1786. The root mean square deviation of ten measurements of a certain quantity is 1 cm. With what probability can we assert that the representativeness error in absolute value is less than 3 mm?
Solution. The result of the measurement can be considered as a normal random variable, and the sample is small. Therefore, by formula (2) we have:

$$
P(|\hat{X}-\bar{X}|<3)=\Theta_{10-1}\left(\frac{3 \cdot \sqrt{10-1}}{10}\right)=\Theta_{9}(0.9)=0.608
$$

If we had used formula (1), we would have obtained

$$
P(|\tilde{X}-\bar{X}|<\varepsilon) \approx \Phi\left(\frac{3 \cdot \sqrt{10-1}}{10}\right)=\Phi(0.9)=0.632
$$

which is slightly larger than the true value.