Example 14. The quality control department checks 475 items for defects. The probability that an item is defective is 0.05. Find, with a probability of 0.95, the boundaries within which the number $m$ of defective items among those checked will be enclosed.
Solution. We will use formula (4.3.9). First, we determine the number $\varepsilon>0$, and then the boundaries within which the number $m$ is confined. According to the condition $n=475, p=0.05, q=0.95, P=0.95$. In accordance with the condition

$$
2 \Phi\left(\varepsilon \sqrt{\frac{475}{0.05 \cdot 0.95}}\right)=0.95, 2 \Phi(100 \varepsilon)=0.95, \Phi(100 \varepsilon)=0.475
$$

From the table of values of the Laplace function $\Phi(1.96)=0.475$; hence, $100 \varepsilon=1.96$, from which $\varepsilon \approx 0.02$. Thus, $\left|\frac{m}{475}-0.05\right| \leq 0.02$; $-0.02 \leq \frac{m}{475}-0.05 \leq 0.02, 0.05-0.02 \leq \frac{m}{475} \leq 0.02+0.05, 0.03 \leq \frac{m}{475} \leq 0.07$, $14.25 \leq m \leq 33.25$. Since $m$ is an integer, then $15 \leq m \leq 33$.