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The results of a common standardized test used in psychologyresearch is designed so that the population mean is 100 and thestandard deviation is 50. A subject earns...

Question

The results of a common standardized test used in psychologyresearch is designed so that the population mean is 100 and thestandard deviation is 50. A subject earns a score of 90. What isthe z-score for this raw score?z-score =

The results of a common standardized test used in psychology research is designed so that the population mean is 100 and the standard deviation is 50. A subject earns a score of 90. What is the z-score for this raw score? z-score =



Answers

The test grades for a standardized test are normally distributed with a mean of $50 .$ A grade
of 60 represents a z-score of $1.25 .$ What is the standard deviation of the data?

POTUS is not is Sig Medical Toe 8.6, which is overtly so. I don't think hypothesis that you and Baby Sigma not equal to 8.6 as we have used not equal to it is a student test with the very youthful forgiven to be 0.5 Since this is spoken to test the value, Alphaville be split into two parts. The robin 0 to 5 will be the left side and shaded region. In this bigger and 0.25 will be the right side shaded region. Now the grease off freedom do you have equal toe end minus one? There is the size off the simple which is given to be quick. So degrees of freedom equal to minus one that this let me We have to find critical values for degrees of freedom. Equal Do Larrin for right side critical value village see under a fight where 20.25 and degrees of freedom 11. So the right side critical value is 21.920 as seen in the table for the Chi Square distribution. But let's say critical venue BBC and the Alfa But now I'll five in B minus this left side of 0.25 So by minus 0.25 which is equal to 0.975 So we will see under by quarter 0.975 and degrees of freedom 11. We get the critical value to be 3.816 A scene in the table for the chi square distribution. Really? Now finds the best way the test. Will you guys credit equal to N minus one? My tip line by s square There s is the signs off is the standard division off the simple Do you read? A domestic muscular vested minds the standard deviation off the population. So this will be 11 my people I s is given to be 903 11 My reply by 9.3 square divided by 8.6 square Vigilant and affiliated with Gilda Value 8.864 Now we have to see where find a 64 in this figure, there really be. Look today clearly it may be something hurt when 0.86 foot. This uncharted region is our north critical region that this the test value falls in the North critical region, so do not reject the night high POTUS is that this? There is not enough evidence to reject the claim that the standard definition off the ages is 8.6.

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 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.

The key for understanding this question is understanding what a percentile ISS. So the 90th percentile actually means that the area to the left up here percentile value is equal to 0.90 So what we're really being asked is for the set score that corresponds with the area of 0.90 So to do that, we're going to look at our standard normal table, and we're going to try to find the number on this table that is closest to 0.9. So we start up here we look, we see all these numbers are very, very small, so we're gonna scroll down, keep going, keep going. And we look for 0.90 So we see that the closest values on this table to 0.90 is 0.89973 and 3.90147 And this value right here corresponds with us. That score of 1.28 and this one corresponds with the set score of 1.29 So are set scores gonna be somewhere between 1.28 and 1.29 and a natural guess would be 1.285 for the 2nd 1 we're going to have the area. So the left of ours, that score being equal to 0.95 And again we're looking for these That score that corresponds with that area. We're gonna go back towards that score going, keep scrolling, and we're gonna look for a 0.95 and we see that the two values that are closest to 0.95 are these who values right here. So this 1st 1 has is that score of 1.64 and this one has is that score of 1.65 So are are is that score is gonna be somewhere between 1.64 and 1.65 So a good guess would be 1.645 for the final one. The area to the left of the sub score that we're looking for is going to be equal to 0.99 And once again we're looking for he said, score that corresponds with that value. So we go back to his own table, we keep scrolling down and we look for 0.99 and we see that the two values that are closest to 0.99 are right here. So the one on the left corresponds with a Senate score of 2.32 and the one on the rate corresponds with the set score of two point 33 So somewhere between 2.32 and 2.33 and a good guess would be 2.3 to 5.


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