Kassandra Ross

2022-06-20

From a collection of objects numbered $\left\{1,2,....,K\right\}$ 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 $P\left(T\ge t\right)$
I have the following questions:
What are $T$ and $t$?
I think that distribution is uniform, but how do I calculate the $p$-value?

mar1nerne

In a uniform distribution defined between the values $a$ and $b$, the cdf for $k\phantom{\rule{thinmathspace}{0ex}}ϵ\phantom{\rule{thinmathspace}{0ex}}\left[a,b\right]$ is $\left(k-a\right)/\left(b-a\right)$.

Dale Tate

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 $p={q}^{20}=0.015.$.