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The mayor of a town has proposed a plan for the annexation of anadjoining bridge. A political study took a sample of 1700voters in the town and found that 38% of th...

Question

The mayor of a town has proposed a plan for the annexation of anadjoining bridge. A political study took a sample of 1700voters in the town and found that 38% of the residentsfavored annexation. Using the data, a political strategist wants totest the claim that the percentage of residents who favorannexation is over 35%. Testing at the 0.05 level, isthere enough evidence to support the strategist's claim?Step 1 of 7: State the null and alternative hypotheses.Step 2 of 7: Find the value of t

The mayor of a town has proposed a plan for the annexation of an adjoining bridge. A political study took a sample of 1700 voters in the town and found that 38% of the residents favored annexation. Using the data, a political strategist wants to test the claim that the percentage of residents who favor annexation is over 35%. Testing at the 0.05 level, is there enough evidence to support the strategist's claim? Step 1 of 7: State the null and alternative hypotheses. Step 2 of 7: Find the value of the test statistic. Round your answer to two decimal places. Step 3 of 7: Specify if the test is one-tailed or two-tailed. Step 4 of 7: Determine the P-value of the test statistic. Round your answer to four decimal places. Step 5 of 7: Identify the value of the level of significance. Step 6 of 7: Make the decision to reject or fail to reject the null hypothesis. Step 7 of 7: State the conclusion of the hypothesis test.



Answers

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.
Adults were randomly selected for a Newsweek poll. They were asked if they "favor or oppose using federal tax dollars to fund medical research using stem cells obtained from human embryos." Of those polled, 481 were in favor, 401 were opposed, and 120 were unsure. A politician claims that people don't really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Exclude the 120 subjects who said that they were unsure, and use a 0.01 significance level to test the claim that the proportion of subjects who respond in favor is equal to 0.5. What does the result suggest about the politician's claim?

All right. So what do we have this time? The types off raw materials used to construct stone tools they were found at an archaeological site called Casa de Rito are shown to us. They're given to us. We have a table and a random sample off 1486 stone tools was obtained from an ongoing excavation site. Now, what we have to do is we have to use a 1% level of significance. Okay, 1%. Level of significance means what are ALPA is 0.1 okay, and one person level of significance to test the claim that the original distribution off raw materials fits the distributions at the current excavation site. So what is going to be a little hypothesis in this case, Arnel hypothesis is that the regional distribution of raw materials that the regional distribution off raw materials fits the distribution of the current excavation site? It's the distribution. It's the distribution off the current excavation site off the current excavation site. This is what we're going to write in an AL hypothesis. And what do we write for our alternative hypothesis? It will be that the regional distribution off raw materials off raw materials does not fit. Or I can say the reason of distribution of raw materials and the distribution off. The current excavation sake are not the same. Our e can say that they are not similar, that they're not similar. Okay, All right. Now what do we have now? We have a table that is given to us. So let's just look at the table closely. We have raw material. Okay? We have raw material. One of the raw materials that we have The first one is Basil. The next one is obsidian. Then we have welded tuff welded tuff. Then we have paternal short. Or let me just write. This is PC. Okay. And then the last one is the other category. The other category. Okay, then we have reasonable percent off stone tools. All right, then. We have the regional percent off stone tools. Okay, on the stone tools. All right. This is 61.3%. 61.3%. Then we have 10.6% 10.6%. Then we have 11 point 4%. Then we have 13 points, 1% and then we have 3.6%. All right, Now, what is the observed number of tools at current excavation site? So let me just write this as the observed values, the observed values. Okay, these are 906. 906. Then these are 1 62 then these are 1 68 then 1 97 and 53. Now, what is the total? I need this total because this is going to be my sample size, my total sample size. And this student is already given to me as 1486 Right. Okay, so this is 1486 Okay, now, the first step in calculating the guys question district is goingto be to find the expected value e. Now, how exactly do I find the expected value? How do I find the expected values? These are going to be for each category. Expected values are given by the sample size in the sample size that is n multiplied by the probability off its category or the proportion that we have. Right. So the probability the probability off each category off each category. Okay. All right. So what is my sample size? My sample sizes. This oil 1 46. Okay, these are going to be the expected values. Okay, these will be the expected values. All right. So what is going to be the expected value for the first category? If I take my calculator, this is going to be in. That is one for 86 multiplied by now. I want 61.3% off 1486 So this is going to be 61% 61.3%. 0.631 This is 9. 37.666 Or let me describe this as nine. 37.67 Okay, the next one stand 10.6% off for 1/4. 861486 multiplied by 10.6. Divided by 100. This is 1 57.516 1, 57.516 Okay, then we have 11.4% off. 1486 multiplied by 0.114 This is 1 69.4041 16, 9 0.404 Then we have 13.1%. 1486 multiplied by 0.131 Right. That in 0.1% this is 1 94.666 Recognized this as 1 94.67 Yeah, and then I have 3.6% or 0.0 36 multiplied by 1486 This is 53.496 This is 53.496 but I could access 53 point Faith. Okay, These are my expected values. Now I want to find the guys question distinct. This column is going to be for individual chi square values, right? For individual chi square values. Okay, Now, what is the formula for Chi Square statistic? The overall case question The stick is going to be first one. Find the individual high school values of the formula for that is observed minus the expected, or we consider the difference of the observed and expected square upon the expected value. We do this for all the categories and in the end of the some of them, all up, and this will give us the highest question to stick for our problem. Okay, so let us do that. Now. The difference is observed and expected. This is 9 37.67 minus 906 I found this. Now I squared the study 1.6 71.67 and I divide this by 9. 37.67 9. 37 point 67 Okay, so this is 1.696 or I could write. This is 1.71 point 07 Okay, moving on to the next category. This is the difference between 1. 62 1. 57.516 We square this 4.484 and divide this by 1. 57 point 516 which is one point. Sorry. This is 0.12764 I can write this as 0.13 Okay, then we have 1 68 and 1 69.404 We square this, okay? And then we divide this by 1. 69.0 40.1 16 So this is 0.11 Okay, let me just write this much. Many of the difference between 1 97 and 1 94 minus 1 94.67 We square this 2.33 and divide this by 1. 90 4.67 which is 0.278 or regulate this as 0.3 Then we have the difference between 53.5 and 53. We square this and we divide this by 53.5. So this is 0.0 467 reconnected says 0.0 05 All right, now I need to some all of these individuals high school values, right? We saw, were you? The submission is going to give us the final chi square statistic. So this is 1.7 Yes. 0.13 less 0.11 plus 0.3 plus 0.5 Now, this is 1.246 on my guys. Question stick is 1.246 All right, so now we have this s one point 1.246 All right, now, what is do we need now? We need the degrees of freedom that is DF now. This is given by number off categories. Number off categories minus one Know how many categories do we have? If you look away? We have basal obsidian welded tuff than PC or spc Paternal short on the other. So these are the five categories, so number of categories is five five minus one is four. So this becomes four. So my diesel fuel it was for okay. Now you can either use a chi square table and with the help of the table, you will get a P value range off the values. You will not be able to find the exactly value in order to find exactly value. You need the guys for statistic, the degrees of freedom. And along with this, you need a statistical software like Excel or SPS or our python or whatever statistical tools that you're using. Like over here. I'm using an online calculator, so this will give me an exactly value. So my thigh square statistic is 1.26 If I'm not mistaking 1.246 1.2 for six and my degrees of freedom is five minus one, which is four. Okay, My significant level. There's al 50.1 Andi, I calculate this and write P value is coming out to be 0.87 my p value. My peopling is 0.87 My Alfa was 0.1 right? Yeah, I also was 0.1 All right, So what I can say over here is that since my P value is much greater than Alfa right, I can say that I will fail to reject that. I feel to reject my null hypothesis. H not right. So I can say that I do not have I do not half enough statistical evidence enough to this critical evidence to suggest to suggest that to suggest that the regional distribution off raw materials that the regional this line is very important. This is the answer the regional distribution, the reasonable distribution off raw materials. We'll draw materials and the distribution and the distribution from the current excavation site from the current excavation site and the current excavation site are different. Okay, so this is my answer. I fail to reject my inner life. But this is aged not And I say that I do not have enough statistical evidence to suggest that the regional distribution off raw materials and the distribution from the Context Commission site are different. So this is an ally prosthesis. We're saying that it fits, and the alternative is that it does not fit right. So we do not have enough evidence to suggest that the distributions don't fit. This is going to be an answer.

In this exercise, we are talking about bounds for proportion. Confidence intervals, he found, is the maximum margin of error that will exist in the conference interval for a given confidence level for a given sample proportion and for a given sample size. Now for part A were asked that for a given bound for the margin of error of a proportional confidence interval, we want to show that the required sample size for this bound is given by the equation shown below. So let's assume given bound that we want. So that is our margin of error and this is given by this formula, and this has just taken out of the formula for confidence intervals. We should end up with the required equation and so we can see that we have shown what we set out to show next for Part B. We are told that we believe that proportion is to over three and we want to know how large sample size we would need in order for the margin of error to be at most 0.5 assuming a 95% confidence level. So this means that we want our bound to be 0.5 and for 95% confidence interval are critical. Value is 1.96 and although we haven't drawn a sample yet, we're trying to determine the size of the sample that we need to draw. It is believed that the proportion is two out of three, so we can set our sample proportion two of her three. And now we can make use of this equation to find the necessary sample size. So this comes out to 341.48 And since we can only sample integers, that means that we should choose the next highest integer above that number, which is 342. So therefore, in this situation to have a balance of 0.5 on the margin of error, the sample size that we must select is 342 and then for part C, we are asked what is the necessary sample size to assure that the width of a 95% confidence interval is at most 0.1. So if we have a confidence interval like this, this is the width of the interval, and the bound is half the width, so the bound is the upper bound of the margin of error. So this formula pertains to half the width, which means that our bound is 0.5 So in our formula we know the Critical Valley. That's 1.96 because we're talking about a 95% confidence interval. We haven't specified a sample proportion, and we know that the bound is 0.5 to be conservative. We want to specify the maximum and that's needed to have the necessary bound that and will be maximized when this product is maximized. And it turns out that that product is maximized when P is exactly exactly one half. So. Therefore, we have all the parameters necessary to put into this formula. This comes out to 384 0.16 so we round up and we choose and it was 385. And so to ensure that our confidence are 95% confidence interval with is at most point one, we must select a sample of size 385 or greater

We have a sample with N equals 83. And of that 83 members of the sample 64 satisfy a certain condition. Want to meet our equal 64. Now we want to use the sample data to test the claim that p does not equal 2.75 for the population at a confidence level of 5% or alpha equals 0.5 Now that we've identified the confidence level, we can go to the following procedural steps to conduct this hypothesis test first. Is the normal distribution appropriate to use? Yes, it is. Because both N. P and Q. P. R greater than five. B. One of the hypotheses were testing we're testing whether or not H not P equals 10.75 or H a p does not equal 0.75 Ak We're conducting a two tailed test. Next compute P hot. And the test statistic P hot is all over. N equals 20.77 Z is given by the formula on the right, taking in as input P hat P Q n N producing down to Z equals 0.42 Next we compute the P value. We use the table to see the area outside Z equals plus and minus 0.42 as we shaded in yellow and the graph on the right. This P value corresponds to 0.6745 Next we reject H not, no, we do not. P is greater than alpha and we interpret this signing to suggest that we lack evidence that P does not equal .75.

We have a sample with n equals 493 members. And of those 493 members. 136 of them satisfying certain conditions want to meet our equal 136. We want to test to clean the population proportion P is greater than .214 With a confidence level of one of alpha equals 0.01. Now that we've got to find the confidence level, we have to proceed to the following steps to conduct this hypothesis test first is appropriate to the normal distribution. Yes, it is. Because both MP and QPR greater than five. What are prophecies that we're testing I know is that people's 0.214 are alternative. Is that P is greater than 0.214 Ak We're conducting a one tailed test. Next compute P hot. And our test statistic P hot remember is just all over end 10.276 Z is equal to p hot minus pee. All over route PQ over end. As we see on the formula on the right, giving Z equals 3.35 Next compute the P value associated this test statistic the P value. We can identify an easy table corresponds to the area under the normal curve to the right of Z equals 3.35 or P equals 0.4 We've graphed this also on the right to demonstrate how this looks on the distribution. Next we reject asian all based on this P value. Yes, we do. He is listening to alpha and we interpret that to mean that we have evidence that P is greater than .214.


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