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mathematics

0. Series 31. Year - 5.

We are sorry. This type of task is not translated to English.

6. Series 30. Year - S. nonlinear

  1. Try to describe in your own words how and for what purpose nonlinear regression is used (it is sufficient to briefly describe the following: model of nonlinear regression, methods for finding regression coefficients, uncertainties in the determination of regression coefficients, uncertainties in the function being fitted, statistic methods for testing the values of the regression coefficients, how to choose the form of the fitting function). It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
  2. In the attached data file regrese1.csv you may find pairs of valuest $(x_i, y_i)$. Fit these data with a sine function in the form \[\begin{equation*} f(x) = a+ b \cdot \sin (c x + d) . \end {equation*}\] Plot the measured values and the fit and comment on it briefly. It’s not necessary to perform regression diagnostics.
    Hint: Be wary of correct constraints for the values of parameter $c$.
  3. In the attached data file regrese2.csv you may find pairs of values $(x_i, y_i)$. Fit these data with an exponential function in the form \[\begin{equation*} f(x) = a+ \eu ^{b x + c} . \end {equation*}\] Estimate the values of all regression coefficient including their uncertainties.
    Hint: Using graphical method examine homoscedasticity. You may use Huber-White (sandwich) estimator for determining the uncertainties in estimating regression coefficients if necessary.
  4. In the attached data file regrese3.csv find the pairs of values $(x_i, y_i)$. Fit these data with a hyperbolic function in the form \[\begin{equation*} f(x) = a+ \frac {1}{b x + c} . \end {equation*}\] Plot the measured data in the form of means and error bars and briefly comment on it. Perform the regression diagnostics.

Bonus: In the attached data file regrese4.csv you may find pairs of values $(x_i, y_i)$. We want to fit these data with a function too complex to be expressed analytically. Use spline regression to fit these data with appropriately chosen knots and order).

For data processing and creating the plots, you may use the R programming language. Most of these tasks can be solved by slightly altering the attached scripts.

Michal wanted to make the last series as hard as possible.

5. Series 30. Year - S. linear

  1. Try to describe in your own words how and for what purpose linear regression is used (it is sufficient to briefly describe the following: two significant applications of linear regression, least squares method, maximum likelihood estimation, linear regression model, basic graphical methods of regression diagnostics). It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
  2. In the attached data file linreg1.csv you may find the results of a certain physical experiment, in which we measured the pairs of data $(x_i, y_i)$. We want to fit the measured data with a theoretical function in this case a parabola in the form \[\begin{equation*} f(x) = ax^2 + bx + c . \end {equation*}\] Determine the value of the coefficient $a$ and its uncertainty. It is not necessary to use regression diagnostics.
  3. In the attached data file linreg2.csv you may find the results of a certain physical experiment, in which we measured the pairs of data $(x_i, y_i)$. We want to fit the measured data with a theoretical function, in this case a logarithmic function in the form \[\begin{equation*} f(x) = a+ b \cdot \log (x)  . \end {equation*}\] Plot the measured data into a graph with the fitting function and briefly comment on it. It is not necessary to use regression diagnostics.
  4. Suppose we have measured pairs of data $(x_i, y_i)$ and want to fit them with a linear function in the form \[\begin{equation*} f(x) = a+ bx  . \end {equation*}\] Derive the exact formula for calculating the regression parameters. You may use any and all sources, if you cite them correctly. (Actually derive the formula, do not just write it.)

Bonus: In the tasks b) and c) perform regression diagnostics and discuss, whether all necessary criteria (assumptions) are met.

For data processing and creating the plots, you may use the R programming language. Most of these tasks can be solved by slightly altering the attached scripts.

Michal heard somewhere, that linear regression is really easy.

4. Series 30. Year - S. testing

  1. Try to describe in your own words what purpose serves testing of hypotheses and how its done (it is sufficient to briefly describe the following: null hypothesis and alternative hypothesis, type I and type II error, level of significance, test statistic, confidence level, $p$-value). It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
  2. In the attached data file testovani1.csv there are measurements of a certain physical quantity. Using one-sample $t$-test find out whether the real value of the measured quantity is equal to $20$. Then suppose our aim is to show that the real value is larger than $20$. Test this claim using an appropriate modification of $t$-test (be careful which null hypothesis and alternative hypothesis you choose).
  3. In the attached data file testovani2.csv you may find the measurements of two different physical quantities. Assume the measurements to be of the same physical characteristic, just under different conditions (temperature, pressure etc.). Test the hypothesis that the value of said physical characteristic is the same under both sets of outside conditions using the two sample $z$-test.
  4. Use the data from the last task in the first series of this year and using Kolmogorov–Smirnov test determine which of the four data samples comes from uniform distribution and which comes from exponential distribution.

Bonus: Assume you have at your disposal measurements of 2 physical quantities (i.e. two sets of measurements), where all the data are independent. Set up a modified $z$-test, that will test the hypothesis that the real value of the first physical quantity is double the real value of the second physical quantity. It is sufficient to set up the corresponding test statistic and confidence level. (Hint: Use the multidimensional central limit theorem with appropriately selected function $f$, and then proceed analogically to setting up a classical two-sample $z$-test) For data processing and creating the plots, you may use the R programming language. Most of these tasks can be solved by slightly altering the attached scripts.

Michal wanted to test, how difficult problems you can solve.

3. Series 30. Year - 2. hellish

figure

A road and a pathway, both leading to Hell, lie on different sides of a river. We are moving along the river in the direction shown in the picture. Banks of the river are formed by concentric circular arcs. The pathway leads along one bank, the road along the other and the width of the river is constant. Route along which bank of the river is faster? For every arc, we know the central angle $φ_{1},φ_{2},\ldots$ and the radius $r_{a1},r_{b1},r_{a2},r_{b2},\ldots$, where the suffices $a,b$ denote the left and right bank respectively.

Occurred to Lukas on the way to Peklo (Hell).

2. Series 30. Year - 1. beach date

Imagine you are going on a date with your girlfriend/boyfriend and you end up watching the sunset on the beach. The sun above the sea horizon looks very romantic, so to prolong this special moment, you decide to use a forklift to lift you up. The forks of the forklift move up with such speed that you can see the sun touching the horizon at any moment. Determine the speed of the forks.

Dominika vzpomínala na Itálii.

1. Series 30. Year - 4. The world is tilted

An observer is on a ship in the open sea, in the height $h$ above the sea surface. There is a horizontal railing in the distance $d$ from him, in such a position, that when he looks directly, perpendicularly at it, the bottom edge of the railings touches the horizon. However, when he looks at the part of the railing that is distance $l$ to the left of the original point, the horizon appears to be $s±s_{s}$ lower. What is the radius of the Earth.

Lubošek trpí mořskou nemocí.

1. Series 30. Year - 5. On a walk

Katka decided to go for a walk with her pet rat. They arrived on a flat meadow and when the rat was at a distance $x_{1}=50\;\mathrm{m}$ from Katka, she threw him a ball with the speed $v_{0}=25\;\mathrm{m}\cdot \mathrm{s}^{-1}$ and an angle of elevation $α_{0}$. In that moment, he started running towards her with the speed $v_{1}=5\;\mathrm{m}\cdot \mathrm{s}^{-1}$. Find a general formula for an angle $φ$ as a function of time, where the angle $φ(t)$ is the angle between the horizontal plane and the line between the rat and the ball. Draw this function into a graph and, based on the graph, determine, whether it's possible for the ball to obscure the Sun for the rat, when the Sun is situated $φ_{0}=50°$ above the horizon in the direction of the running rat. Use the acceleration due to gravity $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}$ and for simplicity imagine we are throwing the ball from a zero height.

Mirek pozoroval, co se děje v trávě.

5. Series 29. Year - 2. multiparticular

Let's have a container that is split by imaginary plane into two disjunct parts A and B, identical in size. There are $nparticles$ in the container and each of them has a probability of 50 % to be in part A and probability 50 % to be in part B. Figure out the probabilities of the part A containing $n_{A}=0.6n$ or $n_{A}=1+n⁄2$ particles respectively.. Solve it for $n=10$ and $n=N_{A}$, where $N_{A}≈6\cdot 10^{23}$ is Avogadro's constant.

4. Series 29. Year - 3. Save the woods

We have a toilet paper roll with the diameter $R=8\;\mathrm{cm}$ with an inside hollow tube of diameter $r=2\;\mathrm{cm}$. Every layer of the paper has the thickness $d=200µm$ and the layers lies perfectly on top of each other. By how many does the number of pieces of the paper differ had we used a piece of the length $l_{1}=9\;\mathrm{cm}$ instead of $l_{2}=13\;\mathrm{cm}?$ A part of the solution has to be an estimate of the approximation error (if you use one).

Bonus: Calculate the precise length of the spiral the toilet paper makes.

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