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### (9 points)4. Series 30. Year - P. statistician's daily bread

We've all been there, you spread some honey or some preserve on a slice of bread, take a bite, and suddenly, the spread drips through a hole and lands right on your hand. Determine how does the probability that there is a hole straight through a slice of bread depend on its thickness. The model of how does the dough rise is left up to you. (For example, evenly distributed bubbles with an exponential distribution of radii is a good model).

Michal stained his clothes.

### (10 points)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 points)3. Series 30. Year - 1. long film

You are downloading your favourite film with file size 12 GB at 10 MB ⁄ s. Assuming the signal travels along a twisted-pair wire at the speed of light and modulation spreads the transmission speed evenly, that is at 1 b ⁄ s we would have to receive 1 second of the signal to acquire 1 bit of information, determine the length of cable filled by the film's data if it travels along a sufficiently long cable.

Michal's colleague claimed that 100Gb Ethernet frames are smaller than a chip.

### (12 points)3. Series 30. Year - E. reflective snap band

Measure as many characteristics of a high-vis snap band as you can. We are specifically interested in:

• The band contains a piece of metal on the inside, which can be bent lengthwise (when coiled) or along the shorter edge (when straight). What are the radii of curvature of these bents if there is no external force?
• If the band is straight and we start bending it in one place, at what angle will it snap into the bent state? At what angle does it become straight again? (Do we see any hysteresis?)
• What is the torque required to bend the band?
• Is one of the states (bent or straight) more energetically favourable? Estimate by how much. Unfortunatelly, we are unable to mail these bands abroad, we therefore ask that you obtain one yourself and include pictures of the band you used in your solution.

Erik could not bend his …

### (10 points)3. Series 30. Year - S. limiting

1. Try to, in your own words, describe the method for creating interval estimations of expected value of a general distribution of measured data (it is sufficient to describe the following: central limit theorem (CLT), covariance, correlation (Pearson correlation coefficient), multidimensional CLT, law of propagation of uncertainty and its uses.) It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
2. In the attached datafile mereni3-1.csv there are measurements of a certain physical quantity $v$. Assume we cannot be sure whether the measured data have a normal distribution. Find the uncertainty (standard deviation) of the measurements (neglect the type B uncertainty), set up the interval estimations using CLT and briefly interpret their meaning. How would the results (and interpretation) change if only the first quarter of the data was available?
3. Suppose our aim is to measure a physical quantities $x$ and $y$, which we will then plug into the equation $\begin{equation*} v= \frac {1}{2} x y^2 . \end {equation*}$ and suppose that we are certain that all measurements are independent and we already have measured a significant amount of data, processed them and there are the results \begin{align*} x &= (5,2\pm 0.1) , \\ y &= (12{,}84\pm 0.06) . \end {align*} Estimate the value of  $v$ and its uncertainty.

Hint: These equations may come in handy $$\frac{\partial}{\partial x} $\frac {1}{2} x y^2$ = \frac {1}{2} y^2\, ,$$ $$\frac{\partial}{\partial y} $\frac {1}{2} x y^2$ = x y \, .$$
4. Using a computer simulation demonstrate the validity of central limit theorem i.e. generate $n$-tuples (sequences of $n$ real numbers) of independent realizations of a random variable, which does not have a normal distribution (use the exponential, uniform and Poisson distributions with arbitrary parameters) and show, using a histogram, that applying the transformation $\begin{equation*} \sqrt {n}\frac {\overline {x_n - \mu }}{S_n} , \end {equation*}$ to the data will (approximately) yield a normal distribution $N(0, 1)$.

Bonus: Suppose our aim is to measure physical quantities $x$ and $y$, which we will then plug into $\begin{equation*} v= x^2 \sin y . \end {equation*}$ Assume the most general model of measurement (i.e. the measured data do not have a normal distribution and the measurements of $x$ and $y$ may not be independent. In the datafile mereni3-2.csv you may find the results of measurements of $x$ and $y$, determine the uncertainty of $v$ and construct an interval estimation of $v$.

For data processing and creating of plots use the R programming language. In the attached scripts is explained all necessary syntax.

### (12 points)2. Series 30. Year - E. one full fat milk, please

Milk with higher fat content should be „whiter“ – more light is scattered and less is transmitted. Conduct a measurement of the fat content of milk with the help of a color scale (contact us at fykos@fykos.czto get the pdf with the scale – you have to print it yourself). The difference in whiteness is most apparent when you add a dye to each glass of milk. You can use e.g. black ink or any other dye, but with different colors you have to create your own color scale which you have to add to your solution. Use different types of milk and mixture of milk with water. Discuss the reliability of this method of measurement.

Mára byl bledý jako stěna.

### (10 points)2. Series 30. Year - S. guessing problem

1. Describe in your own words the purpose of interval estimation of mean of a normal distribution and explain its physical interpretation (it is sufficient to describe, in your own words, the following concepts: physical interpretation of the estimation of expected value, difference between point and interval estimation, measurement uncertainty). It’s not necessary to state the exact mathematical derivations. It’s sufficient to briefly explain the concepts and their properties.
2. Attached to this task, in the file mereni1.csv there are measured values of a certain physical quantity (assume type B uncertainty of B $s\_B = 0{,}1$). Create both the point and interval estimations of the measured physical quantity and try to interpret their meaning.
3. Suppose we measure a certain physical quantity and we know that due to the method being used, the measured values will have a variance equal to a constant $c$ (ignore the type B uncertainty). How many measurements do we need to make to achieve an uncertainty below $s$?
4. In the attached file mereni2.csv there are data of measurements one physical quantity two different ways (neglect type B uncertainty). Which method used more precise measurement equipment? Which method produced a more precise results Briefly give reasons for your answers.

Bonus: Try to rigorously derive that in a normal distribution the sample variance is an unbiased estimate of the real variance (i.e. the mean of sample variance is equal to the real variance). For the solution of this problem you may use any and all sources (if you cite them correctly).

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 guessed the optimal wording of the problem, let's hope he was right.

### (5 points)1. Series 30. Year - 3. Bouncy bounce

Let's have an ideal bouncy ball (with coefficient of restitution equal to one and negligible dimensions). We throw this bouncy ball down an infinitely long staircase, where a step has height $h$ and length $l$. The bounces happen without any influence from friction. Describe the relation between the maximum height reached (measured from the first step) after $n-th$ bounce and the initial parameters.

Lubošek potkal v městské dopravě Mikuláše.

### (7 points)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.

### (12 points)1. Series 30. Year - E. Pechschnitte

Does bread always falls on the side that has the spread on it? Explore this Murphy's law experimentally with emphasis on statistics! Does it depend on the dimensions of the slice, or the composition and the thickness of the spread? Try to explain the experimental results with a theory. Use a sandwich bread.

Terka má stůl ve špatné výšce. 