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Homework: Proportion Testing for True Proportion Score_ 0 of 25 pls complete) 7.4.13-TSaHW Score: 0%, 0 of 100 ptsCuestion HelpAteseaicn centel claims Inal 3183 adu...

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(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value and identify the rejection region, $(c)$ find the test statistic $F,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, the populations are normally distributed, and the population variances are equal. If convenient, use technology. The table shows the salaries of a sample of individuals from six large metropolitan areas. At $\alpha=0.05,$ can you conclude that the mean salary is different in at least one of the areas? (Adapted from U.S. Bureau of Economic Analysis) $$\begin{array}{|l|l|l|l|c|l|}\hline \text { Chicago } & \text { Dallas } & \text { Miami } & \text { Denver } & \text { San Diego } & \text { Seattle } \\\hline 43,581 & 36,524 & 49,357 & 37,790 & 48,370 & 57,678 \\37,731 & 33,709 & 53,207 & 38,970 & 45,470 & 48,043 \\46,831 & 40,209 & 40,557 & 42,990 & 43,920 & 45,943 \\53,031 & 51,704 & 52,357 & 46,290 & 54,670 & 52,543 \\52,551 & 40,909 & 44,907 & 49,565 & 41,770 & 57,418 \\42,131 & 53,259 & 48,757 & 40,390 & & \\& 47,269 & 53,557 & & & \\\hline\end{array}$$

Yeah. All right guys, the first thing we need to do here is we do identify on are null and alternative hypothesis are no hypothesis. Excuse me being that the variations are going to be the same. And the turn of prosthesis is that they're going to have different variations as you can see right here. Okay, so the next thing I did to solve the problem was I opened a google she and I am putting a lot of the data. You'll notice all the data is and put it right there. Okay. And then, so what I'm gonna do from there, what I'm gonna do from there is this I'm gonna go ahead and calculate the variances for each. The first variance I'm gonna calculate using google sheets. So I'm gonna type equals var open parentheses and then just highlight everything I want. I know have an extra box in there. That's not a big deal. And there is my first variant. Second variants will be found the exact same way it looks like it already knows what I want, which is great. And there is our second variance. Alright, so from there I need to find my f statistic. Okay, so my F statistic is just going to be the greater variance divided by the smaller variance and there we go, there's my f statistic. Once I found my F statistic, I'm ready to find my P value. Actually. Sorry, scratch that. Once I find my F statistic, I'm almost ready to find my P value. But there's one more step in order to find a P value. Using google sheets with an F statistic. First need to calculate degrees of freedom. Okay, very easy to do in google sheets. So degrees of freedom is just all the data, the amount of data you have minus one. So how do you find that? Google sheets is equals count. Just highlight everything. I have noticed how I do have an extra box there. That's not a big deal to subtract one. And that I have 20 in this case 24 degrees of freedom. And in our second case again equal count Open parentheses, highlight everything and I do need to subtract one. And the other case have 15 15 degrees of freedom. Okay, so now I'm ready to find my P. Value. So my P value uh through here is going to be something called an F. Distribution or F. Dist. And then open parentheses. I need to put three values. Number one, my F statistic. The second value is going to be the number of entries from S one squared. I'm sorry the degrees of freedom from s one square in this case is 24. And my last and final value is the degrees of freedom from my S two squared. So that's right here. I close all that up and there is my P value. However, I'm not quite finished yet. Okay, that is a P value for a one tailed distribution. What do I mean by that? That's a good P value. If that was a greater than or less than sign, this is a not equal to sign. Okay. That's not equal to sign. Which means um uh that which means that I have actually had the wrong P value right now. Since this is for a one tailed distribution and I need a two tailed distribution because they're not equal sign, I need to take this value multiplied by two. So equals I click on the value multiply symbol in google sheets. Asterisk hit A two and I'm done. That is my P value. Notice that P value is just a touch above 0.05, meaning I'm going to fail to reject the no hypothesis. I'm gonna fail to reject. And all hypothesis

Following is the solution in number 14 at one way Innova test. Uh and this is about the mean sales prices for three cities. And the null hypothesis here is that the mean sale prices are the same for these certain houses. And then the alternative is that at least one of them is different. The second step is to find the critical value and you need three pieces of information to find the critical value. One is your alpha, Your significance level, that significance level. In this case, that's usually given to you is 10. They also needed the degrees of freedom for the numerator, which is the number of categories in this case, the number of cities minus one. So there were three cities that we looked at minus one is two, so degrees of freedom for the numerator is two degrees of freedom for the denominator is the total number of data values minus the number of categories. So in this case, if you counted up those data values, there were 31 -3 cities that we looked at. So 31 -3 is 28. So that's what we need. So from there you can use a table or you can use software. I'm gonna use software. So I wrote a program and I called it inverse. F. I'm not going to show you how to write this program. You can youtube it if you wish. But um it makes Makes it easier for me. So the area is the alpha value. So we'll put in .10 for that. Degrees of freedom from the numerator was too. And then degrees of freedom for the denominator was 28. And that's going to spit out my f. star my alpha value and my uh critical value which is about 2.503. Let's call it 2.503 is my F. stars 2.503. So anything greater than 2.503. We're going to reject h not anything less than 2.503. And we're gonna fail to reject the null. The next step is to find the f statistic and you can do that manually but especially with bigger data sets that can be really time consuming. So I went ahead and punch this into stat. Edit. And these are my data values. So these are the I think these are in thousands of dollars but these are the mean sale prices and if you go back to stat and then tests and the very last one in nova And you put in your columns just make sure you separate them. This is on the T. 84 by the way but make sure you separate them by commas otherwise it's not gonna read it right so nova for those three columns and that's gonna give us everything we need to. The f statistic is about 0.966 Let's go and write that down. So 0.9 66 which is somewhere over here. So that lands in the non rejection region. So that's actually gonna tell us why our fourth step which is the decision and we're going to fail to reject H not since the F statistic is less than the critical value. Now you can also use the P value method that's what this second piece of information is good for it. Now this other stuff doesn't really matter. Um You can just kind of ignore it because this is really what we need. We need the F statistic and we need the p value and the p values pretty large. It's about 0.39 And what you do is you explicitly compare the P value with your alpha value. So the p value in this case is greater than your alpha value. 0.39 is bigger than 0.10. And any time you're P values greater than alpha, you failed to reject H nine. If it's less than alpha then you you reject. And then the final step is to conclude this, you know, with actual words and bring it back to the question at hand. And so what we're going to say is that there is not enough evidence or there is not sufficient statistical evidence. So there's not sufficient evidence to suggest that the mean sales price prices of houses In the three cities are different. Okay that's the five step in Nova process.

The following is a solution video to number 11 and it compares the well being index for different parts of the country, Northeast, ah Midwest, South and West. So the first step is to state the hypothesis. The null hypothesis is that the mean the means are all equals the mean for the Northeastern people is the same as the Southern people and Western people and so on. So the four means are the same. And then the alternative is that there they're not the same or at least one is different. The second step is that we need to find the critical value, critical value, I called it F Star. And there are three pieces of information we need to know and I wrote it, I wrote it down right here, We need the alpha and that's usually given to you at the .10 in this case. And then we need the degrees of freedom for the numerator and the denominator. So for the numerator, it's the number of categories minus once. There are four categories. The four regions of the country, Northeast, Midwest, South and West. So four minus one is three. So that's the degrees of freedom for the numerator. And then the degrees of freedom for the denominator is the total number of data values in this case. There were 34 of them minus the number of categories. 34 minus four is 30. So those three pieces of information is what you need. Now you could go and look at a chart. Usually most books have them uh chart or uh you can search online or you can program something in your calculus. That's what I did. So I made a program in the calculator. I called it in verse norm. I'm not going to show you how to write this program. You can google it or youtube but if you want. But um it's pretty nice and it's easy to see. So I call it in verse norman now it's gonna ask me for the area, right? And That's 10%. That's your alpha value. And then d. one. I just wrote that as for the numerator. So the degrees of freedom for the numerator remember was was three. And then for the Denominator D. two was 30. And then I press enter again and that's going to give me my Critical value of about 2.276. So f. star equals Uh 2.276. So that's going to go right here. So anything greater than if our test value of R. F. statistic is greater than 2.276. Then we're gonna reject. If it's less than 2.276 then we're gonna fail to reject. Okay so anything to the right? That's where it's in the rejection region. That's that .10. This represents 10% of the distribution. Then we reject. So now we're going to turn to our calculator again or at least I am because doing this manually can be a pain. Um If you had a Staten edit here your data value. So I think this was north Northeast. The L one, L two, I'm pretty sure was midwest L three I think was south and then L four was west. So these are the well being index means and you punch those in. Then you go back to Staten test. It's the very last one where it says innova and we're going to go to a nova and then let's just go ahead and type in all of our columns, all of our categories. We want to test all of them and make sure you're separating those with comma So it's all one comma L two comma. L three, comma. L four. And then whenever you press enter that gives you you know pretty much everything you need. So F equals 7.49 Okay so our f statistic is about 7.5 which is definitely in the rejection zone and that tells us on number four what we do we're going to reject are null hypothesis. So any time it's greater than the F. Star it's reject. Another way to double check. That is you can look at this p value. So this is another good thing about this. One way innova test um This p value here tells you whether to reject or not as well and this is about seven times 10 to the negative fourth. So the p value is not seven or it's not six point 95 It's 6.95 times 10 to the negative fourth. So this is actually a very very small number, basically zero. Any time something gets put in scientific notation it's basically zero and any time the p value is less than alpha in this case it's definitely less than it's less than 00.10. We reject. H not. So that's just a nice way to double check. And then the 5th and final step is to summarize this, you know, an actual plain words. So we're gonna write the sentence. There is sufficient evidence, evidence um to suggest that at least one mean score of well being index is different than the other regions. And we'll see in a later problem. I think it's south, that's quite a bit different than the other regions. So that's the five step process for one way an over test.

For the match pairs given on the right. We want to conduct a sign tested matched pairs testing P does not equal 0.5 at alpha equals 0.5 significance. This question is testing our understanding of non parametric tests in particular how to conduct a scientist of matched pairs. We proceed there steps A through D below to solve. So first, in a we stayed alpha hypotheses, this gives alpha equals 0.5 H. And r P does not equal five. H. A P does not equal five. So the no hypothesis P is equal to have, the alternative is not A and B. We compute the test out. So are signs for these matched pairs are as follows and equals 12 is the total number of matched pairs. So X equals the number of plus or the total number plus and minus which is 4/12. Thus we have seen it equals x minus 5/45 over and equals negative 1.155 Thus the p value is for a normal distribution, P equals two PZ greater than zero equals 248 Thus we conclude that we fail to reject asian off because he is greater than alpha, which means that we lack evidence to support the claim P does not equal five.


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