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Question 113 ptsLet the population proportions of success in Population and Population be P1 and p2, respectively: Based on two independent random samples of size n...

Question

Question 113 ptsLet the population proportions of success in Population and Population be P1 and p2, respectively: Based on two independent random samples of size n each; one drawn from Population and the other drawn from Population the sample proportions are p1 0.6 and p2 0.3, respectivelv- It is known that the margin of error for the 95% confidence interval of P1 Pz is 0.098. Find the value of n (select the closest answer):Hint: | anorc0.05_ loner-F)645 anoringu 625 lower=F)~96192180

Question 11 3 pts Let the population proportions of success in Population and Population be P1 and p2, respectively: Based on two independent random samples of size n each; one drawn from Population and the other drawn from Population the sample proportions are p1 0.6 and p2 0.3, respectivelv- It is known that the margin of error for the 95% confidence interval of P1 Pz is 0.098. Find the value of n (select the closest answer): Hint: | anorc0.05_ loner-F) 645 anoringu 625 lower=F) ~96 192 180



Answers

Consider the following results for independent samples taken from two populations
$$
\begin{array}{ll}{\text { Sample } 1} & {\text { Sample } 2} \\ {n_{1}=400} & {n_{2}=300} \\ {\overline{p}_{1}=.48} & {\overline{p}_{2}=.36}\end{array}
$$
a. What is the point estimate of the difference between the two population proportions?
b. Develop a 90$\%$ confidence interval for the difference between the two population proportions.
c. Develop a 95$\%$ confidence interval for the difference between the two population proportions.

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. 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.

Will be conducting a two proportion Z test and also uh confidence interval for the difference in the proportions. Let's start with our sample proportions in group one. The successes out of the sample size is 15 out of 24.75 in group to our sample proportion is 0.6 and are pooled. Sample proportion is going to be 33 out of 50 or 500.66 We'll use that in our test statistic formula in a minute. Now let's check to make sure our technical conditions are met, we're told these are simple random samples and they are independent. The two successes are both bigger than five and the sample sizes minus their respective proportions or minus their respective successes are also five or greater. So these are also met. Now for our test, the null hypothesis is that the two proportions from the populations are equal. And the alternative hypothesis is the population proportion number one is bigger than the population proportion. Number two, bigger than because we want to write tail test. Now I've plugged in our sample statistics into our test statistic formula and this gives us a Z score of 1.1. And when you reference appendix, the appendix in the back of the book for the normal probability table, the p value is 0.8643 But because this is a right tail test, I'm going to do one minus 10.8643 which will give me a p value of 0.1357 which is larger than our significance level of 0.5 So we are going to fail to reject H. L. Meaning that we still believe that to population proportions are the same. Al for our confidence interval there's our formula. I've plugged in my numbers into the formula. Note here that at 90% are test statistic is 1.645 giving us a margin of error of 0.217 And when I add and subtract that from the difference in the sample proportions of 0.15 we get an interval from negative point oh 672 positive 6720.367 so we are 90% confident. The difference in population proportions lies somewhere between these two numbers.

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?


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