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In a sample; E(X - 1) = -20 and ZX = 20. What is the mean?0020010Oo5...

Question

In a sample; E(X - 1) = -20 and ZX = 20. What is the mean?0020010Oo5

In a sample; E(X - 1) = -20 and ZX = 20. What is the mean? 0 020 010 Oo5



Answers

Suppose that a random sample of size 20 is taken from a normal distribution with unknown mean and known variance equal to $1,$ and the mean is found to be $\bar{x}=10 .$ A normal distribution was used as the prior for the mean, and it was found that the posterior mean was 15 and the posterior standard deviation was 0.1. What were the mean and standard deviation of the prior?

So in this problem, we suppose that a random sample size of 20 is taken from the normal distribution with an unknown mean and unknown variance equal to one, and the main is found to be explore is equal to 10. And a normal distribution was used as the prior for the main, and it was found that the posterior mean, it was 15 and the posterior standard deviation was 0.1. And so were asked what were the mean and standard deviation of the prior? And so we know that we're given X one this morning, so we're giving it a This is uh they're independent, identically distributed random variables. So we have exported to 10 and the prior distribution of the main, it's normal and that the posterior mean 15 and the posterior standard deviation Was your .1. So we need to find the prior main and the prior standard deviation. And so we have these two relations were to post musical to end times you're not plus peter prior and that the um post main sometimes it and our times this came as the prior man plus the firemen. And they're not as times you know her, remember this? And we know that data is equal to one over sigma squared. And this holds for each index prior post or not. And it's called the precision. So from the first equation we know that a prior it's equal to your post minus and times it or not, so that's equal to one over sigma squared. Post minus n over signals were not. It's equal to one over 0.1 sq -20/1 square for 80. Okay. And then this implies that it is equal to one over sigma squared of the prior. Um which means that sigma squared of the prior is equal to 1/80, so Mhm. Mhm. You can find thought This four is your .11, 1. Yeah. And then from the 2nd equation we have that thinner closed from sometimes it or not plus, you know, prior minus and playing. So tonight. Oops. Hi. Mhm. Physical to data prior times prior, which is equivalent to this equation. Mhm. Mhm. And we know that it is not is equal to one and in a prior is equal to 80 that we found earlier. So from now we can find the prior mean just plugging in our values. Mhm. Mhm. Okay, mm hmm. And this will give us 16.25.

When this question, we're told that we have a random Sample size of 2 25 drawn from a population, The population has mean 100 and standard deviation 20, and were asked to compute the mean and standard deviation of the sample means, so the mean of the sample mean is just the population mean, which is 100. The standard deviation of the sample mean is the standard deviation of the Population mean, divided by the square root of N, which is 20 Over 15, which is square out of, then that's 1.33.

Um The central limit limit Children states that that if this sample Bye. And is larger that is a creator or it's called a tatty then there's something distribution distribution of sample means Palestinian normal distribution. Um The minnow sam playing distributions of sample means this meal. Well sample I mean yes knew where newest the population. Me and the population means is 100. And the standard division of the sampling distribution of sample meanies. Ah New reforms Rugova. And then they recognized the population mean an energy center. Right. So we have big night Population mean is 16. And the um Sample sizes for that is eight. Now we know that is a call to. Mhm. Is the courage to X minus mu upon stigma. We have X. S. Um 110 -200 upon heat. There's 1.25. And we have again X. Is 150 0 at once. It 3.75. Um Now Observe the values of X. That is 1.25 and 7.35 In the three table to obtain the area of the left side of the respective Cisco. So the area of left side. Yes 0.0764. And area right side is We're A .9236. No, that is so abstracted the areas to evaluate the total area of this region between. So total area is maybe i is point 9999-. So that is um sorry 9236 -0.076. For that is a call to 0.847. Mhm. Well the that is Recalled to 84.72%. Your sister this ahead? No.

The following is a solution to number four. And this says that a random sample of 19 people, I guess uh from a normal distribution. So that's important. The population distribution is normal. That makes that sample size okay. If it weren't normal, then we would need a sample size of at least 30. So the 19 is okay. And the sample mean of that particular uh Uh there is no context of this problem. So the sample mean of this sample is .8 and then the sample standard deviation is .4, and we're supposed to test that alpha equals .01 if the mean is less than one. Um so first off, because we don't know the population standard deviation, we were never given sigma, were only given the ass, we need to use the T test. Alright, We cannot use the Z test because we don't know sigma's we have to use the T test and it's a five step process, like I said. So first off, the uh the first step is to state our Nolan are alternative hypotheses. So the null hypothesis h not always has some sort of equality and we're testing a population means. So you're always gonna use greek symbols here. So mu is 1.0. And then we're testing if it's less than 1.0. So the alternative is that mu is less than 1.0. The 2nd and 3rd step, I'm actually going to write together because we're going to use technology to get it. So the second step is to find the test statistic and you can use a formula for that. But if you have a calculator, if you have, you know, Excel or some sort of software, um it does a lot of the work for you. So that's what I'm going to use just to save some time. So t is the test statistic and then the p value is also going to be given to me. So let's go to the calculator. Now I'm on ATI 84. And if you go to stat on the T- 84 and go down to tests, it's the second option here. The T test. And we need the summary stats here, so make sure summary stats is highlighted And the mute not, that's the hypothesized value. So in this case it's 1.0 or just one. So it's the null hypothesis. The X. bar is your sample mean, and that was given as .8 SX is the sample standard deviation, which was given us four, And then in is the sample size and that was given as 19. And then you get down here, and this is the alternative hypothesis. So it's either not equal to less than or greater than and if you look back that alternative is less than one. So that's all good. And now we can calculate and that gives you some stuff here. But really this is the T. And the P. Is what I need. So the T. Is negative 2.179. So let's go and write that down -2.179. and then the P value Is .02 0.2 And the p value really is probably the most important thing on that what you do. And step forward as you explicitly compare the P value with the alpha value. And remember the alpha value here is 1.2 is greater than one. And whenever the p value is greater than alpha than we fail to reject the null hypothesis. So failed to reject H. Not had that P value been less than alpha than you would reject H not, but since it's greater than alpha than you failed to reject, so any time you failed to reject your conclusion, which is the last step step is going to be something along this line, there is not sufficient evidence to suggest that The population means I'll just use mu is less than 1.0, so there is not sufficient evidence to say that the alternative hypothesis is true, so therefore we're for lack of a better phrase, were accepting that the null hypothesis is true.


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