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According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches.

(a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall?

(b) What percentage of the adult male population is more than 6 feet tall?

A.$\approx 84.7 \%$

B.$\approx 14.2 \%$

Applications of Integration

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Campbell University

University of Michigan - Ann Arbor

Boston College

Okay, So for this problem, we were told that, dear. According to the health survey, the Iight the height of adult males in the United States are normally distributed was mean of 69. It's not inches and 28 inches. So so we're given mean, so mean We don't use a good meal because 69 inches and then you given the this standard deviation. It's 28 sanitation where you would use sigma because 2.8 inches. So that's how you define. So so the survey is normally be modeled. Relax. Is it because 21 over and in the square to pi times sigma time Miss E to the minus 1/2 X minus mu. It was Sigma squared. And this is true for all ex. This is true for all ex great, and you questions you so that means effort max zero. Every else in that domain you can plug in Miracle 69 and Sigma equals 2.8 inches to this expression here. Now the question would like. The answer is to determine the probability that an adult male shoot at random chosen a random will be between 65 inches and similar three inches tall. Hey, it's similar to say What is P? Alexe is between, uh, 65 and 73 inches tall. So, mathematically, this is you. We can translate this into a little trouble. We will become so limits of integration are gonna be from 65. Consider it. It's free after lex D X. So all right. Now you can plug in old in America about stuff after we see the ready. So mu is 69 signals to plate into f of X. Now, for our into go, um, interval will become one over to two pi and then square root every signal. What is Cigna Signature stamp? Civilian. We used to Quinn eight inches e to the minus 1/2 in the X minus 69 of a sigma, which is 2.8. And then you square the entire thing. The X So that's the interval we need to evaluate. And now limits are from 65 two 73. Is the car into a goal we're supposed to do? You should see that I pushed out this constant fear. One over to Red Troop I, Thomas 2.8. Because of that Constance, I pushes how it automatically So the only part of integral you have to do is this part here, and news board is no elemental way to do it. Um, you can use but war from which is freely available on the Web. Or you can use my people. The people are going to uh, So anyway, So if you put this into an integral calculator, uh oh, are integrating numerically for them. We're using that Simpson's rule. You will get something like 0.8469 And so or roughly roughly rather the 85% 85% now moving on to part B. So before B, we would like to know the percentage of the off the adult male population that is more than six feet six feet. Remember, You know, the units were using the interest, so we needed to hurt six feet. So how many in deciding six feet so six feet. He's in close to 12 every six inches because one foot one foot it was 12 inches. So you might plan six by 12. You get 72 senator two inches. Um, so now that so the question becomes, what is the probability was a problem. X rated integrity, too. On translating this integral, it will be Seamus integrating from 72 to infinity of f of x d x. When we put in everything you should get inta go. And he's going to be the constants that before So one of a square to pie Time of sigma, which was 2.8 and then the inter go from 72 to infinity. Then you have ye to the miners 1/2 x minus 6.69 point zero of a 2.8. Okay, then the whole thing squared dx again. This integral has no elemental form. This is how you do the interval. Now there are trucks of doing intervals like that spirit. I'm not gonna go into that. Um so using Wolfram maple or what about Integral Cog? Yeah. If you should get something like zero point 14 or 14%. 14% questions asking for an answer in percentage to. So you get 14%. And that's how you have that question