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Describe the sampling distribution of p. Assume the size of the population is 25,000_n = 600, p = 0.338Describe tne shape of the sampling distribution of p Choose t...

Question

Describe the sampling distribution of p. Assume the size of the population is 25,000_n = 600, p = 0.338Describe tne shape of the sampling distribution of p Choose tne correct answer below:The shape of the sampling distribution of p approximately norma because n < 0.O5N and np(1 -p) < 10. The shape of the sampling distribution of p not normal because ns0.OSN and np(1 p) < 10. The shape of the sampling distribution of p approximately norma because <O.O5N and np(1 -p) z 10. The shape of

Describe the sampling distribution of p. Assume the size of the population is 25,000_ n = 600, p = 0.338 Describe tne shape of the sampling distribution of p Choose tne correct answer below: The shape of the sampling distribution of p approximately norma because n < 0.O5N and np(1 -p) < 10. The shape of the sampling distribution of p not normal because ns0.OSN and np(1 p) < 10. The shape of the sampling distribution of p approximately norma because <O.O5N and np(1 -p) z 10. The shape of the sampling distribution of not normal because ns OSN and np(1 P)z 10_ Determine the mean of the sampling distribution of p (Round t0 three decimal places as needed )



Answers

Describe the sampling distribution of $\hat{p}$ Assume that the size of the population is 25,000 for each problem. $$n=300, p=0.7$$

So in this question, were asked to describe the sampling distribution of P And we're at all to assume the population size is 25 1000. and here we are given the sample size as 500 and the portion PS find slower. And so here we see that the hen, which is the sample size is less than or equals two final five times many five Thousands values. 1 2 50. So this condition is satisfied the Independence Commission. And then we also check for sample size. So np one minus P U C equals 220 which is more than or equal to tents or sample size condition is also satisfied. So we know that sampling distribution P is an approximately normal distribution with the mean of P. We call 2.4 and the centre deviation of P Equals to P. 1 -7 over and square room Equals to 0.02 two. Approximately normal with mean and standard deviation.

One of the key components. As we get into sampling distributions and using sampling distributions for inference, um, sampling distributions are in fact the building block for inference. And one of the biggest things that we need in order to do, inference is approximating our data to a particular shape curve or a particular shape distribution that we know a lot about. And for the sampling distributions of part of sample proportions we want to know is our sampling distribution going to be approximately normal. That is the shape that we're going to um going to try to match it to, so is approximately normal because if we have a normal curve then we can calculate p values, we can create confidence intervals. We can do a lot of things using a sample to estimate or to test against a population value or a claim that's given about the population based on our sample and for sample proportions, The condition that we're going to use is called the large counts condition. Which what we're going to do is we're going to check is there at least 10 expected successes and 10 expected failures and to do that, we're going to take our sample size and multiply it by the population proportion and see if that's at least 10. Are we expecting to get at least 10 successes with whatever given characteristic that we're looking at? And then also are we going to get 10 failures? Are we gonna get 10 individuals and are a sample that does not have the given characteristic? If both of those things are met, then we can say that our sampling distribution is approximately normal. Which opens up the door for confidence intervals for hypothesis tests by calculating p values and really allows us to use our sample to make predictions or test claims about our population.

This problem we are asked to describe the circumstances under which the shape uh the sampling distribution of peace approximately normal. So the circumstances under which the shape is approximately normal. There's basically two conditions. The first condition is that the sample size and, yes, Basically no more than five of the population size, so that the samples are basically independent. And the second condition is that n. P one minus P should be More than R. equals two. And so we have a large enough sample size. So when these two conditions are satisfied, the shape is approximately normal.

In this problem, we are describing a sampling distribution which is a huge part of statistics. You've probably described distributions before and we're going to do the exact same thing with a sampling distribution based off of conditions and formulas that we know about a sampling distribution. The first thing when we're describing any type of distribution is to identify what the center of the distribution is, so that for a sampling distribution would be the average of all of our sample proportions, which we know is equal to the population proportion, which is given which is seven. And the reason why we know that is true is because random sampling is used. If I were to not take random samples then unless if we did actually take every possible sample size of 300 from a population of 25,000, which would be absurd. Um But random sampling being used means that the average of all of our P hats would be equal to our population proportion without having to actually create every possible sample possible sample of size. 300 from 25,000. So again, if you're tying in those conditions, random sampling gives you that the mean is the population proportion. Okay. The next thing is is our shape approximately normal. And we're going to use the large counts condition for that which is N. P. Is that greater than or equal to 10 and in one minus P greater than or equal to 10. Which are sample size is fairly large here. So that is going to be true 300 times 3000.7. That is 210. Much larger than tim. And then we also have to check the compliment. So 300 times three which would be 90. We are checking to make sure that there are at least 10 are expected successes and 10 expected failures. We would expect to have 210 successes and 90 failures, which is more than 10. So it is approximately normal. It did meet that condition. And then the last thing we have to do is are spread. Again, we don't have to do outliers because it is approximately normal. If you've done that with previous distributions, but we do have to check the spread. So the square root of p which is 0.7 tom's 0.3 divided by in which is 300. And that will give us our standard deviation, which I'm going to round 202 six. And again, we know that is true because the independence condition is met. So, again, you can see all three of these conditions random, independent and normal. Match up with the description of the sampling distribution and the independence condition is that the population is greater than 10 times the sample and clearly 25,000 is greater than 10 times 300. Okay, so that condition is met as well.


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