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Suppose the heights 18-vear-old men What the probability thatapproximately normally cistributed Mean 69 inches standard deviation inches mer 2ceo andom between 68 a...

Question

Suppose the heights 18-vear-old men What the probability thatapproximately normally cistributed Mean 69 inches standard deviation inches mer 2ceo andom between 68 and inches (RourgJecima 03C2erandon samcIyrenty-seven 18-vear-old menected, Wnatthe probability that tne mean height x is between 63 and 70 inches? (Round vour ansiver to four decimal DiacesLompare Vomr Anserparts (a) and (b) the probability The probability Part (6) mucn higher because the FfanFcr higher? Why vrould lurgei for the dist

Suppose the heights 18-vear-old men What the probability that approximately normally cistributed Mean 69 inches standard deviation inches mer 2ceo andom between 68 and inches (Rourg Jecima 03C2e randon samc Iyrenty-seven 18-vear-old men ected, Wnat the probability that tne mean height x is between 63 and 70 inches? (Round vour ansiver to four decimal Diaces Lompare Vomr Anser parts (a) and (b) the probability The probability Part (6) mucn higher because the Ffan Fcr higher? Why vrould lurgei for the distribution_ expect this? The probability part (b) much higher because the mean smal for the distrinution; The probability part (6) much higher because the standard deviation smaller for the distribution Proda bility mucc lo Vel decame standard Vation smaller distobution The probability part (6) mucn higher because the standard deviation larger for the distributiona



Answers

Heights of Men The heights of 18-year-old men are approximately normally distributed, with mean 68 inches and standard deviation 3 inches (based on information from Statistical Abstract of the United States, 112th edition). (a) What is the probability that an 18 -year-old man selected at random is between 67 and 69 inches tall? (b) If a random sample of nine 18 -year-old men is selected, what is the probability that the mean height $x$ is between 67 and 69 inches? (c) Interpretation Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

So we're going to have to distributions in this problem. One is the distribution for individual nails and looking at their heights, and we're assuming the main is 68 inches, with a standard deviation of three inches. So Approximately here is 71 1 standard deviation away, and the next one is going to have groups of size nine And that sampling distribution and then group of size nine, and then you find the main. So our sampling distribution is going to have a smaller standard deviation, still is going to be centered At 68", but now it's going to have a standard deviation Of three divided by the square root of nine, which is one. So our first question asked, what's the likelihood that you sample one person and you get that one person to be between 67 And 69 tall. And we can see that way, that means we would need to convert this to a Z value, so I have 67 minus 68 divided by the standard deviation. Now it's a Z value And I have 69 -68 Over three. And so we can see in both cases that we get uh negative one third, can I get positive one third? So we're symmetrical and I'm going to utilize my normal CDF for this And second and distribution normal CDF, and I'm going to use the lower value is negative .3 Repeating, and the upper is .3 repeating. And leave the mean at zero and a standard deviation at one. And when I do that, I find out that that probability comes out to be about 26%. Part B says now we're looking at this distribution and what's the likelihood of having the mean be between 67 and 69. And when we convert that into Z values again we take the number minus the mean, divided by the standard deviation, which is that one? Yeah. And this month groups this is the 69 minus this, 68 Do I, too? By that standard deviation up here, which is one. And so I get my Z values to be negative 1-1. Yeah, and again, I know that's about 68% but I'm going to use my normal CDF button again, normal cbf use my lower as negative one. My upper is one. And leave the mean at zero and standard deviation at one And we get .68, 2 rounds to seven. Now, part C. Yes, this answer is much larger, we expect that. But why? Basically? Because this standard deviation of this second problem is much smaller because this is the standard deviation that I use For groups of size nine. So we expect for those same numerical values to have a much higher percentage of the X bars being between these two numbers than individual values. So I didn't write all that down. But hopefully you see that this standard deviation smaller, therefore We would expect more values to be in between 67 and 69.

Hi. Problem number 63 for the following Integral 1/3 times route to pie and grow from a to B E to the negative princes X minus Epsilon about two by three squared about about two the X. So we're gonna find what percentage of American men are between 66 72 inches tall. And we want the value the air to be less intense. Negative three. So a cool 66 the equal 72. Using 10 sub intervals, we can approximate it with a Delta X of 0.6, giving our area value of 0.6 82760 And the actual value is 0.68 to 7. So our air is six times since a negative five, which is less than 10 three. So this is our approximation.

This question has a lot to do with graphing utilities on because there is at least 100 different graphing you to tease out there. I'm not gonna answer that far. It just depends too much on the specific. Your sense you used to give a one coherent answer. Um, but there's one part of the question that I can give an answer to, and then is the sort of the second bar questions. See? So let's graph the function they're talking about. Um, if this is, uh, 69 this is 72 that say this is than 66. The function looks something like this. No, my drawings are, of course, absolutely terrible. And the actual function looks difference that the height will change the profile of these like different. But this captures the most important parts. Especially one of the most important properties of dysfunction is that it's symmetric around 69. So the left half of this guy and it right off of this guy are same. Now, the question asks us to explain why the probability that someone is Toller? Uh then, uh, 72 is the same s 1/2 minus the probability that someone is between 69 and 72. And this is a question we can't answer based on this graph. Because what are these? Uh, what are these things in terms of the graph? So the probability that you're taller than 72. That's cool that the red area is this area area photograph to the right of 72. Now, what is the program? Is it that you're between 69 72 inches? That is this green area right here. Now, because this graph is symmetric, the total area of this site, um, equals the area on this side because it's perfectly symmetric around 69. This is one of the more important properties off the normal distribution, which this question is about. That means that that and since it's a probability distribution like the total integral off, his entire function is one. And since the two halfs are equal, the moat must be half, which means that green plus red equals 1/2. And that explains why, uh, the formula given into question, uh, works out because it's 1/2 of the area under a probability function which has total integral well

In question number 32. The sample mean is nobody distributed with me and Sergeant. Division is that the score is the value decreased device mean and divided by the standard division. So we have that set the equal to 70 minus 69.4 over 2.9 over the square. Toto 60 is equal to 1.60 toe to determine the correspondent probability using table for for beef you off x greater than 70 is equal to property off that creature. Them 1.60 is equal to one minus probability off that this event 1.60 we're equal to one minus all points. 9000 452 equal to all went over 548 is equal to 5.48 Person that define answer. Thank you.


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