Hypothesis testing $p$ value

Question

I have to following question:

From a collection of objects numbered $\{1,2,....,K\}$ $20$ objects are picked and replaced. We want to test $H_0: K=100000$ against $H_1 < 100000$, with the highest ranking number $M$ of our sample as test statistic. We find for our realisation for $M$ the value $81115$.

What is the $P$ value?

The correct answer is: $0.015$

I know that the definition of the $p$-value is:

The $p$-value is the probability of getting the observed value of the test static or a value with even greater evidence against $H_0$, if the hypothesis is actually true

or in formula form $P (T \ge t)$

I have the following questions:

• What are $T$ and $t$?
• I think that distribution is uniform, but how do I calculate the $p$-value

Can I get feedback?

Answer 1

HINT: in a uniform distribution defined between the values $a$ and $b$, the cdf for $k \, \epsilon \,[a,b]$ is $(k-a)/(b-a)$.

Answer 2

The probability that one randomly chosen object is at most $81115$ is $q = 81115/100000$. Since the draws are independent (because they are replaced after each pick), the probability that all $20$ choices are less than or equal to $81115$ is $p = q^{20} = 0.015.$