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#6 Gaul Determine the width of the Gaussian distribution function (full width at half maximum) DEUTSCHE BUNDESBANK saukttote HORE UN 10 GN 44 80 100
Let's take a look at how it all started to figure out what we can learn from the data available in China. Based on those data, you can derive a lot of important information already: Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Positive tested 45 62 121 198 291 440 571 830 1287 1975 2744 4515 6061 7914 9926 11977 14642 17393 20641 24581 28284 31500 34886 37588 40620 Dead 0 2 0 3 6 9 17 25 41 56 81 106 132 170 213 259 306 362 427 493 565 639 724 806 910 a) Create a scatter plot for both series in one graph: log(cases) versus days b) Try a linear regression for the time interval in which it looks linear (where we have exponential growth) c) How many days does it take to get a 10-fold increase of cases? d) How many days does it take to infect 1% of the Chinese population if exponential growth continued? e) How many days does it take from positive test until death? 1 What is the ratio of death per positive tested? g) Read the time delay from the graph: How many days after the lockdown was the exponential growth slowed down the increase in the log plot not linear anymore)? h) Can we trust the official data? Not based on your opinion, but just based on the statistical analysis.
#10 Error propagation If you want to determine the number of peas in the jur, you can use the mass method: The total mass of all peas divided by the mass of a single pea gives the number of peas - easy. But what is the mass of a single pea? Which one is representing all others properly? When using the volume method instead, you determine the volume V of a single pea by measuring several radii riassuming a spherical shape. You may calculate the mean value of all radii F. The number of peas N is computed with the total volume Vtot divided by the mean volume of a single pea Vand corrected by the filling factor k (as there are air voids between the spheres). The following equations apply: k. Veot N(k, V. Vted) = N(k. 1. Vor) 4 ? How accurate is this result, the number of peas? It depends on the uncertainty of the filling factor, the uncertainty of the total volume measurement and the uncertainty of the mean radius. Furthermore it depends on the above function relating those parameters. Apply the equation of error propagation for power functions for the above equation! What is the relative uncertainty of N as a function of the relative uncertainties of k, Vox and r? N (k Veet NT' Veer Veot Ver In our particular case, let's assume = 15% uncertainty of = 2.5% and = 12%. What would be the
#11 Sources of errors Think of how to determine the density of one egg (or perform the experiment at home)! Obviously you need to measure the mass and the volume. The mass measurement suffers from an imperfect scale balance and its uncertainty. Try to determine the volume using a cylindrical jar. Fill it with water, drown your egg and observe the rise of the water level. The volume measurement may suffer from the uncertainty of the measurement device and in addition you may make mistakes reading it off correctly. Calculate in terms of error propagation, which relative uncertainty and absolute uncertainty of the egg density you expect. Make a smart guess about the uncertainty of your devices and the water level reading. #12 Gauß broader Let's figure out what the full width at 95% and 99% probability is like. By which factor is that wider than the full width at half maximum?