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Select a vector of temperatures in Centigrade call it X, and convert it to a vector Y representing those temperatures in Fahrenheit: Then show that the correlation ...

Question

Select a vector of temperatures in Centigrade call it X, and convert it to a vector Y representing those temperatures in Fahrenheit: Then show that the correlation between their mean-centered values is 1. (Note that both data sets carry the same information; just in different units ) Note also that their angular separation in data space is Zero.10. Generalize the previous. Pick some numerical values for a data vector X. Show that if Y = mX + bU (so that the entries of Y are all linearly related

Select a vector of temperatures in Centigrade call it X, and convert it to a vector Y representing those temperatures in Fahrenheit: Then show that the correlation between their mean-centered values is 1. (Note that both data sets carry the same information; just in different units ) Note also that their angular separation in data space is Zero. 10. Generalize the previous. Pick some numerical values for a data vector X. Show that if Y = mX + bU (so that the entries of Y are all linearly related to the entries of X) then their mean- centered values satisfy p(AX,AY) = +l. If p = 1 they are called perfectly correlated and if p = ~1 they are perfectly anti-correlated: If p = 0 they they would be called perfectly uncorrelated: Hints First check that AY = mAX How does the sign of m affect the correlation constant?



Answers

Construct a scatterplot, and find the value of the linear correlation coefficient $r$ Also find the $P$ -value or the critical values of $r$ from Table $A$ -6. Use a significance level of $\alpha=0.05 .$ Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section $10-2$ exercises.).A classic application of correlation involves the association between the temperature and the number of times a cricket chirps in a minute. Listed below are the numbers of chirps in 1 min and the corresponding temperatures in 'F (based on data from The Song of Insects, by George W. Pierce, Harvard University Press). Is there sufficient evidence to conclude that there is a linear correlation between the number of chirps in 1 min and the temperature?$$\begin{array}{l|c|c|c|c|c|c|c|c}\hline \text { Chirps in 1 min } & 882 & 1188 & 1104 & 864 & 1200 & 1032 & 960 & 900 \\\hline \text { Temperature ("F) } & 69.7 & 93.3 & 84.3 & 76.3 & 88.6 & 82.6 & 71.6 & 79.6 \\\hline\end{array}$$

So for part A, we're looking at the information for terrorists alarm, and the first thing you notice is that the maximum temperature occurs in February, which is the second month, and that maximum temperature is 81.7. The minimum temperature occurs in July, which is the seventh month, and that minimum temperature is 73 point. Not now, using this week uncalculated value for in which is maximum minus the minimum divided by two. So that would be any 1.7 minus 73.9 divided by two, and that equals 3.9 now. The next thing is, pete is 12 months. So that means B, which is two pi divided by the period. So two part about it by 12 equals pi over six. So next week after let the mean temperature and the mean temperature would actually be the value 40 which would be diverted the shift. And that equals the maximum plus the minimum divided by two. So maximum temperature, plus the minimum temperature right about two. So we have 81.7 plus 73.9, divided by two. So we got a value for D, which is equal to 77.8. Now, again looking in the table, what we find is that the mean temperature. First, others in it's true. And for April T is equal to four and also in April. What we find is that the temperature is decreasing. So going from February to March to April 2, main a temptress decreasing, so the equation would have a negative value for a so this would be wise equal to negative 3.9. Sign off high over six times T minus four plus 77.8 or we air Why is equal to negative 3.9. Sign off pi over six T minus two pi over 30 and then plus 77.8. So that's nutrition that we have four Rossella. Now let's take a look at what is happening with the other City tells key. So it's talking with part B here for Till ski. Okay, so now, for thing is that the maximum temperature occurs in July, which is the seventh month, and that maximum temperature is 43.9 increase. The minimum temperature occurs in January, which is the first month, and that temperature is negative 24.2 degrees Fahrenheit. So from these two week and cal ability values the amplitude a, which is maximum minus the minima divided by two. So we got 43 59 minus negative, 24.2, divided by two, and that comes out to be equal to 34 point or five. Next, the PD again is 12 months, and that will be is equal to pi work. Six. Remember easy to fight over the period. Now the next thing is the mean temperature. So what is the mean temperature attack would give the value for key, which is also the vertical shift, and that's equal to look maximum plus the minimum divided. But so we have 43.9 plus negative 24.2 divided by two, and that comes out to people to 9.75 suggest a vertical shift, and then the mean temperature first occurs in in for for which TZ pulled before. So the equation would be why is equal to 34.45 sign off high over six times T minus four last 9.75? Or why is he quick oot 34.5 Sign off kite over 60 minus two pi over the plus 9.75 So that's your heart beat now for park si. What we see for our salam is that temperature decreases cause capture is DPC. Therefore, rate of change off temperature is negative. And for Parc de, we actually have to find that rate of change of temperature, so that would be equal to the temperature in July, which is 73.9 minus the temperature in January which is 81.3, and divide that by seven minus one. So that comes out to be too negative, 1.2 three beating and it would be degrees Fahrenheit per month. So that's the rate of change of temperature from January to July 4000 and now for part E asking whether TTF the rate of change of after his positive or negative Fortensky so again looking at the table and what we find is that the temperature increases and if the temperature is increasing, then this means that rate of change off temperature is positive and the last thing that we have to capital it is theatrical rate of temperature for till ski so raped off. Change off temperature for people to the temperature in July, which is 43.9 minus the temperature in January, which is negative, 24 point to divided by seven minus one, and that is equal to 11 point city five decrease Fahrenheit per month.

Okay, So I am giving a relationship between no team and this which is, um, on a love flock 15 forces by if so, energy is equal toe a and then no speak or d equals toe into the power Bi e right. It's the same thing. Also be provided with points. One is 25 hours details per second. Do you want people wonderment? And if he Celsius so I two is equal to 40 m per second. Take do is if you don't want drink it into the cells here. So you have to write a doze off points I can write to dies Ellen 25 Oma Island to 10 and and then 14 And then 1 20 right? Neither Michael points. Okay, now let's start with what a So if I need to find this law, it's going to be why you to mind this? Why one divided by its two minus next one which is even do a just don't smoke. So in this case would be law 1 20 My next role to 10 united by north 14. Why you missed it all me. Fine. This movie will divide this 1.19 he was a calculator. Well, we want to the intercept. I mean, which is a good wife. Minds, a X two. Please do while in my experience, one. Right. So we will be routine. Okay. Mhm day minus. You just started like it does one from one. Mom. Mhm 40 which is equal in 1974. Right. So what I get is t equals student. Can you raise to the father? Mhm Dark, sire. Used to the father. Hey, this gives me t equals two years to the power 9.74 into scientist to the father. Minus one point. Do. There's a lot to me, please. Okay, citizens, Part E. When we want to find beef, I have. Do you Line equals 2 85 results is to do equals to 1 75 50 70 years. 23 equals stupid 90 seventies. Okay, so I have side, which is a gluten from this equation D over. He used to the following 9.74 Divided by nine this one point point to technologist substitute. Do you mind? Do you do ankle three. You know, Bob Well, so side one easy food. 50 Tired dividing my ears with volunteer 27 just 30 ft. Meetings per second. 75 leg seven. Thank you. Moved into its perspective. Seven men, you know, a few bodies of size. And right here. Now what does it see? Is, um So the law is that the it is like this one is the most lithe. Hmm? Mm. People living shoots me. Mhm, Mhm. Its influence school. He speaks. Want to one retail? Mhm. Mhm two on the end, Mr. Before Why mhm words do we need to find this enjoyed with 3002 se side? Basically perception she's something because you want to Is your mountain dew age off infection, which is 1 34. Okay, one MP Booth. It is us, I think.

We want to do is we are given um a real life application problem. Um And this table represents um the months january through december. Where month january is going to be marked as time equal to one. And of course I'm a zero of course february will be the first month marked a second. And the maximum is the maximum of temperatures in degrees fair height for a city in pennsylvania and minimum is the minimum temperatures also throughout the year um for the same city. And we are told that the maximum and minimum it can be modeled by this function right here. Where the um constant coefficient is given by 1/12. The integral from 0 to 12 of f f t D T. Um The coefficient for the A one is given by 1 60 integral from 0 to 12. F A. T. Times a coastline up I. T. Over 60 T. And then of course be one is given by 1 60 a girl from 0 to 12 of Fft. Sign up five or 60 DT. Um And we want to actually determine um the maximum um temperature. So we want to come up with some function that represents the maximum temperatures. Okay so we want to find a zero A. One and B. One and we want to do that using um using symptoms rule. Okay we don't really want to integrate. We want to use Simpson's rule and so don't forget that. And a zero which is 1/12 the integral from 0 to 12 of F. F. T. D. T. Using the Simpsons role of course it's going to be 1 12 times B minus A. Which is 12 minus zero over three times how many partitions that we have. And of course we have 12 partitions because there's 12 months and then this is going to be equal to you know FFT basically um or are high temperatures and so this is going to be actually um 33.5. So we'll have this 33 0.5 plus. And then remember Simpson's rule, we do four times that next one, which was 35.4 plus two times. And remember now we alternate um with four times versus two times. So this would be two times 44.7 plus four times. And then there's alternating the 55.6 plus two times the 67 0.4. Um And then we're gonna keep alternating as we go along and then our very last one, we actually want our second to the last one is going to be four times the 38.6 because we want to repeat that january number right here. Okay. And so if we do this all in our calculator, this is approximately equal to 57 0.72 Okay. And then what we want to do is to use the same process to find a one and B. One. So a one would be 16 the integral from 0 to 12 of F F. T. Times that co sign of pie. T over six. Tt. So this is once again going to be 16 times this 12 over three times 12. And then of course this is actually going to be. Now. What we need to do here is we actually need to um do you 33.5 times co sign of zero. And then this will be plus four times 35.4 times co sign of pi over six because that would be t equal to one. And then we have plus two times the 44.7 times co sign of and now it's going to be um T. In this case is too so we have pie or three and then we have plus the four times 55. Ply over 55.6 times now co sign of. And now we have three pi over six which is pi over two. Yeah that should be a pi over two and then we have to times the 67.4 and then this will be co sign and in this case T. Is so this is 0123 T. Is four. So this would be um two pi over three. And then we keep doing it until we can the very last and the very last we're going to have and plus that 33 0.5 times co sign of. And now it's um 12 pi because it's a 12 month um of 12 pi over six which is two pi. And if we do this all in our calculator we should get approximately equal to negative 23.36 And then now we're going to be one and I think you kind of see the process. So now in this case it's 16 the integral from 0 to 12 of F. F. T. And now it's sign of pi T over six T. T. So this is going to be equal to 16 times 12/3 times 12. And now this time we're going to have 33.5 sign of zero plus four times 35.4 times sine of five or six. Um Plus two times whips, 44 0.7 times sign a pi over three. And then of course this keeps going until we get 33.5 sign of two pi. Which then we should get approximately equal to I think negative two point 75 Um and you might get something slightly different um depending on your grounding. So our high temperatures are modeled by or high temperatures are modeled by 57 0.72 minus 23.36 Co sign of pi over 60 minus that. 2.75 Sign of piratey pi over six T. So there is our high temperatures and now we're going to do the exact same thing for our low temperatures. So we're going to actually do the exact same thing but this time we're gonna use our low temperatures and so we're gonna keep doing this and we're and it's basically the same thing in here. But instead of having the 33.5 we'll have the 20.3. So we're going to do the exact same thing for our low temperatures and if you work through that process, the exact same process that we just did um you should get the low temperatures to be modeled by approximately of course. Remember these are always approximate. 40 two 0.4 minus 20.91 So a one so a of zero is 42 point oh four. A one is 20.91 And then of course we have the co sign of pi over six T. And then be one is four point 33 sign of five or six T. So there are both of my models that represent that data. Um And now what we want to do is we want to graph both. We want to plot both of our data and we also want to graph our model for our data. Okay, so I'm going to actually switch to an online graphing calculator called Gizmos and for time purposes I've already got I've already input my data. Um So they developed a scatter plot. Um So of course zero through 11. And my high temperatures are in the black dots and my low temperatures are in the red dot. I mean blue dots. And I'm going to go ahead now and grab both of my, I don't go ahead and graft both of my models. So the high temperatures was mild by y. Equals. And I believe we had 57 points having to minus um 23.36 And we had co sign of I. X. Divided by six and then we had minus the 2.75 About um sign of I. X. Goodbye by six. Okay. And so there is and it nicely fits um my scatter plot. And now what we're gonna do is do the low one as well. So why equals. And this one was 42 point oh four minus that. 20.91 And then we had the minus 4.33 I believe. And that was time to sign. Okay brooks. Okay. And so now we have both of our models um graft as well. They both sit very nicely to their data. Um And so now what we want to do is um what can we determine during what part of the year is the difference between the math and the minimum temperatures. The greatest. And so I am thinking um that um if this is january february March april may june july it looks like the difference is greater during these summer months. So starting at um this is where you notice that in these regions right here I have the biggest um distance between the high temperatures and the low temperatures. So I am thinking that um the difference um temperature difference is greatest between in the summer, probably starting in May to, let's see if I go back to dez mose, so january february March, april may june july august, maybe september so may two september.

So you can get the data. A couple of things come here. So four thing is that the maximum point occurs in the seventh month and the highest temperature is 105.1 and the minimum temperature occurs in January, which is the first month of the year, and that minimum temperature is 65.5 degrees Fahrenheit. So forth thing we can calculate is the amplitude, which is maximum temperature minus the minimum temperature divided by two. So that would be 100 and 5.1 minus 65 point 5/2. So we get a value for a which is equal to 1928 now. The next thing we notice, since it's a year, so the period is going to be 12. So be which is to pine divided by a period would be to pie divided by 12 which is higher work six. So we got a Reebok be now. The next thing that we can calculate is D, which is the vertical shift so D is equal to maximum, plus the minima divided by two. So we have 105.1 plus 65.5 and divide that by two. That gives you 85.3. Now. The next thing we need to do is where we first get this mean temperature. So this D would actually be the mean temperature, so the mean temperature first occurs. I didn't pay pro aan act Mies. That's when T is equal to four. So now we are ready to fight our equation. Which would be why is equal to a, which is 19.8 sign times pi over six. This is B Times T minus four and then plus D, which is 85.3 So expanding that Why is he going to 19 point it sign high over six times T minus two pi over three plus 85.3


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