Mathematical statistics

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Problem: As a result of 10 independent measurements of a certain value \( X \), taken with the same accuracy, the experimental data, shown in the table, were obtained. Assuming that the results of the measurements are subject to a normal probability distribution rule, estimate the true values of \( X \) making use of the confidence interval, covering the true values of \( X \) with a confidence possibility of 0,95 . \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline\( x_{1} \) & \( x_{2} \) & \( x_{3} \) & \( x_{4} \) & \( x_{5} \) & \( x_{6} \) & \( x_{7} \) & \( x_{8} \) & \( x_{9} \) & \( x_{10} \) \\ \hline 5,3 & 3,7 & 6,2 & 3,9 & 4,4 & 4,9 & 5,0 & 4,1 & 3,8 & 4,2 \\ \hline \end{tabular}

20.1 Mathematical statistics

2.57 $

Problem: The technical control department checked \( n \) batches of the same type of products and found that the number \( \mathrm{X} \) of nonstandard products in one batch has an empirical distribution, as given in the table, in one line of which the number \( x_{i} \) of nonstandard products in one batch is shown, and in the other line the number of \( n_{i} \) batches containing \( x_{i} \) non-standard products is shown. It is required to test the hypothesis that the random variable \( \mathrm{X} \) (the number of non-standard products in one batch) is distributed according to the Poisson law, at a significance level of \( \alpha=0.05 \). \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\( n=\sum n_{i} \) & \( x_{i} \) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 1000 & \( n_{i} \) & 380 & 380 & 170 & 58 & 10 & 2 \\ \hline \end{tabular}

20.2 Mathematical statistics

3.34 $

Problem: Consider a random sample \( X_{i} \) from some known distribution. Plot the histogram and a frequency polygon based on the sample, the number of intervals is \( K \). Numerical data: \[ \begin{array}{l} i_{1}=-0,036 ; i_{2}=-0,809 ; i_{3}=0,315 ; \\ i_{4}=-0,265 ; i_{5}=0,471 ; i_{6}=-0,386 ; i_{7}=0,576 ; \\ i_{8}=-0,556 ; i_{9}=0,508 ; i_{10}=0,477 ; K=3 . \end{array} \]

20.3 Mathematical statistics

1.54 $

Problem: Perform the following calculations for the given sample \( X_{i} \). a) plot a histogram, a polygon, a sample distribution function; b) calculate the sample moments and associated quantities (the first, second and third moments, variance, standard deviation, kurtosis and skewness); c) assuming that the sample is obtained from a normal distribution, test the hypothesis that the mean is equal to null when the variance is unknown; the mean is equal to null, when the variance is equal to the sample. \begin{tabular}{|l|r|} \hline\( i \) & \multicolumn{1}{|c|}{\( X_{i} \)} \\ \hline 1 & 0,15 \\ \hline 2 & \( -3,28 \) \\ \hline 3 & 5,13 \\ \hline 4 & 0,19 \\ \hline 5 & \( -40,44 \) \\ \hline 6 & 11,06 \\ \hline 7 & \( -2,17 \) \\ \hline 8 & 0 \\ \hline 9 & 0,26 \\ \hline 10 & \( -7,68 \) \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline\( i \) & \( X_{i} \) \\ \hline 11 & 0,33 \\ \hline 12 & \( -8,03 \) \\ \hline 13 & 0,37 \\ \hline 14 & 23,67 \\ \hline 15 & 44,56 \\ \hline 16 & \( -1,62 \) \\ \hline 17 & 42,31 \\ \hline 18 & 2,62 \\ \hline 19 & 21,84 \\ \hline 20 & \( -1,7 \) \\ \hline \end{tabular} \begin{tabular}{|r|r|} \hline\( i \) & \multicolumn{1}{|c|}{\( X_{i} \)} \\ \hline 21 & \( -0,49 \) \\ \hline & \( -0,2 \) \\ \hline 23 & 0,35 \\ \hline 23 & \( -32,11 \) \\ \hline 25 & 13,72 \\ \hline 26 & \( -0,02 \) \\ \hline 27 & \( -1,95 \) \\ \hline 28 & \( -12,02 \) \\ \hline 29 & \( -7,96 \) \\ \hline 30 & \( -2,97 \) \\ \hline \end{tabular}

20.4 Mathematical statistics

10.28 $

Problem: According to the results of a selective study the distribution of average milk yields from one cow per day in a farm (liters) was found. \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline Interval & \begin{tabular}{c} \( 7,5- \) \\ 10,5 \end{tabular} & \begin{tabular}{c} \( 10,5- \) \\ 13,5 \end{tabular} & \begin{tabular}{c} \( 13,5- \) \\ 16,5 \end{tabular} & \begin{tabular}{c} \( 16,5- \) \\ 19,5 \end{tabular} & \begin{tabular}{c} \( 19,5- \) \\ 22,5 \end{tabular} & \begin{tabular}{c} \( 22,5- \) \\ 25,5 \end{tabular} & \begin{tabular}{c} \( 25,5- \) \\ 28,5 \end{tabular} & \begin{tabular}{c} \( 28,5- \) \\ 31,5 \end{tabular} & \begin{tabular}{c} \( 31,5- \) \\ 34,5 \end{tabular} \\ \hline \begin{tabular}{l} Number \\ of cows \end{tabular} & 2 & 6 & 10 & 17 & 33 & 11 & 9 & 7 & 5 \\ \hline \end{tabular} Find the probability \( P(15,4

20.5 Mathematical statistics

1.28 $

Problem: Consider a random sample \( X_{i} \) from some known distribution and answer the following questions: a) find the estimate of the parameter \( A \) using the method of moments, if it is known that the sample is made from the uniform distribution \( U(-1 ; A) \); b) find the estimate of the parameter \( B \), using the method of moments, if it is known that the sample is made from the uniform distribution \( U(-B ; B) \); c) find the estimates of parameters \( c \) and \( C \) using the method of maximum likelihood estimation, if it is known that the sample is made from the uniform estimation \( U(c ; C) \); d) find (and compare) the estimates of the parameter \( L \) using the method of moments and the method of maximum likelihood estimation, if it is known that the sample is made from the exponential distribution \( E_{L} \); e) find the estimate of the parameter \( m \) using the method of moments, if it is known that the sample is made from the normal distribution \( N(m, 1) \); f) find the estimates of the parameters \( M \) and \( S \) using any known method, if it is known that the sample is made from the normal distribution \( N(M, S) \); g) plot a histogram and a polygon based on the sample, the number of intervals is 3 ; h) in each of the points (a) - (f) estimate the proximity of this theoretical distribution to the empirical one

20.7 Mathematical statistics

25.69 $