# Computing sample size and Best Critical Region

#### Cynderella

##### New Member
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

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

approximately.