Using the standard normal curve and $z$ :
a. Find the minimum score needed to receive an A if the instructor in Example 6.11 said the top $15 \%$ were to get A's.
b. Find the 25 th percentile for IQ scores in Example $6.10 .$
c. If SAT scores are normally distributed with a mean of 500 and a standard deviation of $100,$ what score does a student need to at least be considered by a college that takes only students with scores within the top $7 \% ?$
a 85.6; b 89.2; 648
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okay for this problem were asked Teoh show our normal curve and Z score understanding to calculate a few different things. So the general procedure is going to be Ah, draw our home. See table are I'm sorry? A normal distribution and Ah, and look at the percentiles percents they want us to figure out. And that will give you use an apple. It too. Figure that out. So per day. Um, we're talking was referring back to example 6.11 and so we want to figure out the top 15%. Get a So we're looking at the top 15% of our data. So, uh, it's gonna restate that as a probability that X is greater than 0.85. So that means that's why I like to draw my diagrams out here. So just for a little throwback here because we need the information, you look back at the problem. It's said that the mean for that test was 72 and the standard deviation for that waas 13. So for considering 85 72 plus 13. Yeah. So we're looking at a above one standard deviation above here. Such of the right basically is what's looking at. You want the 10.85 percentile or the top? 15% is what's represented in that diagram. Let's go over here and use our Creeley available Apple it. I, like stop wits, have a lot of different applications, and we're kind of normal problem. Normal distributions. So our stop list and give this a little nicer diagram, and we know the meaning of standard deviation for this problem, we can see it. Um, And if you want to skip spit out Hagen positive video and see the difference Dinner deviations. Um, but we care about values to the right of a certain point. In this case, we care about values to the right of no area. Second with him, score at wording, working backwards. Okay, I think out loud. So working backwards from that, we're still using the same, uh, diagram, but reaction I can find in the area under the normal curve. We're going to work backwards to find the value that goes with that area. So really already winners, we're basically looking. If the area is the 15% of here, we're looking for the value that goes with that, so and we still have the same mean standard deviation. But I've seen just to the area calculation. So generally thinking we had to do there waas knowing that top 15% working backwards, this 150.85. So that's going to give us back Thea Value. That goes with this very point, um, and so it looks like 85.4 with our answer. We want for that. So to get that top score, you had to have at least a score of 85.4 seven, so we'll save 85.5. So that's part a two days. Do you have a table to or work from the graphing calculator and do something called Inverse Normal? You could do that there as well. So let's look at Part B is a different set of scores that refers to example 6 10 So this one was the 25th percentile. So again, the same process so we're looking at are diagrams, and we let's represent with the 25th percentile. Looks like for a que ah, 25th percentile would be kind of looking over here. What's make it green 25th percentiles here. So we're looking for us if the 25th percentile, we want to know the value that goes with that. Um, so we want to know Is Thea score? If you're the 25th percentile, it's the same thing working backwards. Now we have different. Ah, different normal distributions. So we do want a name that what we have. Look, looking back for the problem, I Q scores are centered at 100 and plus or minus. Ah, standard deviation is 16. So we could still do the same thing. Work backwards here. We need to change our mean. It's Jenner deviation to match the idea of set for I Q 116. We're gonna leave this alone. We have the area. We want the value. Um, it's gonna be a left tail area again. So, uh, let's work backwards. But we want the 25th percentile. So to get that value, I keep score. That goes with the 25th percentile. It is 89.2. Okay. And look apart. See another distribution. That's normal. But as a t scores so that s a T scores and me and 500 standard deviation 100. I want to be in a college. It takes the top 7%. Okay, so let's go over here. Top some percents. Gonna be a very slim appear. Must be a predictive February picky college. Okay, so they told us in this problem that 500 is the mean in the there s a T scores and the standard deviation is closer. Riotous 100. Yes. We want to know this four that goes with the top 7% top 7% score. So really, what I want to think about is the the 93rd percentile. So let's go over here and ah, give me the change or information because we have a different distribution of blue on our experience. So we know the mean for these s a T scores versus the A. C T scores is 500. Move and hold. That 100 is the standard deviation of scores, and we want to be the top 7% which means that you're the 93rd percentile or above the 93rd percentile. So you calculate that still gonna be left tail. So you it means you need an S a t of 6 47 0.57 Okay, so for that? I mean, the S A t not sure they have. Ah, decimal values for S a T scores. So let's call it the um 6 47 point 37 Hollywood is best wants. It's constant. 48. Okay, since you 6 47.5