20.4 Mathematical statistics

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}