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[Solution] Let X_1,X_2,....,X_n be N(μ ,σ ^2) independent random variables. Let X̄=1/n∑limits_i=1^nX_i and α ∈ (0,1) Assume that n

Question: Let ${{X}_{1}},{{X}_{2}},....,{{X}_{n}}$ be $N\left( \mu ,{{\sigma }^{2}} \right)$ independent random variables. Let $\bar{X}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{X}_{i}}}$ and $\alpha \in \left( 0,1 \right)$

Assume that n = 25 and $\Pr \left( |\bar{X}-\mu |>2{{z}_{\alpha /2}} \right)=\alpha $. Find ${{\sigma }^{2}}$
Assume that n = 25 and that ${{\sigma }^{2}}$ is unknown. Let ${{S}^{2}}=\frac{1}{24}\sum\limits_{i=1}^{25}{{{\left( {{X}_{i}}-\bar{X} \right)}^{2}}}$. Find c such that $\Pr \left( \frac{5|\bar{X}-\mu |}{S}>c \right)=\alpha $

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[Solution] Let X_1,X_2,....,X_n be N(μ ,σ ^2) independent random variables. Let X̄=1/n∑limits_i=1^nX_i and α ∈ (0,1) Assume that n

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