A casino dealer suspects that his die isn't a fair die. He thinks that the die gives more chances to the even numbers (the probability for an even number is larger than $0.5$). To test his suspects he decides to have an experiment, he writes down the outcome of $60$ independent die rolls. The dealer would say that the die is unfair if the number of even numbers in those $60$ rolls will be larger than $35$.

I'm asked to:

1. Formalize the dealer's assumption.
2. What is the rejection region ($C$)?
3. What is the probability for both mistakes ($\alpha$ and $\beta$)?

I started by saying that $X$, the number of even results of a fair cube $X\sim B(\frac{1}{2}, 60)$, and thus $X \sim N (30, 15)$

$H_0: \mu = 30$
$H_1: \mu > 30$

So this is the dealer's assumption. The rejection area should be derived from the fact that the dealer will reject $H_0$ if and only if the even numbers on the experiment will be less or equal to $35$. I'm having a hard time finding $C$ and calculating $\alpha$ and $\beta$.

A casino dealer suspects that his die isn't a fair die. He thinks that the die gives more chances to the even numbers (the probability for an even number is larger than $0.5$). To test his suspects he decides to have an experiment, he writes down the outcome of $60$ independent die rolls. The dealer would say that the die is unfair if the number of even numbers in those $60$ rolls will be larger than $35$.

I'm asked to:

1. Formalize the dealer's assumption.
2. What is the rejection region ($C$)?
3. What is the probability for both mistakes ($\alpha$ and $\beta$)?

I started by saying that $X$, the number of even results of a fair cube $X\sim B(\frac{1}{2}, 60)$, and thus $X \sim N (30, 15)$

$H_0: \mu = 30$
$H_1: \mu > 30$

So this is the dealer's assumption. The rejection area should be derived from the fact that the dealer will reject $H_0$ if and only if the even numbers on the experiment will be less or equal to $35$. I'm having a hard time finding $C$ and calculating $\alpha$ and $\beta$.