Computing sample size and Best Critical Region

Let \(X_1,X_2,\ldots,X_n\) denote a random sample from a normal distribution \(N(\theta,100)\).

Show that \(C=[(x_1,x_2,\ldots,x_n):c\leq \bar x=\frac{\sum_1^n x_i}{n}]\)

is a best critical region for testing \(H_o:\theta=75\) against \(H_1:\theta=78\).

Find \(n\) and \(c\) so that

\(P[(X_1,X_2,\ldots,X_n)\epsilon C;H_o]=P(\bar X\geq c;H_o)=0.05\)
\(P[(X_1,X_2,\ldots,X_n)\epsilon C;H_1]=P(\bar X\geq c;H_1)=0.90\)



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