506. Find the minimum sample size for which with a reliability of 0.975, the accuracy of estimating the mean $a$ of the population using the sample mean is $\delta=0.3$, given that the standard deviation $\sigma=1.2$ of the normally distributed population is known.
Solution. We will use the formula that defines the accuracy of estimating the mathematical expectation of the general population by the sample mean: $\delta=t \sigma / \sqrt{n}$. From this,

$$
n=t^{2} \sigma^{2} / \delta^{2}
$$

According to the condition, $\gamma=0.975$; therefore, $\Phi(t)=0.975 / 2=0.4875$. From the table in Appendix 2, we find $t=2.24$. Substituting $t=2.24, \sigma=1.2$, and $\delta=0.3$ into (*), we obtain the required sample size $n=81$.