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Determine Hz and &; from the given parameters of the population and sample size.p=84,6=18,n=81...

Question

Determine Hz and &; from the given parameters of the population and sample size.p=84,6=18,n=81

Determine Hz and &; from the given parameters of the population and sample size. p=84,6=18,n=81



Answers

Determine $\mu_{x}$ and $\sigma_{x}$ from the given parameters of the population and the sample size. $$\mu=52, \sigma=10, n=21$$

Given parameters from our population that we're pulling our samples from. How can we calculate the population means of the sampling distribution of the sample means? And the population standard deviation of the sampling distribution of the sample means. Well, with our information we saw that the population means of the sampling mean, The sampling distribution of the sample means is equal to the same as the population mean of the individual numbers. We also had that this population standard deviation of the sampling distribution of the sample means can be calculated by taking the standard deviation of the individual numbers and dividing it by the square root of our sample size. So from our given information, the population we were pulling our samples from had a population of 52. So that's going to be the same as the population mean of our dis sampling distribution of our sample means. So musa Becks bar is 52. Now sigma's of X. Car. The standard deviation of our sampling distribution of our sample means is calculated by taking the standard deviation of the population that we pulled our samples from, divided by the square root of how many were in the sample that we took when we calculated our standards are sample. Um me And here the score to 21 is not 21 is not a perfect square number, so we are just going to go ahead and calculated calculate that out and round to two decimal places and we get that are sigma's of ex far is 2.18. So arm use of expire is 52 And our Sigma setbacks bar is 2.18.

All right. So, we are given some parameters of a population and its sample size and were told to figure out the mean of the sampling distribution and the standard deviation of the same. Bill, the distribution. But we have an issue because we don't know anything about the population. And if we knew something, if the population were normal, then we could make some claims about this. But because we don't know uh much about anything about the population distribution, we can't do much because the central limit theorem says that regardless of the population distribution is as N increases, the sampling distribution becomes approximately normal. So, I've done other videos where we have some distribution that looks something and you can even do stuff like this where it's like I've seen 11 of my staff classes, we had something like a It's a problem that was under this had an area of one and this is not normal. But if you sample from it, uh the sampling distribution as and increases it becomes normal. It's pretty cool. No. Very cool. Very, very cool. However, For in practice, and needs to be greater than 30 for this thing to hold true for this, this to hold true. So here, obviously, and is 130, so it's not 30 or greater. So we can't really make this claim. But let's just for the sake of of being clear, will say um if the population is normal, if the population open pop as abbreviation pop is normal. Yeah, yeah, then and if that's true, then the sampling distribution means is mu and the standard error of the mean, also knows the standard deviation of the sampling distribution is sigma over route N. So Then we could say that μ X bar is 27 and sigma expire is 15. Excuse me, six over 15, which is approximately one route 55. So, um and it says as as it increases these these sampling parameters get closer to mu and closer to sigma over root end. So, um again, we just have to be careful so we run through this for the sake of practicing the calculations, but I'm gonna put a little start here with a little be warning. I'll do this star in red to say, be aware that there's an issue here because and is supposed to be greater than 30 for this to all. True. And here we have. And it's not so just be cautious of that if you're working with as you're working through these exercises, so they're going

What are we are asked to find? The population means of the sampling distribution of the X. Bars, denoted musa Becks bar. And the population standard deviation of the sampling distribution of the sample means denoted sigma sub X. Bar. Well, from the work that we did in the section meuse of X bar is the same as mu. So in other words, the population mean of the sampling distribution of the sample means is the same as the population. Mean that we pulled our samples out of. So the population mean of the individual numbers. And the standard deviation of our sampling distribution of our sample means can be calculated by taking the standard deviation of the individual numbers in that population that we pulled our samples out of, divided by the square root of the number in our sample, so square root of N. Mhm. So from this particular question arm use of expert that I've been asked to find is the same as the population means. So it's 27 and sigma's of X bar is calculated from the standard deviation of the individual number six divided by the square root of how many were in the sample size, so 15 And simplifying that. Well 15 is not a perfect square number, so we are just going to divide that out and round to two decimal places and we get that, that's about 1.55. Yeah, So our use of expert is 27 and our sigma's of expert is approximated to be 1.55. If they wanted you to give it exact, you would have answered it in the previous lines form.

All right, guys, so for this one, this time for this problem, we need to find the pot sample. I'm sorry, the population variance and standard deviation is a little different from the previous two questions here. All right, So the population variance here is I'm going to do that. I'm gonna go ahead and use a google sheet. You can do the same thing with Excel or you can do it by hand, you can write your text book if you like to do by hand. So the first thing I'm going to find is the population variance. Here's how you find that on um google sheets of a. R. And then piece of a RP. Got it for variance of an entire populations. Like click on that. I'm then going to use my cursor to highlight all the numbers in that I want I hit enter and it's 16. All right, let's continue on with population standard deviation. So, first I'm gonna get my equal sign. I'm then going to start typing in standard S T S T dev. That's good. But that's for just a sample. I need to find standard deviation for a population right there. Okay. So then I'm going there. I'm going to highlight all the numbers I want and I get a nice four. Okay. And then a great check. Is this standardization square to be the variance is four square at 16. You betcha, we're done.


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