# Category Archives: Testing

## Clinical trial

Let $$X$$ be the number of patients in a clinical trial with a successful outcome. Let $$P$$ be the probability of success for an individual patient. We assume before the trial begins that $$P$$ is unifom on $$[0,1]$$. Compute

1. $$f(P \mid X)$$
2. $${\mathbf E}( P \mid X)$$
3. $${\mathbf Var}( P \mid X)$$

## Random Digit

Let $$D_i$$ be a random digit chosen uniformly from $$\{0,1,2,3,4,5,6,7,8,9\}$$. Assume that each of the $$D_i$$ are independent.

Let $$X_i$$ be the last digit of $$D_i^2$$. So if $$D_i=9$$ then $$D_i^2=81$$ and $$X_i=1$$. Define $$\bar X_n$$ by

$\bar X_n = \frac{X_1 + \cdots+X_n}{n}$

1. Predict the value of $$\bar X_n$$ when $$n$$ is large.
2. Find the number $$\epsilon$$ such that for $$n=10,000$$ the chance that you prediction is off by more than $$\epsilon$$ is about 1/200.
3. Find approximately the least value of $$n$$ such that your prediction of $$\bar X_n$$ is correct to within 0.01 with probability at least 0.99 .
4. If you just had to predict the first digit of  $$\bar X_{100}$$, what digit should you choose to maximize your chance of being correct, and what is that chance ?

[Pitman p206, #30]

## Mark-recapture

A common problem in ecology, social networks, and marketing is estimating the population of a particular species or type. The mark-recapture method is a classic approach to estimating the population.

Assume we want to estimate the population of sturgeon in a section of the Hudson river. We use the following procedure:

1. Capture and mark $$h$$ sturgeons
2. Recapture $$n$$ sturgeon and you find that $$y$$ of them are marked
3. The estimated sturgeon population is $$N = \frac{h n}{y}$$.

Motivate statement $$3$$ using the hypergeometric distribution.

## Leukemia Test

A new drug for leukemia works 25% of the time in patients 55 and older, and 50% of the time
in patients younger than 55. A test group has 17 patients 55 and older and 12 patients younger than 55.

1. A uniformly random patient is chosen from the test group, and the drug is administered and it is a success. What is the probability the patient was 55 and older?
2. A subgroup of 4 patients are chosen and the drug is administered to each. What is the probability that the drug works in all of them?