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Problem 61 Hard Difficulty

6.61 Final averages are typically approximately normally distributed with a mean of 72 and a standard deviation of
12.5. Your professor says that the top $8 \%$ of the class will receive an $\mathrm{A} ;$ the next $20 \%,$ a $\mathrm{B} ;$ the next $42 \%,$ a $\mathrm{C} ;$ the next $18 \%,$ a $\mathrm{D} ;$ and the bottom $12 \%,$ an $\mathrm{F}$.
a. What average must you exceed to obtain an A?
b. What average must you exceed to receive a grade better than a C?
c. What average must you obtain to pass the course? (You'll need a D or better.)

Answer

a 89.56 ; b 79.29; c 65.45

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Video Transcript

okay for this problem were given some test information or swamp some score information. Um, and we're talking about normal distribution, and we always want to show our sketch. And we know that these test scores air really distributed with a mean of 72 a standard deviation of 12.5. So, um, and there's kind of some information about how a Z B's and C's are on this distribution. So the professor says, uh, the top 8% get in a so just a layout or do a couple of different steps in this problem toppy presented in a So we really want to do is you want to find out the score that goes with the 92nd percentile toppy presented in a over here and for per peas. To think I had a little bit Ah, so you want to get better than a C? So and they said the top a present get in a the next 20% get a B, um, and then they give some more information there, too. So you're better than to see you gotta be or in a so there's a percent listen there 20% so really what you do need to know. For percentile wise, you get 70 to 72nd percentile for that one and became for the word in the last one. The last one wants us to plan ahead, get better than a D in the course. And so they lay out all the information in the talk about the top a present, and then the next 20% Ah, getting a B in the next 42% get a C and the next 18% get a D in the bottom, 12% get an f. So But Questions C says you need a D or better eso. What we didn't do is look at the last little bit of it. So if you want I d or better, we just need to get a above the 12% to give you 12% of the bottom. So, um, 12 percentile. So I gotta find the scores ago with each of these percentiles, and you could look at Table three work backwards to do that, or we use a nice apple. It and I prefer step with a lot of free APs, and we're gonna use the normal distributions here and work backwards s instead, If I'm area under the normal curve, we're gonna work backwards to find the value cause corresponding to an area. So the area is like the percentile score is the big thing. We have to know so But the apple it needs to notice, like the table information. We need to know what you mean. And standard deviation is that's like it drizzy scores and percents proportions correlate to those. So this is how it looks here, A little nicer version that what I have drawn there is let's find this red area first. So the top 8% Uh oh, that means the 92nd percentile there. So let's look and see. We get to work backwards from there. And if we know the point 92nd percentile, certain boundary value, it's gonna tell us that's 89.5. So your score to get in the in a in the class for this information is a symphony to be after above that number. So let's say 89 point six, you're in a all right, so let's go with their see years of 72nd percentile. So let's look over here. 72nd percentile don't need the change information cause it's a still the same normal distribution and working backwards to find the Valerie value that goes with the 72nd percentile. It's a 79.2, the case that you have to have a score that's greater than 79.2. All right, so that gives us our C and then to not fail. Lead the deer better. So, way down at the 12 percentile. All right, so just for the purposes of doing this. So the 72nd percentile I was like here, and that was kind of looking. Now, that's a battery value series, which really includes both, um, and then this is really gonna be all these numbers to not fail. Failing means the blue. So not feeling really means above here. So we're gonna sketch that out. Let's stop it, do the work for us. Um, and we're going to say, Hey, what country value goes with this? So we have a 0.12 and these need to score higher than a 57.3 to get a D or higher. And so there we have it. So a little bit of percentile work and a little bit of working backwards and interpreting what information is given to us