# Smallest µ such that H0 is rejected (hypothesis test, unknown mean)

#### lollerendk

##### New Member
Let X1, X2, . . . , Xn be normal-distributed random variables with unknown mean µ and variance 1 and let X^- =1/n sum_{n=i}^n be their sample mean. Consider the one-sided hypothesis test with hypotheses H0 : µ0 < 0 and H1 : µ0 > 0 with parameters δ = 0.01 and n = 100.
What is the smallest µ such, that H0 is rejected with a probability of 90%?

So, if I understand it correct, I have to find the sample mean X^- which satisfies that P(H0 reject|H0)<δ=0.05 and P(H0 reject)=0.9. I simply fail to realise how to incorporate the last information, that P(H0 reject)=0.9, in the hypothesis test.
As α = Level of significance = P(Type I error) = P(Reject H0 | H0 is true) = 0.01, s^2=1, n=100, µ0<0, I have tried to use the formula
z<sqrt(n)*(X^- - µ0)/s, for z=2.326 (as P(Z<2.326)=0.99), then 2.326 < 10*X^- <==> X^-=0.2326, as µ0<0. Thus, the found mean is 0.23.
There is, however, no way this is true. I have not used all the information.
I am totally stuck, and would appreciate any nudge in a better direction greatly. I hope the above text is not too confusing/badly written. Thank you 