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Finite [Ji( (crence Apprximariont Tno Spice Dimtnsian(orm of (4.9.1). Thus we Yrilc with 0< 8, < | and 0<8*1.J M (4.9,2)(. 2v,." ",+l,"-14&#...

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Finite [Ji( (crence Apprximariont Tno Spice Dimtnsian(orm of (4.9.1). Thus we Yrilc with 0< 8, < | and 0<8*1.J M (4.9,2)(. 2v,." ",+l,"-14'(4,t" 26,.4.1 WL)(u;.1 26,." "1,-1)(7" 20,.1' "-1).bclore,selectwc shall obtain approximation (termed the classic explicit)explicit finile difference(4.9.)0(", ",-" 06.3014,,4)[classic explicit]Note that (4.9.3) uses the five points laheled 0, 1, 2, % and in Figure I0 t0 calculate Obvio

Finite [Ji( (crence Apprximariont Tno Spice Dimtnsian (orm of (4.9.1). Thus we Yrilc with 0< 8, < | and 0<8*1. J M (4.9,2) (. 2v,." ",+l,"-14' (4,t" 26,.4.1 WL) (u;.1 26,." "1,-1) (7" 20,.1' "-1). bclore, select wc shall obtain approximation (termed the classic explicit) explicit finile difference (4.9.) 0(", ",-" 06.301 4,,4) [classic explicit] Note that (4.9.3) uses the five points laheled 0, 1, 2, % and in Figure I0 t0 calculate Obviously. the method of cakculalion involves point-by - point evaluation plane using Ihe points on the planc , Given the initial condilions for the plane by plane evaluation follows As special case the choice yields (4.9.4) ",44,64 46 JMi ",..-) we thercby eliminale the MeTi Stability Ihe classic explicit approximation (4.9.3) can Acetaaincd using von Neumann" mcthod: The extension of (4.5.5) for the one-dimensiona cusc nov bccomes Elx,Y. Y)ze"e" "e When subsliluled into (4.9.3) and common lerms canceled, there results (4.9.51 amplification (actor = &** = | ~ 4p(sin? Bk +sin? 44 Since |e/ <1 required for stability, follows that ~Isl-4p (sin? B* +sin" @#)si Pumbolic Panitl Dillarentinl Fquation But 8, and B, are arbilrary and follows thal (4.9.6) P < 2(sin? Bzk +sin'&k We scc thal (4.9,4) is taken thc uppcr limit of the stability bound (or (4.9.3). Finally: had not uscd k =kz but relained k + kz- the stability bound would b (4,9.7) h<- V



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Flight formula for Indian spotted owlets The following table shows the average body mass $m(t)$ (in $g$ ) and average wing chord length $w(t)(\text {in } m m),$ along with the derivatives $m^{\prime}(t)$ and $w^{\prime}(t),$ of $t-$week-old Indian spotted owlets. The flight formula function $f(t)=w(t) / m(t)$ which is the ratio of wing chord length to mass, is used to predict when these fledglings are old enough to fly. The values of $f$ are less than $1,$ but approach 1 as t increases. When $f$ is close to $1,$ the fledglings are capable of flying, which is important for determining when rescued fledglings can be released back into the wild. (Source: ZooKeys, 132,2011 ) $$\begin{array}{lrrrr} t & m(t) & m^{\prime}(t) & w(t) & w^{\prime}(t) \\ \hline 1 & 23.32 & 41.45 & 10.14 & 14.5 \\ 1.5 & 50.59 & 64.94 & 20.13 & 26.17 \\ 2 & 82.83 & 57.95 & 36.7 & 39.86 \\ 2.5 & 105.13 & 31.08 & 58.92 & 47.11 \\ 3 & 115.48 & 12.48 & 81.55 & 41.38 \\ 3.5 & 119.4 & 4.44 & 98.99 & 27.94 \\ 4 & 120.76 & 1.51 & 109.75 & 15.74 \\ 4.5 & 121.22 & 0.51 & 115.5 & 7.99 \\ 5 & 121.37 & 0.17 & 118.34 & 3.85 \\ 5.5 & 121.42 & 0.06 & 119.69 & 1.8 \\ 6 & 121.44 & 0.02 & 120.32 & 0.84 \\ 6.5 & 121.45 & 0.01 & 120.61 & 0.38 \end{array}$$ State the units associated with the following derivatives and state the physical meaning of each derivative. a. $m^{\prime}(t)$ b. $w^{\prime}(t)$
c. $f^{\prime}(t)$

First one. The test for a unit root in series you rate unemployment rate using the usual dickey fuller test with a constant yeah. And the augmented dickey fuller with two legs of change of unemployment rate. I find that seven both times we are unable to reject the now hypothesis that unemployment rate series is a unit fruit. The legs are not significant. However, the significance of the legs matters. So the outcome of the unit root test, we will repeat what we have done in part one two series vacancy rate and report the result in part two. I guess similar result. So the rate is a unit root. Well part one and two. I use package the R. Package A. T. S. A. And the function is a D. F. Dot test. R. Three. We assuming that unemployment rate and vacation re rate are both integrated of level one. We test for co integration using the angle grandeur test with no legs. So the step the steps are as follow. We first regress, you read on the rate then we yet the residual and we run the key fuller has on the residual to see whether the residuals our unit root. I find that you're right and we rate Arco integrated at the 5% level. Yeah Heart Forest. I get the leads and lacks estimator of the change in vacancy rate and I did note that uh CB rates up minus one. This is for the lack and plus one is for the lead. This is a regression result. So the usual centered errors are in green and in round brackets, the robots that Iran's are in blue and in square brackets you can see that the main estimate on vacancy rate is highly significant. This one is not correct. So the centered errol the usual one for the estimate of the first lack of change in vacancy rate is 164 In all cases except for the estimate of the lead of C. V. Right. The robust standard Iran's are larger than the usual standard errors. This is usually the case it happens but rare that the robot standard errors are smaller than the usual standard errors. The r square of this regression is 0.77 So for the rate, because the robot standard error is larger than the usual standard error. So we will get a wider confidence interval if we use a robot standard error and for confidence interval you will run this function in our count in and you impose the name of the regression. It was spits all the 95% confidence intervals for all explanatory variables. The default version is the 95% interval. But because the standard barrel of this estimate is are very close, two versions are very close to each other so the confidence intervals should be roughly equal. Yeah. Last part. What you could say about real business of the claim that you rate and the rate are co integrated. Yeah. When I run the test and good grandeur, the results are not consistent across alternative types of process. In one case I can reject the notion that the residuals are united and for all the cases I cannot reject. So I conclude that the claim that you rate and be rate our co integrated is not robust.

Part one. The test with strict exhaust Janet E gifts is row hat equal to minus 0.97 and the T value of minus 2.41 The regression that includes growth of minimum wage and growth of C P. I gives row hat equal to minus point 098 and it is statistic of minus 2.42 Roughly the same T values and roll had value. Therefore, we find evidence of some negative serial correlation and it does not matter which form of the tests we use. Yeah, this is the regression equation for part two and three with three types of standard. Erin's Yeah, note that the estimates are not changing. Only the standard errors change. The first line of standard error is the old LS the usual and probably incorrect type. The second line is the new E West standard payroll type, and the last line is hetero skate elasticity robots standard errors we have over 600 observations and the are square are the same for three types of standard herons, which is 30.293 So comparing the new e West standard error and the usual L s standard error for the variable growth of minimum wage. We find that the new E West than it Errol is much larger than the old l s one roughly four or five times larger. But for the variable growth of C P I, the new E West standard error is actually smaller than the old l s one that is the answer for part two. And for part three, we consider the hetero Scholastics city robots standard error with the newly West standard error, we don't find much difference. Yeah, So the difference between the two type of standard error is that, um, the last one only controls for different variants of the Errol terms. That's why it's called hetero Scholastic City. Robust, but new E was standard. Errol does more than that. The new E West Standard Erin's our robots to both hetero Scholastic City and serial Correlation, as we find little difference between the two, is probably because the negative serial correlation adjusting the standard error on, um, CP, I actually reduces it. Hetero Scholastic city does not have a major effect on the growth of CP I standard error. That is part three and part four. We run a BP test, We get F statistic equal to to 33.8. Yeah, which means the P value is almost zero. There is a very strong evidence of hetero ski elasticity. Part five, The usual F test is 4.53 with a P value of pointing 058 So, in the static model using the hetero Scholastic City Roberts T Statistic lead Teoh a less significant minimum wage effect. Okay, Actually, this one is there. This is for the usual F test and this is through the hetero stick elasticity. Robust test. All right, so the hetero see elasticity robots test for the legs show a very strong significance of the wage effect. Part six, the new we West version of theme F statistic is about 7.79 which show even mawr Statistic. Significance, then just the hetero Scholastic City. Robust statistic. So at just in the F start for hetero scholastic city or hetero ski elasticity and 12 order serial correlation leads to the conclusion that the LAX are very statistically significant. March 7 with 12 legs, the estimated long run propensity is about 0.198 and without the legs. The estimated L R P is just the coefficient on the growth of minimum wage. Yeah, which is 0.151 So when we include the LAX, L R P is about 30% larger using the new We West standard error the 95% confidence interval for the l r P. ISS from yeah, hauling 111 to you Point Thio 84 which easily contains the estimate from the aesthetic model.

Were given the set of data points listed at the top of this whiteboard X fly. And we want to use these data points to answer the following questions. A through F. Starting off with part A on the left, we want to produce a scatter plot of these data points. I've already included the scatter plot. As you can see where the data points X. Y are demarcated by the black crosses or exits next to the right and part B. We want to compute the sum is relevant to the state to as well as the Pearson correlation coefficient. R The sums are given by following the forms exactly. So some access to some of the X values. Some why is some of the individual Y values and so on. To compute are we use the following formula which takes us input, our sample size and and the Sun is just computed. This gives our equals .9126. Next below. In parts you want to find the equation of the line of best fit which requires finding these parameters first are simple mean X bar and a sample mean Y bar are given by the sum of our X values about it by n 6.25 And some of our Y values over M 32.8. Yeah, we can find the parameters for our best fit. Line being a. As follows. The slope B is given by the equation here, which takes us input and the sample size and the sums we found above Plugging In. We get the equal 22 and then plugging in. Ry bar be an X bar to our A equation on the right gives us intercept negative 104.7. This means we have equation for the line of best fit why hat equals negative 104.7 plus 22 X. Next part Do we want to return to the scatter plot on the left and graph ry hat. Doing so we want to make sure we include our X. Men and women, which looks like this next in the bottom right part. You want to calculate? The coefficient of determination are square and interpret its meaning. This is simply the square of the correlation coefficient 0.83 to eight. We interpret this to mean that roughly 83 of the variation of the data can be explained by the corresponding variation and excellently squares line 17 of the data accordingly cannot be explained by this. Finally, in part, after the bottom, we predict y where x equals 6.5 Plugging into our white hat, we obtain 38.3.


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