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?