My answer:

1a) To minimize, you make the decision rule being, reject null hypothesis if x-bar is more than 1, or less than 0. Since there is 0 probability of happening, you never make a type 1 error

b) 1 - p-cubed

c) 0.9 cubed = 0.729

However, I don't exactly understand the concept of this question. So we know the probability of x bar being 1 is p cubed, however, is that even the same as the probability of a type 1 error being made? Also, am I missing some parts to the third part of the question? its 45 marks and I dont think its as simpe as p-cubed

Also, I just noticed the fact that the question asked for the testing of null of p=1 and p<1. I don't even know the answer to 1a now. Is it just minimize the significance level? So confused.