Hey liedora.
Can you show us what you have tried? Can you state the definition of your Type I errors with respect to the significance region (in terms of your c)?
If x1,x2,...,xn ~ Normal(theta,1) and B(theta) is the probability of rejecting the null hypothesis H0 : theta <= 0 in favor of the alternative hypothesis H1: theta > 0. Consider a statistical test that rejects H0 if x/(1/sqrt(n)) > c, where c is some positive number. Show that the power is B(theta) = P(Z > c - theta/(1/sqrt(n))) where Z is a standard normal variable.
Then if theta = 1, find a sample size n and a value c that will allow the probability of a type 1 error to be at most .05 and the probability of rejecting the null hypothesis to be at least 0.8 if theta = 1.
Thanks