Edit: Was able to figure out question by myself.
I am struggling mostly with part a, but also part b. I think because of the way the question is worded, I am having a hard time finding the null and alternative hypothesis.
a. For part a, I was thinking that the null hypothesis and alternative hypothesis would be: H0: u=10 (n) HA: u<10 (n)
or I was thinking it would be: H0: u=5 (n/2) HA: u>5 (n/2)
I honestly am not sure what the null and alternative hypothesis are because of the wording of the question. Also I'm not sure what it means to find 'the test with significance level 0.1'. Does this mean to find a test statistic? If so, how would I do this for this question (not sure)
b. For part b, type II error is when you fail to reject the null when the alternative is true.
So for this case would I use a binomial probability so that we are not in the rejection region and so that p=0.75.
Thanks.
I'll try to get you started--without a thorough discussion of (b), (c), and (d).
(1) The random variable $X \sim \mathsf{Binom}(n=10, p)$ counts the magician's successes in ten trials. If he has no ability and is strictly guessing then $p = 1/2.$ So let $H_0: p = 1/2.$ Then then it seems from the wording in the rest of the parts that the alternative should be $H_a: p > 1/2.$
A good estimate of $p$ is $\hat p = X/n.$ If the magician gets a 'relatively large' number correct, then we should reject $H_0$ and admit he has some ability to call the result by listening to the sound. The question is where to draw the line. The significance level is based on the 'null distribution' $\mathsf{Binom}(n=10, 1/2).$ Given that the magician has no ability, you want to be fooled into believing him only rarely. So you will want him to be right "much" of the time. For the significance level $\alpha = .1,$ you want to find $k$ (called the 'critical value') such that $P(X \ge k\, |\, p = 1/2) \le .1.$ For example, $P(X \ge 9\,|\,p=1/2) = 0.0107.$ So it seems you can select a smaller $k$ and still get a significance level 0.1. Maybe try $k=8.$ Use the formula for the binomial PDF to find what critical value to use.
(2) Here you want to know the chance you will accept $H_0$ (call him a liar) when he really can call the right answer with probability $p = 0.75.$ You will use the same critical value as in (1).
I have given you some help formulating the null and alternative hypotheses. I hope that is a start, you have more work to do.