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Numerade Educator



Problem 57 Hard Difficulty

According to ACT, results from the 2008 ACT testing found that students had a mean reading score of 21.4 with a standard deviation of $6.0 .$ Assuming that the scores are normally distributed,
a. find the probability that a randomly selected student had a reading ACT score less than $20 .$
b. find the probability that a randomly selected student had a reading ACT score between 18 and 24
c. find the probability that a randomly selected student had a reading ACT score greater than 30
d. find the value of the 75th percentile for ACT scores.


a .4078; b .3821; c .0759; d .25.42


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Video Transcript

okay, if this problem were given some information about easy T scores and they want us to find some, ah, probability for certain distributions of that 2000 c. T. Students. So the main score we're told is the mean score that years 21.4 and the standard deviation is six point now. Okay, So we're gonna get our plan here is to use a normal distributions and work backwards. Well, he's an apple. It's sort of a nicer sketch here in a second. But I still like to hand sketch out what's going on. So there's a 21.4 and then one standard deviation to the right would be 27.4. But I will let the apple into the picture for a nicer picture for us. But again, Ah, steps. I like to draw the diagram. Second step. I like to write what I want to find Ah, using probability notation. So we want to find the probability that the A C T score is less than 20. Given these criteria and parameters that we have, it is less than 20 cents. There's gonna be a little bit too here in less than looking to the left. Okay, so my sketch for that one going to go over here and get my free apple, It's he, uh, vacillates online. You could also use cable three, um, my like using step lit because I can't see what's going on. And it's doing the same thing. Like reading me the Z score values and getting the probabilities for you. So, um, we know that we have a mean of 21.4 and a standard deviation of six. So here's our distributional. Nice cleaner value what I showed over here. So I recommend you write that out on your paper for you. Teacher likes to see the, um all the standard deviations gives us a general idea. So I'm gonna change this from between two values. Toe the left since we're looking to the left for less than 20. Okay, so we're gonna look four. What? The probability is looking to the left of 20 here, So let's see what we get. So that says that's a 41. About 41% probability. More precisely. That's about a 410.407 each. That will get a score that's less than 20 So there's part a party, kind of the same thing. We're gonna middle score. So Part B asks us between 18 to 24 so that the probability of the value of being between 18 and 24 just to get an idea here, we're gonna look, uh, was 20 so 18 a little bit over there and 24 is 2124 going to be in the middle. So when I sketch up green, um, gonna go off to the side here and clean that up to make a look a little bit nicer. So let's go over here, go back to our distribution. Um, and we're going to find a different. So you here, it's in this case, we're going to find numbers that are between two values. So this gives us a nicer version of what we have drawn in green. Um, so this case, the left is 18. A little bit to the left of the mean and the right boundaries. 24 a little to the right of the mean. So that tells us our area represented in the picture. There is 0.321 Okay, so about a 38% chance it's between those two values. I said No, we've answered the second thing, that probability that's between those two. And thirdly, they want to ask us, uh, what if we had a CT scored rate of them 30 Nice high a C T score. So it's, uh, first thing showing the direction to be above 30. That's quite a bit above are mean, So let's see what that ISS. So there's 2127.4 30 is gonna be over here to the right, so we're looking for distribution to the right. Let's go back to my apple. It and we want to look to the right of the value of 30 for the entire mean and standard deviation are still the same. And that telling us that that is about 8% chance more precisely 0.75 mine that we score, Um, that high of a C T. If the mean is 21.4, that makes sense. And finally, there's a party for this problem. It's gonna give us a little bit more algebra work to dio, but not too much more. Um, actually, it's not too much. Algebra is the 75th percentile. Okay, so what you want to find here is ah, what is the value for the C t for the 75th percentile? So and this guy's gonna work backwards. So if I was using the graphing calculator, I call that inverse norm, but really, what we're finding here, I'm not going to sketch a doctor's gonna get too sloppy, but we're saying the 75th percentile, which should be around here if we sketched all that. And what is the value that goes with that? Some straight. This is a question mark, and I will answer it. So going over to her applet and we're changes working backwards here. So instead of calculating the area under the normal curve, now we're gonna find the value corresponding to an area. So again, it's the same distributions we have drawn is a 21.4 and a six. Um, but now we're looking for the still gonna be a left tail. Okay, We're looking to the left of Saudi fifth percentile. So and then we're gonna put some fifth percentile for reading off of a table, which you do is find 0.75 on the table and then find the value that goes with that score. So now we're going to let the apple do some work for us. So that says, Oh, well, that would have on a C T value of 25.45. Yes, it will answer the final question. That means we have an easy T value of 25 1 45 for these 75th percentile. All right. And if you worked off a table, you work backwards and get dizzy. Scored a little bit of algebra show that.