Who Ate My Mochi? (Type I and Type II Errors)

A beautiful day, I made some fresh mochi and left them on the table for dinner dessert.

After that, I went to the supermarket to get some cooking ingredients to cook dinner. However, when I came back from the supermarket, they were already gone. I wonder who did this among the children at home. Interestingly, I bought some cupcakes from the store. So I decided that the one who ate all the mochi must also eat all the cupcakes but with ketchup and salad dressing over as topping. He he… he he…

Now if I am to establish a hypothesis testing procedure to help me conclude if Collin, the biggest foodie among the children, was the one who ate the mochi, then I would establish the null hypothesis as:
: Collin was innocent (i.e., he didn’t eat the cake)
And this is the hypothesis that I want to refute. I want to support the alternative hypothesis:
.

Now note that even though I want to support and refute , if there is not enough evidence against , I would not refute anyway. This is because I don’t want to force Collin to eat cupcakes with ketchup toppings if he was innocent. It would be very terrible to let that happen. Also, I need enough evidence to convince everyone as well. Such a mistake, if happened, is called a “type I” error. More specifically, type I error is the error that we commit when we reject the null hypothesis (i.e., convict an innocent) while the null hypothesis is true (i.e., the person is not guilty).

On the other hand, type II error is not rejecting the null hypothesis (i.e., accepting the person is innocent) while the alternative hypothesis is true (i.e., the person is guilty).