20.2 Mathematical statistics

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}