Assignment Schedule for Mat375

1. What have you tried, or considered?
2. What else would you like to know?
3. Who's better? (Your best guess so far)

1. Verify my calculations of $\alpha$ in the final report.

2. Use Riegel's model to estimate Thad and Thomas's times on the 100 meter dash, and on a marathon (42.2 km).

3. Use the same procedure discussed in class to find a power rule for the world-class women close to what Riegel found.

For Friday, 1/31: your group will write a three-page report (or so), typed, on Dave's stick: what is it, and why is it marked the way it is? You should be able to reconstruct any marking on the stick.

1. compute the center of mass $(\overline{x},\overline{y})$.
2. the equation of the regression line -- all "by hand" (i.e. by calculator).
3. Plot -- by hand -- the data, the center of mass, and the regression line.
4. What is the minimal value of the sum of squared residuals?

Using the most recent Keeling CO₂ data, and using linear regression,

1. produce (and evaluate) models of monthly CO₂ trend that are

   • linear in time (decimal date),

   • quadratic in time,

   • exponential in time,

2. Then produce and evaluate a model that is (your favorite of the three types above), with the addition of terms of the form $\sin\left({\frac{t}{T}}\right)$ and $\cos\left({\frac{t}{T}}\right)$ (where $t$ is again decimal date). These sinusoidal terms will capture the oscillations. What value should you choose for $T$?

3. Ultimately you should determine the decimal date at which the model predicts CO₂ to first surpass 450 ppm -- the point at which we lock in 2 degrees C of global warming (per the National Academy of Sciences).

You can see some of the 10 foot markings on this scanned image, to verify your calculations.

We need to evaluate our work. For the project ("Dave's Stick"), please hand me one to two pages, typed, and which will be kept strictly anonymous (i.e. no one else will ever see it):

1. Evaluate your own report, especially comparing it to other reports.
2. Compare your partner(s) and your contributions to the report. Perhaps you took different roles, which you can describe.
3. Feel free to compliment your partners, if that's appropriate.
4. Suggest how you would divide the effort put into the report by each member (divide 100% -- e.g. Bob, 20%; Lulu, 50%; Stan, 30%).

• You all have different backgrounds and skills. Some of you have already done some advanced statistics, or perhaps some math modeling. I'm not focused on that so much, as to make sure that everyone is contributing, and that everyone is doing good quality work (which will only improve as we go along, I'm quite sure).

• I will continue to ask these questions of you and your partners for the rest of the course. You have at least one more small group project (Keeling), plus the final project -- with different partners. So you will be evaluated by different people.

  First, be a good partner yourself!:)

  Second, do a fair and honest appraisal of others.

  I will issue you a participation grade at the end, which reflects only the synthesis of the reports, and I won't share individual comments to protect reviewers.

  If you ever felt that I were being unfair in this assessment, you could request an audit by any faculty member from our department of your own choosing, who would review my methodology, and whose grade would overrule my own, if it differed.

• There is exam redemption! You all can retake exam 1, at your leisure, and I'll average your two scores.

• You may discuss the problems with each other, and with me.

• These will be due Friday, 2/28.

• Enter the data into a spreadsheet with 366 entries (due Friday, 2/28).

  You should triple enter the data -- that is, each person should enter it separately, then compare to make sure that it was accurately entered.

  You might build a common spreadsheet (like this perhaps), so that it's easy to compare.

Please also include confidence intervals for amplitude $A$ and phase $t_0$.

I'll grade based more on the final paper than on the original.

1. Use Newton's method to find roots of the following, starting from the value $x_0$ given.
   1. $f(x)=\ln(x)-1.23$, with $x_0=1.2$.
   2. $f(x)=\cos(x)-.23$, with $x_0=1.2$.
   3. $f(x)=x(x^2-1)$, with $x_0=.4$.
   Graph the functions in the vicinity of $x_0$. Compute 4 iterates, by hand (show your work, but use a calculator). Use software to solve for the "true solution", and compare your answer to the value obtained. Are the iterates approaching the solution, and, if so, how (and how quickly)?

2. Use non-linear regression to fit the models \[ at^2+bt+c+A\sin\left(2 \pi \frac{t-t_0}{T}\right) \] and \[ at^2+bt+c+(\alpha + \beta t) \sin\left(2 \pi \frac{t-t_0}{T}\right) \] to the NOAA Keeling data.
   1. This file will load the data into Mathematica for you, to get you started.
   2. Discuss both models. How well do they fit? Compare. One model is a generalization of the other, of course.
   3. Are we justified in taking $T=1$?
   4. Are we justified in believing that the amplitude is changing in time?

If the globe were warming, then one thing we concluded is that we should see more MaxMax years-of-occurence on the recent side (toward 1992), and fewer on the earlier side (toward 1854).

So in order to test a null hypothesis of "No climate change" against an alternative of "Global warming", we could test the distribution of MaxMax years-of-occurence by decades, and see if it's uniform: all decades are the same.

Please do the following (on your own, but feel free to share ideas and results within your group):

1. Describe what you would expect for the other series -- MinMax, MaxMin, MinMin.

2. For your time series, create a histogram showing how many years occurred in each decade (1854-1863,1864-1873, etc.). Draw this by hand.

   Jon and Proctor -- you did the first three months of the year. Please do the same, but for all extremes in the first three months of the year.

3. Climate is usually defined as the average of thirty years of weather. Also create a histogram using three-decade-long bins (example: 1854-1883,1884-1913,1914-1943,1944-1973,1974-1992 -- check out that last one! How would you adjust for it?). Draw this by hand.

4. Describe how one might decide if your histograms are "abnormal" (and so we might "reject the null"), given the null hypothesis of no climate change.

Carry out labPP.html, and submit the results to me by Wednesday, 3/25.

Carry out labPP2.html, and submit the results to me by Friday, 3/27.

Due Monday, 3/30!