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6. A random sample Xi' Xn is selected from the shifted exponential distribution; and the sample mean is X B-x x 2B f(x) = x < B Is X - 1 an unbiased estima...

Question

6. A random sample Xi' Xn is selected from the shifted exponential distribution; and the sample mean is X B-x x 2B f(x) = x < B Is X - 1 an unbiased estimator of 8?

6. A random sample Xi' Xn is selected from the shifted exponential distribution; and the sample mean is X B-x x 2B f(x) = x < B Is X - 1 an unbiased estimator of 8?



Answers

Suppose that the random variable $X$ has the continuous uniform distribution $$f(x)=\left\{\begin{array}{ll}1, & 0 \leq x \leq 1 \\0, & \text { otherwise }\end{array}\right.$$ Suppose that a random sample of $n=12$ observations is selected from this distribution. What is the approximate probability distribution of $\bar{X}-6 ?$ Find the mean and variance of this quantity.

It is important to get a solid foundation of the vocabulary for sampling distributions. For us to start this next part. Now, a sampling distribution is different than a sample distribution. So when we're talking about a sampling distribution, you're repeatedly taking samples out of a population with a specific size end and you're calculating some statistic about that sample. And then you're actually looking at the distribution of that statistic and how we can apply that later in our inferential statistics. So our first thing with vocabulary is that if we're looking at the probability distribution of all the possible values of the random variable X. Bar. Remember ex far as a sample mean? So I grab a sample and I find the mean of that sample. And then I grab another sample size, same size end, and I find the mean of that sample and I do that over and over and over. And then I look at the probability distribution of those um sample mean values. And that expert is calculated from a sample of size and from a population whose mean is new and whose standard deviation is sigma. That process produces a probability distribution known as the sampling distribution of exper.

There will be a lot of uh beginning questions that ask you about the distribution of the sample mean X. Bar. For that statistic. Like what is the sampling distribution of expire later on. We'll also look at what is the sampling distribution of P. Hat. So when they ask you what is the sampling distribution of a statistic. They want you actually to address three issues first to describe it, shape, second to say what its center is and then third to say what its spread is. So shape center spread is what you want to answer when it says what is the sampling distribution of a statistic or describe the sampling distribution of it? That sort of thing. Now shape when we're talking about the sample mean X. Bar. The distribution of the sampling distribution of X. Bar will be normal. If we pulled our samples out of a population that was normal, no matter what sample size we take. If we take our samples from a population that we don't know is if it's normal or not or that we know it's not normal, then we need sample sizes that are larger and usually 30 size 30 or greater is what we would need in that situation. So here it says that our samples are taken from a population that has a normal distribution. So the shape of our sampling distribution of X. Bar is also normal since the samples were taken yeah from a population with a normal distribution. So that's a shape next. What is its center? Mhm. So with the center, what does it mean? So the mean mu of the sampling distribution of exper we denote that time use of X. Bar. What we saw in working with this that the center of the sampling distribution of the sample means is the same as the center of the individual numbers. Um That population that we've pulled the samples from. So the population mean of the sample means is exactly the same as the population mean of the individual numbers, how about spread will spread? Is your small case sigma's of X bar. And we find the standard error of the mean by taking the standard deviation of the original population and dividing it by the square root of our sample size. So now how do we get for the center and spread specific numbers will back up in the question. It told us that the population had a mean of 30. So my center my meuse of X bar is the same as that new. So it's 30. And the spread, the standard deviation of our sample means we calculate by taking the standard deviation of the original population we pulled the values samples from. So in this case that's eight And divided by the square root of how many were in our sample. And we were taking sample size 10. So then when you go through and calculate that we get Um spread. Sigma's of expert of about 2.53. So what is the sampling distribution of expert? The shape is normal, the center is 30 and the spread is 2.53.

All right, So we have a a random sample of size 10 that's taken from the population that's also normal. Um With a with a given mu and sigma. So I don't know the sampling distribution of X. Bar well, because the population is normal, the sampling distribution of X. Will be normal. It is normal. Normally distributed. Yeah. Mhm. Yeah, you ted. Um however if the population was not normal then the sampling distribution would not be normal. Um But just because the central limit theorem is super awesome and pretty amazing. Yeah. Mhm. Yeah, the central limit through and I'll put this over here because it's sort of separate. And aside here from this exact question, uh the central limit theorem says that no matter what the population Distribution is, if you have a size of the sample that's greater than 30, the sampling distribution is normal, which is pretty cool. So you can have some weird distribution. That's the exponential ones kind of kind of popular in the world of the world. So you have something like this, if you sample from this distribution. Mhm. These samples will make a normal distribution, which is just I mean, come on, that's awesome, That's just cool. So they go

Yeah that's probably have been given the cumulative distribution function. Capital F of X is equal 20. If X is less than zero. 0.2 Acts effects is between zero and five and one. Mhm. If X is greater than or equal to five and we would like to find a few different probabilities from us. Yeah, I would first like to find the probability That X is less than 2.8. Well this is just the definition of capital f. 2.8 mm Which is 0.2 times 2.8. We just plug it infrared. What's going on here? All we're going to do is take .2 times 2.8 and that's going to give us this probability and when we do This gives us .56. So these probabilities .56? Yeah, on b We would like to find the probability X is greater than 1.5. Well this is equal to 1- the probability X is less than or equal to 1.5 Which is 1 -101.5 Which is 1 -0.2 times 1.5. Mhm. It's always that 1 -2 times 1.5. And this gives us .7. On c we want the probability X is less than -2. Again this is just capital f of -2. And from the chart since this is less than zero this is just zero. Mhm. And um do you want the probability X is greater than six. Yeah Which is 1- the probability that X is less than or equal to search Just one -F of six. If such as ones, this is 1 -1 which is zero.


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