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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) $$\begin{array}{|l|l|l|l|c|l|}\hline \text { Chicago } & \text { Dallas } & \text { Miami } & \text { Denver } & \text { San Diego } & \text { Seattle } \\\hline 43,581 & 36,524 & 49,357 & 37,790 & 48,370 & 57,678 \\37,731 & 33,709 & 53,207 & 38,970 & 45,470 & 48,043 \\46,831 & 40,209 & 40,557 & 42,990 & 43,920 & 45,943 \\53,031 & 51,704 & 52,357 & 46,290 & 54,670 & 52,543 \\52,551 & 40,909 & 44,907 & 49,565 & 41,770 & 57,418 \\42,131 & 53,259 & 48,757 & 40,390 & & \\& 47,269 & 53,557 & & & \\\hline\end{array}$$

Part one. The variable T O T F A T R T e is defined as the total fatalities per 100,000 population. Mhm, the average of this variable in the years 1980 is 25 point 495 In the year 1992 is 17 point 158 and in the year 2000 and four is 16 point 7 to 9. Yeah, The next part of these section is that you run a regression of this variable on the year dummies and you can find mhm. Even the base here is 1980 shit, and you find that the dummy of the year 1981 is not significant, even at the 10% level. But other estimates the other dummies are highly significant. At the one person level. Yeah, they are also negative. This result implies that, except for the year 1981 any other year has a lower fatality rate than 1980. Driving seems safer after 1980 and 1981 but we have not controlling for other factors. In Part two, we add more controls through the regression B, A, C 10 and B, A C eight our blood alcohol limit and estimate on these variables are minus 1.85 Okay, um, minus 1.76 respectively. The standard Iran's are 0.5 chu and 0.71 You can easily calculate the T statistic to B minus three and minus two, respectively. These variables are highly significant. Okay, Next, we examined the laws. The per se law has an estimate of point on 63 with a standard barrel of point 394 and SB brim okay has an estimate of poin 541 and a standard barrel of point 686 So, as you can see, these variables have centered. Iran's greater than the estimates. They are not significant implying that these laws do not have a negative effect on the fatality rate. Sir, we don't find any evidence. Yeah, For the effect of these laws or three, we re estimate the model from part two using fixed effects at the state level, and I will only report their variables of interest for the beta the sea. So BC 10 should have minus one point 063 b a C eight minus 1.4 37 per SE minus 1.15 152 and SB Prim minus 1.227 Their standard Iran's are as follows 0.269 0.394 0.234 and 0.343 Mm hmm. So they are all highly significant. Usually the three stars denote yeah, significance at the 1% level. Comparing to the Poland L s estimate, we find that the blood alcohol limit the first few have weaker magnitude, but still negative and highly significant. The estimates on the laws increase in magnitude and become highly significant. Yeah, All of these estimates agree with our intuition about the relationships between laws and fatalities. Part four, we have the variable represents the number of miles driven per capita mhm. Given that these variables this variable increases by 1000, it means if each resident dries 1000 miles, okay, more above the average trend of the state. We need these phrase the average trend of the state because we used fixed effects at the state level. So we are using the variation within the state to explain the fatality rates. Get ahead of this very boy is 0.0 nine four. And the standard barrel is also very small, making these parables this variable significant at the 1% level. So if this change happened, then the fatality rate over that state is expected to rise by 0.9 per 100 1000 population. Part five. I will re estimate, um, part three, but I went cluster the standard barrel at the state level. Okay, so clustering doesn't change the estimate on the variables. Only the standard Errol's change. So I will report the standard error, the new one only for B A C 10 women get 0.487 b A C A. T 0.815 per se 0.43 nine and s be prim 90.553 So the first variable significant at the 10% level. His second parable. Oh, this one should be 5%. The next one at 10%. Mm. Her say law at also 5% and the same four SB prim. Yeah, we see that the centered Iran's all increased. Yeah, The estimates used to be significant at the at the 1% level, but now B A. C. Eight barely significant. And the other three are significant at the 5% level, so the effect of heterogeneous um, a real terms it's quite substantial.

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

You will be out of luck. Apple gives up its are one you or one And aren't you by the corresponding average rate of our when puts, todo wants us to find different variants or different confirmations given the distance is equal to 80 on the AC tree are able to that's called 80 miles and 48 MPH. So for our own or she problem, it asks us to ultra very clear. Verify that are two is equal to 20 are one over R one minus 20. So it's by copying down our corresponding of equations that are two is equal 20 arm one over R one months 20. Okay, so, um, we know our one is D over t one and R two over t two. Oh, and Leslie d equals 80 articles. 40. Okay, so let's start by plugging teat and to our, um ah rate equation so are equal. Oh, well, first we need to isolate t. I guess are in this case Penis multiple by both sides. Divide both sides by our one for our anti plugging this into our arbitrate enough r is equal to well to over cheap one plus t two post aren't that will be over by deep over our over our bust meat over a far too about sequel to Perlis. Multiply both sides by the denominator of the right hand that I d over. But there are if people t d anarchy wanted Bisley are to to confirm our verification So that are they have or for tea. What it I slipped the look tiny Are you first to t d um yes or he over arm and bring here. We want to multiply both sides by art by nominator on the left hand side by both sides by the barrier on the hands so that would give us both are are on over are what from here Let's factor out the B and do not move, are I? And this gives us, um, looking anything are 40. So it's started what's multiplied and he talked denominator by our one for five farm on that is arm or and we could double by a little bit biting both in the murder and the nominee to when you are one over it. One over is equal to Archie. Sounds pretty familiar. I think this is our final four Archie Boles, 20 are one over. Our one month 20 Archie goals are one over our rent one minus 20 so that it's verified. Okay, so that's the first portion of our problem from now. We need to don't want to find our now we want to find our awesome told me. So let's put this into graph. Oh, so, um, just yeah, this little icon side is not only but why. So it's called that part. All right, Well, the variable on the right hand side is only excellence, but that just so it's in a form where kind of more used to ah, we don't the zeros or singular zero of our right hand side would be here When the numerator is equal to zero, we know the left hand side will be both as big girl. I'm gonna go. Are one cannot equal 20 is That would mean we're dividing. So that would be a nascent tote on as our one approaches infinity, Immuno are a few would equal 20 he could of life is like in and even see their budget money. So these are a few ass. And that's what will the graph look politic as well. No, um it's gonna cross through. That's two points. And it can't touch other look, something this key. And as R approaches infinity, it would be positive number with something like that. And we want to know what our assets on the money and Botham our correspondents you rates. That kind of just tells us my other great only well 20 MPH Where? Okay, the, uh oops. So far, Because basically would be impossible. So if try to plug it in? No, our goal to, um, to over t one plus two. Um, and we substituted in are to algebraic right or R two and R one ultra Brickley and from there and found a 20 in the denominator evident algebraic substitution is it's possible to do it isolating either r two or r one because both of these terms are basically symmetrical. That makes sense. So, um, it tells us neither are one. Nor aren't you? Well, 20 mission. Um, but that's just kind of just showing entre Brickley Oh, you won't find the third portion of although it's part seat. It asks us to verify Then, when arching was greater than art one or the speaker attorney would be greater than it's going. Um, that over drinks. Oh, the going speed with the between. Um, wait. Yeah. When the speeder training with greater than the speed going So are to be group within our one. Oh, Rachel. Uh huh. We are one. Okay, so when le going speed is greater than the returning speed. So I wrote it down wrong. Um, then the going speed would be between 20 and 40 MPH. She's even, and we chose up. So that means we're basically solving inequality already here. I want our Once, he and earlier we found the algebraic equivalent of our chew. So it's like that in the 20 are one. It's our man. 20 are one over R one minus 20. From here, we can move everything to one side and creative function than inequality. So in this case, I think I'm just gonna multiply. Both sides by the nominee are are in length. It wasn't funny. Are are ones finest are one. Money are ones with a very one sided, which gives us are ones. One of our 1.2. Its body is greater than throw from here. Let's all or one just so It's a little bit more comfortable with Paul as it's What's you X squared minus 40 clacks on. Then he walked, um, at City zero. Our easy Barone's let me off like you just quote X. That would give us X experts for me, Chelsea zeros of our functions are gonna throw on a horny look at the inequality from above. We have this same asking photos before because our one cannot equal 20 are one is able to x a strict theater that look Arab or zero that are also boat is gonna 20 okay on as FX Negative, Bennett He won't be a negative times a negative, which would give us positive. It is odd. Multiplicity time change as X approaches. But affinity, it would be a positive. So that would bust like that. Want to own when? After Brexit is going to be less than zero. Um, so that's gonna be between zero and 40. So right now we have two options. We could go from 0 to 20 or 20 to 40 But, um, what houses decide here is the initial problem. So he said, Are one is greater than our to or ultra. Frankly, the speed earning would be greater going anyways. Are one is greater than, aren't you? Yes. So that means our r or X value work. Ah, reader set of solutions. So the portion of bath clips here to the right hand side which could test from 20 to 40 or wrong plug, are one in for expense. Moores B. When anything is less than our one, which is less than 40 which confirms what we were trying to find in our initial problem over here. Just to reiterate our answers, um, were part a We loved it and 12 algebraic part. He figured out that neither of the rates could equal 20. So that gives us a sense for both wrote. And since both of the on rates are almost symmetrical than photograph, it's, um, the origin, which explains why both the horizontal and vertical as my toes were equal to 20 un purport. See, we just wanted to buy the conditions are one greater than our to or the speaker training would be speed going to boundary. It's which found using the polynomial inequality passes that were practicing throughout this whole traffic. So this is just an example of one. You It's a polynomial inequality. But process, for instance, Find your gas mileage. Friends. Oh, yes. Manage plans with your friends. What you're planning for trick or something like that. So yeah, that's why what we're learning.


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