(Solution) - Let X1 Xn be iid n 2 where both parameters are -(2025 Original AI-Free Solution)

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Let X1,...,Xn be iid n(?, ?2), where both parameters are unknown. Simultaneous inference on both ? and ? can be made using the Bonferroni Inequality in a number of ways.
a. Using the Bonferroni Inequality, combine the two confidence sets

into one confidence set for (?, ?). Show how to choose a, b, and k to make the simultaneous set a 1 - ? confidence set.
b. Using the Bonferroni Inequality, combine the two confidence sets
into one confidence set for (?, ?). Show how to choose a, b, and k to make the simultaneous set a 1 - ? confidence set.
c. Compare the confidence sets in parts (a) and (b).