Help with homework. Decision rule to minimise a type 1 error of a sample mean, with probability unknown

[Towls]

Question link:

http://imgur.com/Cw3UiZx

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.

 

[Towls]

Judging by how the question is focusing on mainly the probability side of things, I feel like I'm not supposed to calculate any test statistic, or use it as a point.

 

[j58]

I'm inclined to agree with all your answers. For part (b), you should probably calculate the numerical value.

What might be confusing you is that the Type 1 error rate is defined as P(reject null | null is true). That is, it is a conditional probability, calculated under the assumption that the null hypothesis is true.