Thread: Hypothesis Testing: Power Function

My attempt:
a) Test statistic: Z = (X bar - 13) / (4/sqrt n)
Let μ = μ_a > 13 be a particular value in H_a
Power(μ_a)
= 1- beta(μ_a)
= P(reject Ho | H_a is true)
= P(reject Ho | μ = μ_a)
= P(Z>1.645 | μ = μ_a)
= P(X bar > 13 + [(1.645 x 4)/(sqrt n)] | μ = μ_a)
I also know that X bar ~ N(μ, 16/n)
I am stuck here...How can I proceed from here and express the power in terms of μ and n?


Thank you very much!


Note: this topic is also under discussion in SOS mathematics cyberboard

Now I have a little more progress:

Power(μ_a)
= P(X bar > 13 + [(1.645 x 4)/(sqrt n)] when μ = μ_a)
And X bar has the distribution N(μ, 16/n)

Here, we know the distribution of X bar, so I believe that we can express P(X bar > 13 + [(1.645 x 4)/(sqrt n)] when μ = μ_a) as an integral of the normal density and replace μ by μ_a, but how can I integrate it and get a compact answer?

Thank you!