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You may need to use the appropriate appendix table or technology to answer this question_A researcher reports survey results by stating that the standard error of t...

Question

You may need to use the appropriate appendix table or technology to answer this question_A researcher reports survey results by stating that the standard error of the mean is 20. The population standard deviation is 400_(a) How large was the sample used in this survey? 400(b) What is the probability that the point estimate was within +35 of the population mean? (Round your answer to four decimal places:)0.5468

You may need to use the appropriate appendix table or technology to answer this question_ A researcher reports survey results by stating that the standard error of the mean is 20. The population standard deviation is 400_ (a) How large was the sample used in this survey? 400 (b) What is the probability that the point estimate was within +35 of the population mean? (Round your answer to four decimal places:) 0.5468



Answers

A researcher reports survey results by stating that the standard error of the mean is $20 .$ The population standard deviation is 500 .
a. How large was the sample used in this survey?
b. What is the probability that the point estimate was within $\pm 25$ of the population mean?

We have a population that has a mean of 400 it has a standard deviation of 100. Now, we don't know what the shape is of this distribution, and it actually doesn't matter because we're taking a sample that is huge sample size of 100,000. And then we're going to take these numbers and then find the mean of those numbers and we want to find in part a what the expected value is or what the mean is for the X bar distribution. And that would be 400. And the idea is, if you continue to take the samples of size 100,000 and get an X bar and take another sample of size 100,000 and get an X bar and continue to do this over and over again, the mean of these X bars and do this infinitely many times the mean of these X bars would end up being about 400. And how will these X bars vary? Well, they're going to be actually very close together because the standard deviation of X bar will end up being because of the central limit theorem. We'll end up being the standard deviation of the individual population divided by the square root of em. And when you hit that on the calculator, I believe you get about 0.316 So about 0.3 and then part, she says. What is that sampling distribution going to look like? Well, the sampling distribution will be approximately normal because of the central Limit theorem. So that's because of the central limit theorem. Okay and Central Limit Theorem says. As that sample size gets larger and larger, larger, that these X bars will tend to form a normal distribution. And I'm actually gonna draw a picture of that distribution that distribution of X bars and remember, each one has to come from a sample size of 100,000. It's going to end up centering at 400 and then a standard deviation is where this inflection point takes place. So this would be about 400.3. This would be about 400.6, and this would be about 400.9 three standard deviations higher, and this one would be 399 0.7. This one would be 399 point four, and this one would be 399.1. And so that's the shape of the distribution will be approximately normal, centered at that 400 with this as a standard deviation. And what does that say about it? It basically that if you continue to take samples of size 100,000 that the X bars you get, it's highly unlikely you'll ever get an X bar that is lower than 399.1 or one that is higher than 400.9 and it will follow. Those X bars will be 68% of the time Will be this between the 3 99.7 and the 400.3. So as long as the sample sizes continued, we can predict how those X bars from the sampling distribution will look and they'll they'll have a normal distribution

In 31. It says a simple random sample of 400 individuals provide 100 yes responses. So I have that written over here to the side. In this 400 X is going to represent the number of yes responses, in which case we have 100 Part A says What is the point? Estimates of the proportion of the population that would provide yes responses. So the point estimate is pretty simple. In this case, point estimate is the number of yeses over the total, which is 100 over 400. You can write that is 1/4 or as 0.25 in Part B were asked, What is your estimate of the standard air of the proportion? So our standard error is going to be square root of P hat times one minus P hat. You can also call that Q hat over in p had is a point estimate. We know that from earlier, So we have 0.25 times 0.75 over 400 all under the radical, and that is 0.217 And lastly, four hour Part C says compute the 95% confidence interval for the population proportion. So our confidence interval here is our point estimate. Plus and minus are critical value times that standard air. So a point estimate we found earlier to be 0.0.25 plus and minus are critical value. In this case, we have a, um z Asterix. Um because we have a proportion we're gonna treat all of our critical values. Azeez So is the critical value in this case. Because it's a 95% confidence interval. We're gonna use 1.96 and over time, as you do, many of these problems will memorize that a 95% confidence interval uses a Z critical value of 1.96 times that standard air we found in B. We get to five plus or minus 50.4 24 So subtracting those two numbers, we get 20.2076 and adding those two numbers, we get point 29 to 4 and that is our 95% confidence interval.

This question, We're told that he normally distributed population has a mean of 57,800 and a standard deviation of 7:15. And were asked for us for the probability, since this is a normally distributed population probability that a single randomly selected element is between 58 1,057 1000. So we convert this to our standard normal random variable And we when we do that conversion, we get c between .27 and -1.27 And we look for these two values in our table, so we get .6064 -14-3, Which gives us .4641. We're party part B find the mean and standard deviation for samples of size hundreds, So are mean Is basically our population mean, which is 57,000 800. Our standard deviation is basically 750 over squared of 100 which is 75 Part C. Find the probability that the mean of a sample of 600 Between 58,057,000, and we're going to use, our answer is calculated in part B to convert our X to the standard normal random variable, so we can see between 2.67 and -10.67. That's just probably dizzy, less than 2.67, Which is .9962.

All right. So we're given that this is a sample of the population were supposed to find the point estimate for the mean and standard deviation. So let's start with a mean, so that's just gonna be we're gonna add these up. And then there's six data points, so this equals five full status. 13. 13% is 23 30 40 54 over six. This equals nine. All right now, to find our standard deviation. Ah, that's got to be similar to our standard deviation calculation for populations. So right, he's all out. All right. Now, if this r X and we find X minus mu, you are not mu it be ex bar. Actually, in this case, you refers to a population. All right, so five minus nine. Negative. Four negative. 11 negative. 215 All right, Now we square it. So that's 16 114 Uh, sorry, hon. 25. All right, we add these up. He's at up to 16 17 18 22 23 2025 23 is 48. Ah, we divide that bye and minus one. Since this is samples of 48 divide by six minus one because 48 over five, which is Ah, 9.6. Take the square root of that. That's approximately equal to 3.0 984 And there you have it.


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