Question

Question 4. Suppose for i=1,...,n both the mean and variance are unknown. Based on n=100 sample data, we would like to test vs

a) at a type 1 error level $\alpha$, find a sample statistic T and the rejection region R that correctly controls $\alpha$ exactly, i.e., find T and R that satisfy (must be exact in distribution not approximate).

b) Compute the asymptotic power of T, i.e., what does $Prob(T(X)>R|\mu_x<0.5)$ converge to as sample size goes to infinity?

Question 5. Following question 4, except that the test now becomes $H_0:\mu_x\ge 0.5$ vs

a) does your previous statistic T still control the type 1 error level correctly for the new hypothesis? Namely, is it true that $Prob(T(X)>R|\mu_x\ge0.5)\le\alpha$ ?

b)Compute the asymptotic power of T in this case

r, Ni.d Normal(μι, σι) o : μ1-0.5 HA : μ| RI,11-05)

Answer 1

Two ways to increase the statistical power of an experiment overlap in the two sampling distributions for null hypothesis and alternative hypothesis Larger the sample size, the higher the power 2. Decrease random error - Reduce the variance: smaller variance yields higher power. Power is higher when the standard deviation is smaller than when it applied consistently.