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Use the given data to find the minimum sample size required to estimate population proportion or percenlageMargin of error: 0.03; confidence level 90%; and unknown(...

Question

Use the given data to find the minimum sample size required to estimate population proportion or percenlageMargin of error: 0.03; confidence level 90%; and unknown(Round up lo Ihe nearest intoger: )

Use the given data to find the minimum sample size required to estimate population proportion or percenlage Margin of error: 0.03; confidence level 90%; and unknown (Round up lo Ihe nearest intoger: )



Answers

Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, $x,$ for the sample size provided. Lower bound: $0.853,$ upper bound: $0.871, n=10,732$

Mhm. Confidence intervals for a population proportion have the form of the point estimate. P hat minus the margin of error. E being the lower bound to the point estimate. P hat plus the margin of air being the upper bound. And that's important information to know when asked a question such as this Says a confidence interval of a population proportion. So it is asking us to find a confidence and has the confidence interval of a population proportion has a lower bound of 0.051. So they're telling us the lower bound and an upper bound of 0.074. So they're also telling us the upper bound And it has a sample size of N is equal to 1120. Now that information is given to us in this particular question and from that it wants us to find first the point estimate of the population proportion P hat 2nd the margin of error. E and then third the number of individuals in the sample with that attribute or characteristic, which is denoted with the um notation X. So since our confidence interval for population proportion is P hap minus E. Is your lower bound in this problem, That is the 0.51 So this lower bound is 0.51 To the upper bound is the 0.074. Now, since we have that information and the first thing it wants us to find is the point estimate of the population proportion. This p hat, well it's P hat minus C. To get the lower bound P hat plus E. To get the upper bound soapy hat is just going to be the value that's in the middle of the lower bound and the upper bound. So to find P hat, we're going to take that 0.51 Add it to the 0.074 and divide by two. And we see that p hat. When we do that calculation comes out to be 0.625 So that's your first part. You're a part of the calculation. Yeah. Now the next thing it wants us to find is the margin of error. E. And to find just you know what this part A. To find part B. We noticed that P hat plus E is equal to that 0.74 Mhm. Yeah, so our P hat plus E is equal to the 0.74 But from part a we know P. is 0.0625. So we are going to plug that in for the p hat into our equation. And solving for E. We just subtract the 0.0625 from both sides and get our answer to be is that E. Is equal to 0.074 zero Point 6-5. And we get 0.0115. So that's our answer to part B. And then the last part C. Is asking us to find the number of individuals in the sample with the attribute X. Well, this requires us to think back to what the definition of P head is and P hat is X divided by N. The number in the sample with that characteristic divided by the number in the sample space. But we're going to plug in the numbers that we know and solve for what we're trying to find. P hat. Remember we found in part A. That is the 0.625 is equal to X they want us to find. So we're just going to write the letter X. And then they told us Was 1120. So solving for X, we will multiply both sides by 1120. So 0.0625 times the 1120 is what we get for X. And as we calculate that through our X will be equal to 70. So our point estimate P yet Is 0.625. Our margin of error E is 0.0115. And the number in the sample with that characteristic is 70.

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.

In this problem, you're being given a confidence interval and being asked to find the margin of error and the sample proportion just to recap. Remember that the confidence interval goes from the sample proportion, minus the margin of error to the simple proportion, plus the margin of error represented. Visually, we have here we have this. So let's say this area is your confidence interval. At the very middle, you have your sample proportion, and on either side you have your margin of error. Now we're given the confidence interval of 0.245 to 0.475 and defined the sample proportion. We can see that it's exactly in the middle of your lower bound and your upper bound. So we simply find the average, which weaken do by adding 0.245 plus 0.475 and dividing by two, which gives us 0.36 now to find your margin of error. You can see that within the confidence interval, there are two times the margin of error. So if we find the difference of the upper and the lower bound by doing 0.475 minus 0.245 and divide this by two. Since we know there's two times the margin of error between them, this tells us that the margin of error is equal to 0.115 and we can verify that these answers are correct by constructing the confidence interval from these values. So if we reconstruct the confidence interval, okay, because we'll see that we do in fact get the confidence in the role that we started with. So 0.36 is your sample proportion and 0.115 is your margin of air.


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