Question

3. You will have just a single observation of X on which to base your choice between Ho: X has a Normal distribution with mean u = 5 and standard deviation o = 2 а VS. H1:X has a Binomial distribution with n =25 and p = 0.20. Consider the rejection rule "Reject Ho if X is an integer" Find a P(Type I Error) and B = P(Type II Error). Justify your answer.

Answer 1

$H_0: X \sim N(5,2^2) \:\: vs \:\: H_1: X \sim Bin(25,0.2)$

Given : rejection region = {X | X is an integer }

P[Type I error ]= Probability of rejecting H₀ when H₀ is actually true

=P[ X is an integer | X ~ N(5,4) ] = $\sum_{x=-\infty}^{\infty} P[X=x]= \sum 0=0$

The normal curve is between $x= -\infty \:\:to\:\: x= \infty$ , so betwwen this interval there will be all the integers included. But each integer will be a point on the axis and area corresponding to each of these points will be nearly 0. So, sum of

P[Type II error] = Probability of accepting  H₀ when H₀ is false

= P[X is an integer | X ~ bin(25,0.2) ]= 1

since, all of X's realisations are natural numbers and hence, integers.