Question

I want to test the hypothesis that a die is fair by rolling it over and over, independently, until the third time I see any single number. I will conclude that the die is loaded (not fair) if it takes four or fewer rolls for any single number to come up three times.

What is the significance level?

Answer 1

A die will be said to be fair if for n number of rolling, each of the 6 sides occurs with equal probability, i.e, probability of occurrence any of the 6 sides is 1/6.

Here the null hypothesis is .

We'll solve this problem using p-value.

We'll reject the null hypothesis if, within the first 4 or fewer rolls, any single number occurs thrice.

Let X be a variable denoting the number of rolls until any single number occurs third times.

So

Let denotes the number of rolls until i occurs third times, i= 1(1)6

Under H0,

Thus Observed p value

= Probability that any one of the is less than or equal to 4

If we reject the null hypothesis if any single number occurs thrice. within the first 4 or fewer rolls, then the above p value in (i) must be less than the significance level.

Therefore the significant level of the test must be more that 0.04, that is the significance level may be 0.05.