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Question 311ptsFor particular population; sample ofn = scores has standard error of 4.For the same population; sample of n 4 scores would have standard error ofoM-8...

Question

Question 311ptsFor particular population; sample ofn = scores has standard error of 4.For the same population; sample of n 4 scores would have standard error ofoM-8OM = 6oMe4oM-2Question 321ptsFor normal population withu 40 and 0 = 10,which of the following samples has the highest probability of being obtained?Ms42for a sample of n =Ms44for a sample of n =Ms42 for sample ofn 100Ms44for sample ofn = 100

Question 31 1pts For particular population; sample ofn = scores has standard error of 4.For the same population; sample of n 4 scores would have standard error of oM-8 OM = 6 oMe4 oM-2 Question 32 1pts For normal population withu 40 and 0 = 10,which of the following samples has the highest probability of being obtained? Ms42for a sample of n = Ms44for a sample of n = Ms42 for sample ofn 100 Ms44for sample ofn = 100



Answers

Conduct the appropriate test. A simple random sample of size $n=40$ is drawn from a population. The sample mean is found to be $108.5,$ and the sample standard deviation is found to be $17.9 .$ Test the claim that the population mean is greater than 100 at the $\alpha=0.05$ level of significance.

In this problem. First time calculating the population proportion so it is given by P, which is equal to U L C plus L C L by two. So just putting the value here. I can write 0.249 plus 0.201 by two, and on solving it, I get the final value of 0.225 This is the required population proportion now calculating the margin of leader rated to be given by e. Margin of error is equal to U. C L minus p. So just putting the value 0.249 minus 0.225 which is equal to 0.24 So this is the required margin of federal now calculating the value of X. So as we know that B is equal to X Brian. So just putting the value here so I can write the value of access 1200 multiplication and 0.225 on solving I get to 70 as the required value effects

When working with confidence intervals for population proportions, they have the form that the lower bound is P hat minus the margin of error E. And the interval goes to the upper bound of P hat plus the margin of error E. Where P hat is the point estimate for the population proportion. So when we're given an um problem where they give us a confidence interval for a population proportion giving us the lower bound value and the upper bound value also the sample size, they can ask us to actually find the point estimate of the population proportion P. Hot to find the margin of error. E. And the number of individuals in the sample space with the specified attribute or characteristic X. So how do we go about that when we are given the information of the lower bound and the upper bound like we have in this example. Well if we notice that P hat minus C is our lower bound, but they told us that the lower bound is 0.462 So this P hat minus C is 0.462 To our upper bound is 0.509. Soapy hat plus C is 0.509. Yeah. Now that means P hat is the value that's exactly in the middle of these two. So to find our part A are point estimates of the population proportion which is symbolized by P hat We are just going to take the lower bound 0.46 to add it to the upper bound 0.509 and divide that by two to find the value that's right in the middle. And when we do that calculation, our P. comes out to be 0.4855. So that answers the part A of our question. Now part B is asking us to find the margin of air E. And we can either use the information from the lower bound or the information from the upper bound to do this. I'm going to do the example by using the information from the upper bound. Okay, so when I go and use this from the information from the upper bound, I have P hat plus E is equal to that 0.509 But in part a we just saw that P. is 0.4855. Yeah. So plugging that in for the p hat 0.4855 plus E. Is equal to 0.509. And when we subtract mhm The 0.4855 from both sides, we get that E. is equal to 0.0235. So that is our margin of error. And lastly, part C is asking us to find the number of individuals in the sample with a specified attribute X. So when we want the number of the individuals with the specified attribute, that number of individuals acts, we want to recall that our point estimate P had. If I didn't have the numerical values of the lower bound in the upper bound, I would have found by the number of the individuals that had the attribute divided by the total number in the sample end. But in part A we found that P hat is 0.4855 Is equal to x divided by. And they told us in the problem that N is 1680 Solving for X. We're going to multiply both sides by 1680. And we get that our X. is equal to comes out to be 815.64, but X is a number of individuals that have it. It's got to be a whole number. So we are going to round that 815.64 up to 816. Your X has got to be a whole number and these three values then answer each of the parts of the question asked in this example.

Mhm. When we look at a confidence interval for a population proportion and they give us the lower bound and the upper bound of the confidence interval, we want to recall that the confidence in a vote for a population proportion was formed by taking P hat your point estimate of your population proportion minus E. Representing the margin of error as your lower bound to P hat. The point estimate of the population proportion plus E. The margin of air. So your lower bound is that P minus E. And the upper bound is that P hat plus E. So when they give us an actual value for our lower bound of our confidence interval and our upper bound of our confidence interval of our population proportion, then we can be asked to find the point estimate and the margin of error through recognizing this way that it's organized to develop that confidence interval. So are lower bound. The 0.853 Is our P -E. to our upper bound of 0.871 is from the P hat plus E. So notice that that means that you started with P hat you subtracted E to get the lower bound and you added E to get the upper bound. Soapy hat must have been the value that's right in the middle of those two numbers. So to answer part A of this question, find the point estimate of the population proportion and that point estimates denoted P hat we can find P hat when we know the lower bound in the upper bound by just taking the lower bound, adding to the upper bound And dividing by two. Because that allows us to find the value that Right in the middle. So that gives me a value of P. is equal to 0.862. And that is the first part of our answer to this question. Now, how about Part B. Part B asks us to find the margin of error. Well, we notice that for part B to find the margin of air, I could actually either work with the lower bound information or work with the upper bound information. And I'm going to work with the upper bound information just as a matter of that's what I'm choosing to do. And it is a little bit more straightforward with that. So this P hat plus E being the upper bound and them telling us that the upper bound is 0.871 gives me the equation. The p hat plus E Must equal 0.871. But we know from part A. The P hat is 0.862 So when I take out the p hat and put in the 0.862 then plus E. Is equal to 0.871 I can solve for E by subtracting The 0.862 from both sides. And we get that E. is equal to 0.009. And that is our answer to part B asking us to find the margin of error. E. Then finally for part C. For part C, we need to go back to the definition of how we normally find a point estimate for our population proportion. If we just no information about the sample and we don't quite have the confidence intervals lower bounder upper bound yet. So recall that P hat is found by taking X. The number in your sample that has the indicated characteristic and dividing it by N. The number in your sample. Now from the work that's already been done in this problem, P. is 0.862 X. Is what we're trying to find in parts C. So we'll still call that X. And then they told us in the setup of the problem and Is our 10,732 right here in the set up of the problem so we can come down to our equation that we have. It's Our .862 is equal to X over 10732. And to solve for X, we are just going to multiply both sides by 10,732 and get that are X is equal to And it comes out to be 9250.984 but X has to be a number of individuals in our sample that have that characteristic. So we have to round that to a whole number and we get that that is 9251. Mhm. And these three pieces together make give you the solutions to this problem. Uh huh. What?

All right, This question wants us to look at how the standard air changes based on our sample size. So recall that the standard air of X, it's just the population standard deviation divided by the square root of the sample size. So heart, eh has a sample of 50. So our standard air equals 25 which is our population standard deviation over the squared of our sample size. And that works out to be 3.535 Then Part B asks for an equal to 100 which again, our standard error Is there a population sigma over the square root of our sample size, which is in this case equal to 2.5. Then for an equal, the 1 50 same thing Sigma divided by the square root of our sample size, which this one works out to be. 2.412 Then finally, Part D wants us to bump up the sample size to see 100. So standard error is our population standard deviation bye bye squared of our sample size, which is in this case 1.7678 So now let's look at the trend So if we look we start at 3.5, then is our sample size goes up, goes down to 2.5. That goes down to two point. Oh, then it goes down to 1.76 So what can we say as the sample size increases the standard air decreases? Which makes sense because as you get more and more samples, you expect them to be less variable overall. And this will come into play later when we look at calculating probabilities from sampling distributions and why we like bigger sample sizes because they're more representative of the


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