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The article "Determination of the MTF of Positive…

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Problem 152 Hard Difficulty

The article "Characterization of Room Temperature Damping in Aluminum-Indium Alloys" (Metallurgical Trans. $1993 : 1611-1619$ ) suggests that aluminum matrix grain size $(\mu m)$ for an alloy consisting of 2$\%$ indium could be modeled with a normal distribution with mean 96 and standard deviation $14 .$
(a) What is the probability that grain size exceeds 100$\mu \mathrm{m}$ ?
(b) What is the probability that grain size is between 50 and 80 \mum?
(c) What interval $(a, b)$ includes the central 90$\%$ of all grain sizes (so that 5$\%$ are below $a$ and 5$\%$ are above $b ) ?$

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Video Transcript

all right. Now the question is proposing again on the normal distribution here, So they want to use the normal is commission from that Here, the mean is given as, ah 96 and, uh, the standard division synchronized given us for being. So I'm going to write down the formula for sale, which is a standard, normal variable here. So it's given by X minus mu upon sigma. Okay. And if I say absolutely values off humans and movie are X minus 96 a born for being. So the first part is asking us to find the probability that the greens exceed Oh, you know, the size of 100 0 So in that case, I'm under present the greens as eggs and they're saying exceeding 100. So it means it is greater than 100. So we want to find this. Probably did I know. So at X equal to 100 and one to first find the value off there. So it will be 100 minus 96 upon Fuding so you could simplify this part. The value that I'm getting here is ah, Wind. Wait five. So which I actually inviting us going tonight now. So in terms of is there is there for the prisons that it is that greater than 0.29 Now, the greater than probably we can write this as one minus B off their less than equal toe point line. So far, though, this is one minus this property of less than equal. So I can write as fi off 0.9. So now, from the standard normal institution table that we have, we're going to look for this value. So we're gonna take the roll off 0.2 on the column off 0.9 So then that is the table value is actually 0.6141 She performed a subtraction. I'm getting down. So here as a 0.3859 So is the first part that we have done now moving on to the next part for the bar media, saying that the greens should be off the size between 50 and 80 and we need to find the probability of that. So holder present. This part as 50 is less than equal to X is less than equal to 80. So this is what we're supposed to find here, which means that excess ranging from 50 to 80. So at these two values off X, I'm going to find out the values of their also. So when excess 50 Okay, I can write down there will be 50 minus 96 upon 14. So this actually gives me minus three point to night and then further, when X is equal to a T. I am again findings there. So this is a T minus 96 a born 14. So this is actually giving me minus one point 14 So, in terms, I was that Therefore I can't represent this, probably as this is from minus 3.29 minus 1.14 Not whenever we have such a probably where it is raining from one. Well, with the other value, we can directly right this as this is fire off the second value, which is minus one point 1/4 minus the David value for minus 3.9. So we're going to look for these two values from the table for the first value will go for the road 1.1 and a column off 0.4 So that value is 0.1271 minus. For the next value, we're going to look for the Ruoff minus 3.2 on the column off 0.9 So I'm getting that value A 0.0 You know, five Nuffer perform the subtraction and getting the answer as 0.1266 So we're done with them, B, but also now moving on to the next part. The artsy don't question states that, uh what in double baby includes the central 90% of all greens, so that 5% below and 5% above be so now, in this case, I'm going to active. You draw a diagram, so this could be understanding a better way. So for normal distribution of this is the center okay, this is normal. Exception is their normal institution called that we have someone take any random values that they have store to find the value of A and B, which is a double. So there are David easily and Zach value and B B B. All right, so they're sent that the center 90% is this one. This is the central 90 person. Okay, So it means that the remaining is 10% which we will divide on both the sides. So then you say that different? This is five wasn't. And this part is also right, wasn't. And this is the center, 90% which has been given to us. So there's actually becomes. Now what? Reverse problem? Because we've been knowing the probability from Cerritos. That'd be so. I can therefore considers. See? I reckon right on that. 90% off the central area. Okay? In terms off decimal, I contrived a size 0.9. Okay, so therefore the remaining area. I want to get laid. It's actually going to be even minus 0.9 because we have 100 minus 90. Just end. So in decimals is one minus 10.9. So this actually gives me point. What? No 0.0.1 is the total amount. So But we have point when again, divided by two game, because it's going to be on both sides. So this actually gives us 0.5 which is actually work that is the area on both sides. That is, life can really in just the right person. Comes as it'll point, you know, fight. All right, so now at this part from the table beautiful in the reverse order because 0.5 is the percentage that we already have. So from the table, check out the valley, which is near to 0.5 and the divorce order for the Ruin column. We need to find the value of said So if I talk about the value since 0.5 is on both sides, it means it's symmetrical. We don't think the negative and positive about the values off there. So for this value, we can say therefore, act 0.5 probability and getting there is equal do minus and plus 1.64 case. I can therefore see that suppose that is minus 1.64 and since then, be right inside to this, plus 1.64 Now, since I'm going with the values for that for being double, I can find out the X values also. So since we know the formula now, find out eggs we have that Zen is equal to X minus mu upon sigma, so I can therefore write X is equal to commune bless their new signal. So for ex d you is not of the same 96 but that will be that A So this is minus 1.64 into signifies for being too for simplifying this part. Beginning is 73.4 and next time finding xB. So since 96 plus now is that these positives 1.64 indoor for being so simplify beginning this value as 100 any dean 1000.96 So I can therefore now say that the interval a comma be what their masters is actually 73.4 comma when 18.96 So this is the final answer.

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