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The 2014 race for Governor was a tight one. A pollof 802 registered voters conducted in January 2014 hadthe incumbent Governor at 47% and his challenger at 41%. The...

Question

The 2014 race for Governor was a tight one. A pollof 802 registered voters conducted in January 2014 hadthe incumbent Governor at 47% and his challenger at 41%. The pollalso revealed that voters have strong views of the incumbent: 49%viewed him favorably and 44% had unfavorable views. The challengerwas not familiar to the majority of voters polled, but 12% had afavorable view of the challenger and 18% had an unfavorable view.The poll had a margin of error of plus orminus 3.5 percentage points. A

The 2014 race for Governor was a tight one. A poll of 802 registered voters conducted in January 2014 had the incumbent Governor at 47% and his challenger at 41%. The poll also revealed that voters have strong views of the incumbent: 49% viewed him favorably and 44% had unfavorable views. The challenger was not familiar to the majority of voters polled, but 12% had a favorable view of the challenger and 18% had an unfavorable view. The poll had a margin of error of plus or minus 3.5 percentage points. About 40% of the poll was conducted on cellphones. Compare the margin of error to what you obtain from the Margin of Error Rule of Thumb. (Assume that the level of confidence is 95%, and round your estimate for the margin of error as a percentage to one decimal place.) The result for the margin of error is?



Answers

A poll for the presidential campaign sampled 491 potential voters in June. A primary purpose of the poll was to obtain an estimate of the proportion of potential voters who favored each candidate. Assume a planning value of $p^{*}=.50$ and a 95$\%$ confidence level.
$$\begin{array}{l}{\text { a. For } p^{*}=.50, \text { what was the planned margin of error for the June poll? }} \\ {\text { b. } \text { Closer to the November election, better precision and smaller margins of error are desired. Assume the following margins of error are requested for surveys to be conducted }} \\ {\text { during the presidential campaign. Compute the recommended sample size for each }} \\ {\text { survey. }}\end{array}$$

The following is a solution to number 25. And this looks at the 2004 election. This is before It occurred. And there were there's always pollings with, you know, a presidential elections and this that George W. Bush at 49% of the popular popular vote And John Kerry at 47% of the popular vote. But they both had a margin of error of 3%. And that's typically what you're gonna see on any sort of election. But especially whenever it comes time every four years for the presidential election, it's always such and such a percent plus or minus a margin of air. And we're supposed to use the confidence interval to explain why this particular race at the time was too close to call. And um, a few ways that you could kind of go about this, but the Bush confidence interval, let's just make a confidence in the real world quick for the popular vote. So 49 plus or minus 3%. That's what that margin of error is, means that we can be however much confident, I don't know what their confidence level was, that the popular vote will be between 46% And 52%. So if you notice that Uh does not give a conclusive now, you can go into arguments about electoral vote or popular vote, but 46%, that's less than 50%. So there's, you know, at least an okay chance that he wasn't going to win this, or at least it didn't look like he was going to win this in 2004. Or he could, you know, the other side of that, he could go up to 52%, which means John Kerry would get 48%. So then we have the Kerry john Kerry confidence interval And his was 47 Plus or -3%. So 47% plus or -3%. So that confidence interval 47 -3%, that's 44%. And then 47-plus 3% would be 50%. So you can see here too, you know, he may get 44% of the popular vote and then Bush would get 56%, you know, I guess depending on other candidates but probably about 56% of the popular vote. Or he could go all the way up to 50% of the popular vote which would basically make it a tie. And also no or just look at there's some overlap here So they both contain 50%. So that's probably the main reason here. There. Yeah, is overlap in the two confidence intervals. Both mm containing 50% of the vote, 50% of the popular vote. So it's too close to call. Okay, so that's that's kind of the reason. So there is some overlap, but they also they both contain 50%, which would be the majority of the vote for the popular vote. Um, so whenever you, whenever presidential season comes along, there's always going to be, you know, poll after poll after poll, you know, probably a couple times a week that you'll see it if you watch the news and there's always that margin of error in there. And, um, oftentimes these pollsters don't get it right to, so you always have to look at these with a certain level of uncertainty whenever there's there's not a good scientific poll out.

Sometimes we want to see whether we can say that one population proportion is significantly larger than another population proportion. Just basing it off of our estimates of our proportion. Using the idea of a confidence interval For this example, I have a candidate, one who has a point estimate of 0.49. A second candidate that has a point estimate of 0.47. And we have a margin of error. That's three Translating that into symbols. We could say that E because that's a symbol for our margin of error is equal to 0.03, changing our percent into its dismal. Now, let's see what would happen if we found a confidence interval for each of these candidates. So the confidence interval for uh population proportion is P have minus E for lower bound and P hat plus E for our upper bound For candidate one, that would be a 0.49 -3.03. My lower bound and 0.49 Plus the 0.03 for the upper bound. So I would have a lower bound of 0.46 And an upper bound of 0.52. That means candidate wants population proportion. We have whatever percent confidence it would be for these values Is between our 0.46 And are 0.52. So there was a specific confidence level that they had when they went through and calculated the margin of error. They just didn't present it in the question. Now let's do the same thing for candidate to remember. The lower bound as P hap minus E. And the upper bound is p hat fussy. Mhm. So for candidates to I have 0.47 -0.03 and The 0.47 plus the 0.03. Which gives me a lower bound for candidate two of 0.44 And an upper bound of 0.50. Now, why would they say that this election is too close to call that we can't give one of these kind of being a significant lead from the other. Well these two confidence intervals overlap. So when you look at the start of the first candidate at 0.46 the start of the second candidate, 0.44 And then the upper bound of the first candidate is 0.52. And the upper bound of the second candidate is 0.50. So they have a big portion of their intervals that overlap. And so you can't say that one candidate has a significant lead over the other, it's too close to call. Mhm. Uh huh. Mhm. Yeah because, Mhm. Yeah. Yeah. The confidence intervals overlap. Yeah. Yeah. Uh huh mm. Yeah. Yeah. Yeah. Yeah. Yeah.

So the first part is going to gain 10 points and the second point we're going to gain two points. Part three will gain three points and then part for will gain one point. So we act this up so I will add this up for so that would be six and then plus 10 they'll be 16 points, so this will not be.

This is a problem that's looking at, uh, a survey from 1936. They're not a survey. An opinion poll. Um, looking at who was gonna win the presidency. This was right before the 1936 election. This is when, uh, FDR ended up winning the presidency. He was re elected, but there is a magazine titled The Literary Digest that was releasing the results of the opinion poll. And they said that the Republican candidate was gonna win and not FDR. So if we look into more detail into the actual opinion poll, okay. They sent postcards to about 10 million people, okay, That that were potential voters, right? And these voters were found from various lists of things, so they found people that had subscribed to the magazine itself. They looked at some phone lists, um, automobile lists and club lists. I'm a variety of different places, and they sent out all these postcards to voters. As a result, they got about $2.3 million. Um, not million dollars million people to return them. Okay. And the question is, um, just kind of analyzing the sampling right in the process and seeing which which problems may exist. So the first question is looking at the context right of being in the 19 thirties in the United States. So thinking about historical events that were happening during that time, this was kind of right in the middle of the Great Depression. Ah, lot of people were struggling financially, right? And not great situations. So if we think about the sample chosen from places like magazine list for a literary magazine, right in academic, um, Mawr academic based magazine, right? Automobile list. These are all things that are kind of signs of wealth, right? If you belong to these, um, you you had a lot of money in your family to be a part of these different clubs to have your have a phone in your house that was working, right? Thio have your own automobile. These were all big deals, especially in the midst of the Great Depression. So the amount that the sample right of people that they were choosing was not representative of the population, and the reason is because the population, the majority, were struggling. Right? So the choosing just kind of the the list of people from all of these more higher class types of groups. Um, is not representative of the entire population. Okay, So looking at at our response for a three ideas that in the middle of the Great Depression, right, mhm, uh, there were many financial struggles and be less used to books used to sample, um, hold from groups. Not only be wealthy, be a part of. Okay. So, again, contextually, we're in the middle of the Great Depression, right? We have a lot of financial struggles in the nation, so pulling from things that would be a sign of wealth, Right? We're not representative of the entire population were pulling from a very small group of people. How does the low response rate relate to reliability? So we got a little over two million responses of this. 10 million, um, 10 million postcards. Right? That went out to all these people. Because it's such a low response rate. It affects the reliability, right? So the lower the response rate, we could say it this way. So, um, weaker reliability. Right? So because we don't have as many people responding, even though the sample isn't great, right? Um, the lower people, we lower amount of people we have responding, the weaker It's gonna be right. We can't really rely on this And the data. Once we we try to analyze the data and come up with some conclusions and results, it's gonna be pretty severely skewed and again not representative. Because of the few amount of people that responded, Um, the following question is then looking into okay, is this a sampling error or non sampling error? So there's, um, some information in your textbook, um, about these two ideas, this is looking at, um, the idea of issues with sampling, right, would be sampling errors. So if you don't have a big enough sample or there's, um, errors and where you hold your sampling from right, those types of things would be sampling errors. Um, really non sampling errors is just something may be related to the process. So maybe there's some sort of issue with, um, the device that was counting the data or in calculating the actual results. Right. But the actual sampling was okay. Eso in this case, um, this would be an example of a sampling error, and the reason is we have a low response rate, right, and it's non representative of the population. Because of those two things, we kind of have a sampling error, right? It's not non sampling because it's related to the group of people in the responses we got. Um, And then it talks about a poll by Gallup, where they use something called quota sampling. And as it's described in the question is looking at quota sampling is getting, um, survey results from specific subsets of the population. So the way this is described in the textbook is stratified sampling. I'll type it in here. Stratified sampling. So stratified sampling is this idea that, um, we group the population into different strata? OK, eso from each of those we take a specific number or group of people. Hey, so the idea with the Gallup Poll is that they were obtaining survey answers from specific groups of the population, So they took the population divided into groups and then, from those groups, took a certain number of people from each of those groups to be ableto have a sample representing all of those groups of people within the population. So that's exactly what stratified sampling explains. Right? Stratified sampling is the idea that we divide into the strata. And then we take a group of people or an amount of people from each of them. OK, Thea, other things mentioned, which are not the case here would be cluster sample and systematic sample those of the other types of sampling that were discussed in this lesson. Um cluster sampling was you again kind of divide the population into different groups or clusters, but then you only select certain clusters, right? So rather than taking a few, um, not a few. A handful of people from each of the cluster's right. They're only taking a couple of the clusters and using all of them rather than, uh, taking a group from each of them. Okay. And then systematic sample would be be the idea of just kind of taking a list and picking like every third person on the list, right to be a part of your sample. Eso again, That's not the case here. So the case here with quota sampling that Gallup did with his poll of voters, was the idea that we're taking everybody in the population, right? We divided them in different subsets, but we're making sure that we pull from each of those subsets. So as it's described in this module, that's the idea of stratified sampling


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