Let X be a single (i.e. n=1 ) Gaussian random variable with unknown mean μ and variance 1 . Consider the following hypotheses:

Question

H0:μ=0vsH1:μ≠0.

Define a test ψα:R→{0,1} with level α that is of the form

ψα=1{fα(X)>0},

for some function fα:R→R .

We want our test ψ above to satisfy the following:

symmetric in the value of X

its “acceptance region" is an interval. (The acceptance region of a test is the region in which the null hypothesis is not rejected, i.e. the complement of its rejection region.)

Specify the function fα(X) in terms of α below.

(Type alpha for α . If applicable, enter abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α -quantile of a standard normal distribution, e.g. enter q(0.01) for q0.01 .)

f(X)=?

Answer 1

Hi Ram J, these are very interesting questions. When you have the opportunity, can you please show your work for any questions you have attempted to answer, and let us know where you are getting stuck? Thanks.