The measurement error \(X\) of a measuring device has mean value \(E(X)=0\) and variance \(\operatorname{Var}(X)=0.16\). The random outcomes of \(n\) independent measurements are \(X_{1}, X_{2}, \ldots, X_{n}\), i.e., the \(X_{i}\) are independent, identically as \(X\) distributed random variables.

(1) By the Chebyshev's inequality, determine the smallest integer \(n=n_{\text {min }}\) with property that the arithmetic mean of \(n\) measurements differs from \(E(X)=0\) by less than 0.1 with a probability of at least 0.99 .

(2) On the additional assumption that \(X\) is continuous with unimodal density and mode \(x_{m}=0\), solve (1) by applying the Gauss inequality (5.4).

(3) Solve (1) on condition that \(X=N(0,0.16)\).

Data from 5.4

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4 P(|X xm| ) [ + ( xm)], [+(-xm)], &> 0. 982