what-is-the-probability-that-the-sample-of-kindergarten-children-in-example-76-has-a-mean-height-of-

Name: what is the probability that the sample of kindergarten children in example 76 has a mean height of
Uploaded: 2020-06-07
Duration: 3 min 27 s
Channel: Numerade
Description: What is the probability that the sample of kindergarten children in Example 7.6 has a mean height of less than 39.75 inches?

Question

What is the probability that the sample of kindergarten children in Example 7.6 has a mean height of less than 39.75 inches?

Gus Steppen · Accepted Answer

okay for this problem? Were you referring back to a previous example about heights of kindergarten Children? So I&#x27;m the previous example. We were told that, uh, for this problem, we&#x27;re told that the population height for these kindergarten kids is 39.0 inches. Okay. And the standard deviation of the population was to and in the example, they looked in between problem, they looked at a the probability the mean value would be between 38.5 and 40. But for this problem, we want to find out is we want to find out What&#x27;s the probability? I&#x27;m just going to restate this using probability rotation way. Want with probability that the height, the X we&#x27;ll call it random variable is less mean. Height of the Children is less than 39.75 inches. So similar deal. We know that it&#x27;s a normal distribution problem. We can use these scores and find some information on the table. And we know at the end of the day the sample size makes a difference in terms of our variability in. So let&#x27;s write that down to sample size was 25. Okay, so first things first you do your Z score? Ah, over here. Plan to the score formula. So we&#x27;ll do. Um, the 39.75 you plan these two? These were formula here. This minus 39.0. Divided by the standard deviation, which is adjusted for the sample size of 25. So if we do a formula substitute simplify, we get 0.75. I was gonna give you a whole number. There&#x27;s a 0.75 divided by 2/5. And so that gives us 1.875 in the end. So we&#x27;re going to say approximately 1.88 so we can read off the table. So that gives us a positive Z score. 1.88 which makes sense to good chunk above the have the mean there in the population. Mean So then over here, I have my table ready to go. So this is similar standardized table of Z sports and values. So I want to look at a Z score of 1.88 so I&#x27;m gonna go over here to 1.8 and 1.8 and 1.88 is down here. So 1.88 looks like it&#x27;s 0.9699 So about 0.970 s. So in the end, we&#x27;re going to say the probability. It&#x27;s a 97% probability garrison from the table, it&#x27;s, ah, 0.970 probability. So figured out our Z score and with the table and the table will tell us the probability.

Gus Steppen · Answer

okay for this problem. We&#x27;re going back to some school data of the kindergarten Children. So, uh, going to reinvestigate this problem about Children that are normally 39 inches on average and a standard deviation that is two inches average kindergarten heights. So And reports ABC Andy, we&#x27;re terminating an individual child like, Okay, so I&#x27;m just gonna emphasize this enables one. So the big thing about this chapter, we&#x27;re talking about doing a bunch of samples and find the sample mean and then we&#x27;re gonna compare that, in contrast, contrast that in this case with just taking a single sample. So we&#x27;re looking at the probability that the student height is between 38 and 40 inches. So, um, you could show that over here. So it&#x27;s kind of close to the middle there. So 38 9 Is there some 34? We&#x27;re kind of getting my sliver in the middle, So 38 40. So I&#x27;m gonna use the calculator since we have it up. We&#x27;ve done enough disease scores off the tables, and we just know this is a distribution. The biggest thing is using your showing a direction by using the scores were showing off on the white board, So it&#x27;s looking normal. CDF We&#x27;re talking about wanting to look its wars between 38 forties of the lower boundary is 38. Upper boundary is 40 and we know on average the mean the population Means says it&#x27;s 39 and the variability is represented by the standard deviation of too. So really, we&#x27;re just using all this information to get Z scores and disease where they get turned into probabilities. And we&#x27;ve done enough now in kind of see what we get. So the probability from one single sample of getting a single student that&#x27;s that height is, uh, point 383 Now it&#x27;s like a party. So now we&#x27;re talking about Part B. We&#x27;re talking about not taking Weinberger&#x27;s. And what if we take thirties sample of 30 students and look at the average? We&#x27;re gonna take those and average them? So what&#x27;s the class mean going to be? If we do that, well, that&#x27;s gonna get us closer to that truth values. It was probably a higher likelihood, but because of the end, we&#x27;re going to see that it should be a higher number. I&#x27;m just gonna rewrite it for effect. Like to show em my student, show it and then show it symbolically. And then with the calculator, we&#x27;ve pretty much define that. We want to look between 30 and 40. We cannot use the calculator, which is just taking what you&#x27;ve gotten from your tables and we&#x27;re looking at the cumulative cumulative distributive came out of density frequency of between 38 forties. That&#x27;s gonna be the same. Same insane, insane. But here&#x27;s the big key thing here. Instead of having, uh, the two inch plus or minus the center deviation of two inches, we&#x27;re actually gonna take the two inches, and we&#x27;re gonna divide that by the square root of the sample size. So in this case, they said, the sample size is 30. So are very building its lesson because we&#x27;re dividing by the square of 30 and you can actually do that right under calculator. If you know how to click. Right button so squared of 30. Should that up, just a big decimal. But it&#x27;s in the end. It&#x27;s gonna have a probability. That&#x27;s the big kick right there. It&#x27;s word of 30. Your teacher want you to show that out you can show that on your page that puts us all the way up. 2.993 is almost 99% chance. If we take those numbers and averaging, that will hit that middle there. So point my 94 will even say so. That&#x27;s if we took 30. Has sample of 30 an average debt that will get close to that middle number. It&#x27;s definitely between 30 and 40. If individual kindergarten child is selected at random was a probability, it&#x27;s taller than 40. So blue here is now. We&#x27;re talking about a little more extreme, not in the middle. So makes sense. I&#x27;ve gotta mean in the middle. But now the problem ceases for a single pick. Just emphasize that a single pick n equals one. What&#x27;s the probability that the height is greater than 40? Now we&#x27;re talking about tall guess who are getting a single kid That&#x27;s that tall. So let&#x27;s see. Is it? That&#x27;s 30. So 40 is kind of a boundary over here. I&#x27;m just going to use the blue to talk about this because it says probability is greater than 40 is really what we&#x27;re talking about here. So we&#x27;re come about the right side over there, So let&#x27;s see what we see. Let&#x27;s look at our distribution. Gets a single sample so normal CDF between Instead of saying between 30 and 40 were going to say that it&#x27;s 40 is the lower end in the upper boundary is really infinity as far as your calculator is concerned. I was ready. You nines there. The meaning is still supposed to be 39 a standard deviation served up since we&#x27;re talking about a single pick is, too. So no adjustment for sample size. So it says if you take a single pick, there&#x27;s a likelihood that you&#x27;ll get someone over 40 30% chance, so there&#x27;s a decent probability. And if you look at the diagram, that seems reasonable. So it&#x27;s 300.30 nine that you get a single student that&#x27;s over the 40 inch, right? Look again. So party talks about Well, same thing. What&#x27;s the probability of the class mean? Okay, so the probability about the will make this emphasis just act will say X bar X bar, the class mean is over 40 have actually much less likely because we know it&#x27;s supposed to be between closer to 39 so getting the average to be that high should invest a lower number. But let&#x27;s live the calculator. Do us in disease support translations for us. So we know we&#x27;re going to look at a normal CDF Ah, from 42 positive Infinity 39. Supposed to be the mean And here&#x27;s the kicker here, the class mean of sample of 30. So instead of just dividing by the standard deviation, we&#x27;ll take the standard deviation adjusted for the sample size, which we&#x27;ve been told is 30. We need to affect our variability, so it&#x27;s gonna be like, much less likely. So there we go. So probably I get an average. That is, that much higher, above which the average should be is pretty low, so we can see the probability of getting a class size mean of over 40 is only 400.0 three

Gatjang Deng · Answer

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&#x27;s not inches and 28 inches. So so we&#x27;re given mean, so mean We don&#x27;t use a good meal because 69 inches and then you given the this standard deviation. It&#x27;s 28 sanitation where you would use sigma because 2.8 inches. So that&#x27;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&#x27;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&#x27;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&#x27;s the interval we need to evaluate. And now limits are from 65 two 73. Is the car into a goal we&#x27;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&#x27;re using that Simpson&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s how you have that question

Noraney Ocampo · Answer

So then you have your normal distributed data. So your meanness 50. That&#x27;s what goes in the center off your curb. And your standard deviation is 515 So you&#x27;re adding and 5.5 to the right is gonna be 85 with five. This is gonna be 61. This is gonna be 66 point five. And that last part was going to be 72. And to find your values on your curve on the left side, we are going to subtract my 0.5 because that&#x27;s your center deviation. So 50 minus my 0.5 is going to give you 44.5 on their 44.5. Minus five is gonna give you 39 and I&#x27;m gonna stop there because that&#x27;s what they&#x27;re asking or let me make thes number. Niedere will be 4.5. So they&#x27;re asking you what percentage. Uh, what&#x27;s the probably to do that a number is greater than 39? So they&#x27;re basically asking you for the percentages. So again, remember, when you have a normally distributed data, the center section the one standard deviation, then you make the curb again. My normal curbs weren&#x27;t perfect. But what I need is the percentage. So once under the beach, in a way, is 34% toast under divisions. Away is 13.5% close under divisions. Way is 2% actually, three standard deviations, that&#x27;s 2%. And then the last part, this syrup 0.5 off the data is here, Mr 0.5 off the miracles here. So they&#x27;re basically asking you what&#x27;s the probability that your number is greater the 39. So from 39 higher, we&#x27;re adding all the percentage is or the other Whether you could do it, subtract those two. So you know, all the data is 100% name your going to subtract the 2% and then the syrup 0.5%. And that&#x27;s faster, then getting the other way around, adding, but it should give you the same answer. So if you are doing 100 minus, so that&#x27;s 98 minus 980.5. That gives you 97.5 percent or if you want to double check, we&#x27;re adding the percentages. So 30.5 plus 34 plus 34 plus 30.5 plus two, plus Sarah 20.5 also gives you 97.5. So again, either you add the percentages or you subtract from 100. Since this represents all of the data and you only we are concerned about their night or higher. So you just saw a truck roll 100 this little tiny piece already wanted that will check they do to disconnect the percentages.

Joe Lesueur · Answer

for this problem. They claim that basketball players height is 73 inches for the average of each player. Now we can set up the null hypothesis of H not equals meal, which is our population average. Although we&#x27;re checking a sample, make sure you always have the population variable. In our no hypothesis, we&#x27;re gonna set our null hypothesis equal to 73 the amount of inches in their height. The alternative hypothesis states that mu are average height is less than 73 inches. Study says there was 40 subjects and that the P value was very close to zero. That you can assume it is zero with an elf value, we&#x27;ll assume of 0.5 Since the P value of zero is much less than the Alfa of 0.5 we can reject the null hypothesis and clean that there is sufficient evidence that best ballplayers height, on average is less than 73 inches

Luuk Verhoeven · Answer

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&#x27;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&#x27;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&#x27;s graph the function they&#x27;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&#x27;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&#x27;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&#x27;re taller than 72. That&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s 1/2 of the area under a probability function which has total integral well

Problem

A random sample of size 36 is to be selected from…

Problem 32 Hard Difficulty

What is the probability that the sample of kindergarten children in Example 7.6 has a mean height of less than 39.75 inches?

Answer

0.970

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