Question 1 (5 points) ISAT Math scores are Normal with a mean: /l; of 500 and standard deviation 0,of 91 If one student is selected at random: what is the probablity their SAT Math scorel is above 650 Your Answer:



Answers

The national average SAT score (for Verbal and Math) is 1028 . If we assume a normal distribution with $\sigma=92,$ what is the 90th percentile score? What is the probability that a randomly selected score exceeds $1200 ?$

For this question has been given by sample size and strolled. This means our degrees of freedom. This 11 and we're supposed to calculate on 99% confidence interval for the population Syrians. Since we're creating a 99% confidence interval, Alfa is 0.1 now. Since we have been given the sample, we can use this formula to calculate the sample standard deviation or you could use is off versus as many to have or your calculator. If you calculate the sample standard deviation for our sample, it comes to 124.9. Now we can. The next thing we do is fine. The two critical values using G table, the critical values that guy's for outside. And that's where one minus else. All right, I square helpful by doing our case. A spice Where. 0.5 with degrees of freedom. 11. If you take the G table, this value comes to 26.757 thanks, square one minus, also by two. In our case, this guy square 0.995 degrees of Freedom 11. Checking the G table. This value constitutes 1.603 Now that we found the critical values, we can move on to calculate the confidence interval for our population variance to do so. We used this formula here. This is the expression for the confidence interval for the population variance something in the values we have. We get the following expression simplifying this expression further we get the lower limit off our confidence interval as 6411 and the upper limit as 65,898. So this here is how you calculate the confidence interval for a population very

So we're assuming we have a normal distribution and we are given that they mean it's supposed to be 498 with a standard deviation of 100. So this would be a score of about 5 98 right here, and this would be a score of about 3 98 there. And we want to find what's the likelihood of having a score for this particular type of? I think it was an exam between 405 100. We're supposed to assume a normal population again, so we would convert both these into Z values. We take the score minus the mean yeah, divided by the standard deviation and the score minus the main, divided by the standard deviation. And that will give us our Z values and that differences negative. 98 negative night. Excuse me. Negative 98 divided by 100. So that's going to be a Z score of negative 1000.98 and this is the score is only two divided by 100. So 1000.2 and picture wise that 500 would be about here. So it's again very close to a zero Z value and the 400 is going to be about here. So we're finding that area, really? And I'm gonna use my calculator. I could use a table as well, but I'm going to use my normal CTF. And when I hit on my calculator second and distribution normal CDF, I'm going to use this as my low number negative 0.98 My upper number is going to be 0.2 and I'm leaving the mean for the standard normal distribution at zero and standard deviation of one. And when I do that, I find out that that probability is 0.34444 So I don't recall if it said percent. If it said percent, we'd get at about 34% chance of that happening or the probability is about 340.34 Then we had to look at Part B, and it said if you had a total of 300 people, how many of those would you find would have a score that's over 700? So let's find the probability of getting a score over 701st, and we can see that it's not real likely. Here's at 600 roughly and 700 would be about here. This would be at 6 98 so I can see that that probability I'll kind of color in blue is not going to be very big, which means you're not going to have a huge number here. So let's convert this into a Z score again and our Z score. What we got minus what we're assuming and think that was 4 98. It wasn't it. Yep, 4 98 and then divided by 100 and so 700 minus the 4 98. That difference is two oh two. And then we're dividing that by 100. So it's going to be two point Oh two. So that's our Z value, or about two standard deviations higher than the mean. And again I'm going to use my normal CDF here. Second and distribution go to normal CDF, and I'm going to use this as the low number 2.2 You could also look this up in a Z table and then for the upper value. I'm just going to put in 1000 and leave the mean and standard deviation at zero and one. And when I type that in. I find out that that percent is 0.217 approximately. And so now I'm going to multiply that by 300 to find out what 2.17% is of 300 times 300 and I find out that that comes out to be 6.5, so there would be probably six or seven, six or seven people I would anticipate to have a score that is greater than, uh, 700.

So for this problem, we're using theme normal distribution from problem 13 which means that our population mean is for 88 the standard deviation is 1 14. So for part A, we want to see what percent of people, um ah, score less than a 600. So we're going to need the Z score for 600 to get that we use this formula. So we have 600 minus the population mean, which is 4 88 divided by our standard deviation, which is the 1 14. And you saw this and you should get a Z score of 0.982 And now we can use to see score to get the probability that, um, somebody scores less than a 600 because the corresponding value on a Z table for a Z score gives us theory a to the left of it, which represents the probability that somebody will score less than whatever scored this represents. So you look up 0.982 in the Z table, we get a probability of 0.837 and we wanted to. I represent this is a percent, so we just multiply it by 100 and that gives us 83 0.7%. And this is our answer to part A. Now, for part B, we need a new Z score. This time we're looking at 500 so we can just replace 600 with a 500 for X and then re calculate this Z score on. We should get a new Z score of 0.105 Now, Like I said, before we look this value up on our table, this gives us the area to the left. Since we're looking for the right and this represents a probability, we can flip this and get the area to the right by just doing one minus this corresponding probability. So we have one minus, then look up, 0.105 on RZ table. You should get a probability of 0.5418 Subtract this and we find that the probability is going to be 0.458 And now we're asked to look at it. How many people we would expect to score over a 500 if our sample size in a sample of 1000, so we can just multiply our probability by 1000 and this gives us 458.2. Sorry, there was a two on here. A swell 4.4582 That's where I got the two from. Yeah, And since we are it just said to the nearest person or about how many people. So our answer is about 458 people.

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.


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