1702. How many times does one need to measure a given quantity, the true value of which is equal to $a$, to be able to state with a probability of at least 0.95 that the arithmetic mean of these measurements differs from $a$ in absolute value by less than 2, if the root mean square deviation of each measurement is less than 10?
Solution. Let $X_{i}$ be the result of the $i$-th measurement. From the condition of the problem, it follows that $E X_{i}=a, D X_{i}<10^{2}=100$. Therefore,

$$
\frac{E X_{1}+E X_{2}+\ldots+E X_{n}}{n}=\frac{a+a+\ldots+a}{n}=a
$$

We need to find $n$ such that

$$
P\left(\left|\frac{X_{1}+X_{2}+\ldots+X_{n}}{n}-a\right|<2\right) \geqslant 0.95
$$

Since

$$
P\left(\left|\frac{X_{1}+X_{2}+\ldots+X_{n}}{n}-a\right|<\varepsilon\right) \geqslant 1-\frac{C}{n \varepsilon^{2}}
$$

where $C=100, \varepsilon=2$, the previous inequality will certainly be satisfied if

$$
1-\frac{C}{n \varepsilon^{2}} \geqslant 0.95
$$

From this,

$$
1-\frac{100}{n \cdot 2^{2}} \geqslant 0.95 ; 0.05 \geqslant \frac{100}{n \cdot 4} ; \quad n \geqslant \frac{100}{4 \cdot 0.05}=500
$$