Under otherwise the same assumptions as in exercise 5.12, only 6 seedlings had been planted. Determine

(1) the exact probability that the arithmetic mean

\[\bar{X}_{6}=\frac{1}{6} \sum_{i=1}^{6} X_{i}\]

exceeds \(\mu=32 \mathrm{~cm}\) by more than \(0.06 \mathrm{~cm}\) (Erlang distribution),

(2) by means of the central limit theorem, determine a normal approximation to the probability

\[P\left(\bar{X}_{6}-32>0.06\right)\]

Give reasons why the approximation may not be satisfactory.

Data from Exercise 5.12

After six weeks, 24 seedlings, which had been planted at the same time, reach the random heights \(X_{1}, X_{2}, \ldots, X_{24}\), which are independent, identically exponentially distributed as \(X\) with mean value \(\mu=32 \mathrm{~cm}\).

Based on the Gauss inequalities, determine

(1) an upper bound for the probability that the arithmetic mean

\[\bar{X}_{24}=\frac{1}{24} \sum_{i=1}^{24} X_{i}\]

differs from \(\mu\) by more than \(0.06 \mathrm{~cm}\),

(2) a lower bound for the probability that the deviation of \(\bar{X}_{24}\) from \(\mu\) does not exceed \(0.06 \mathrm{~cm}\).