637. Using the Pearson criterion, at a significance level of 0.01, determine whether the discrepancy between the empirical frequencies $n_{i}$ and the theoretical frequencies $n_{l}^{\prime}$, which are calculated based on the hypothesis of a normal distribution of the general population $X$, is random or significant:

| $n_{i}$ | 8 | 16 | 40 | 72 | 36 | 18 | 10 |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| $n_{i}^{\prime}$ | 6 | 18 | 36 | 76 | 39 | 18 | 7 |
Solution. Let's find the observed value of the Pearson criterion: $\chi_{\text {obs }}^{2}=\sum\left(n_{i}-n_{i}^{\prime}\right)^{2} / n_{i}^{\prime}$. We will create the calculation table 19.

From table 19, we find the observed value of the criterion: $\chi_{1 \text { obs }}^{2}=3.061$.

Using the table of critical points of the $x^{2}$ distribution (see appendix 5), at a significance level of 0.01 and degrees of freedom $k=s$ $-3=7-3=4$, we find the critical point of the right-sided critical region $\chi_{\text {cr }}^{3}(0.01 ; 4)=13.3$.

Since $\chi_{\text {obs }}^{2}<\chi_{\text {cr }}^{2}$, there is no reason to reject the hypothesis of normal distribution of the general population. In other words, the discrepancy between empirical and theoretical frequencies is insignificant (random).

T a b l e 1 9

| $i$ | $n_{i}$ | $n_{i}^{\prime}$ | $n_{i}-n_{i}^{\prime}$ | $\left(n_{i}-n_{i}^{\prime}\right)^{2}$ | $\left(n_{i}-n_{i}^{\prime}\right)^{2} / n_{i}^{\prime}$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  |  |  |  |
| 1 | 8 | 6 | 2 | 4 | 0.667 |
| 2 | 16 | 18 | -2 | 4 | 0.222 |
| 3 | 40 | 36 | 4 | 16 | 0.444 |
| 4 | 72 | 76 | -4 | 16 | 0.211 |
| 5 | 36 | 39 | -3 | 9 | 0.231 |
| 6 | 18 | 18 | -3 | 9 | 1.286 |
| 7 | 10 | 7 | 3 | 9 |  |
| $\Sigma$ | $n=200$ |  |  |  | $\chi_{\text {obs }}^{2}=3.061$ |