a) You minimise the probability of a Type 1 error with a lower significance level, i.e. 1% instead of 5%. So, to answer this, would I pick out the degrees of freedom (3 - 1?) and the critical value under the 1% column on the t-table?
b) P(Type 1 error) = the significance level. How do I use the information in the table to calculate this?
c) Would it be $$0.9^3 = 0.729?$$
I haven't seen a question like this before, so I'm trying to build some intuition with this kind of problem. Any help appreciated!
$\displaystyle 1^{st} \text{ thing you need to know is that } P\left( T(X) \in \text{Rejection Region}\right)_{H_0} = P(\text{ TYPE 1 ERROR})$
$\text{where T(X) is the test statistic with which you conduct the testing}$
In your case T(X) is $\bar{X}$
And since $H_0:p=1 \text{ against } H_1:p\le 1$ your critical region will look like $T(X)\le C \Rightarrow \bar{X}\le C$
So if you want to minimize $P(\bar{X}\le C),\text{ you will have to reduce C}$
If previously C was set to be $\frac{2}{3}$ changing C to $\frac{1}{3}$ will reduce your type 1 error.
Hope this helps