Example 6.37. At a significance level of $\alpha=0.05$, test the hypothesis of a normal distribution of the general population, given the empirical and theoretical frequencies:

| $n_{i}$ | 6 | 12 | 16 | 40 | 13 | 8 | 5 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $n_{i}^{\prime}$ | 4 | 11 | 15 | 43 | 15 | 6 | 6 |
Solution. The number of different variants $m$ is 7, so the number of degrees of freedom of the $\chi^{2}$ distribution is $7-3=4$. According to the table of critical points of the $\chi^{2}$ distribution, at the significance level $\alpha=0.05$ and 4 degrees of freedom, we find $\chi_{\text {crit. }}^{2}=9.5$. We will calculate $\chi_{\text {obs. }}^{2}$, for which we will construct a calculation table.

| 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $i$ | $n_{i}$ | $n_{i}^{\prime}$ | $n_{i}-n_{i}^{\prime}$ | $\left(n_{i}-n_{i}^{\prime}\right)^{2}$ | $\frac{\left(n_{i}-n_{i}^{\prime}\right)^{2}}{n_{i}^{\prime}}$ |
| 1 | 6 | 4 | 2 | 4 | 1 |
| 2 | 12 | 11 | 1 | 1 | 0.09 |
| 3 | 16 | 15 | 1 | 1 | 0.061 |
| 4 | 40 | 43 | -3 | 9 | 0.21 |
| 5 | 13 | 15 | -2 | 4 | 0.27 |
| 6 | 8 | 6 | 2 | 4 | 0.7 |
| 7 | 5 | 6 | -1 | 1 | 0.17 |

Since $\chi_{\text {obs. }}^{2}<\chi_{\text {crit. }}^{2}$, the null hypothesis about the normality of the general population is accepted.