EXAMPLE 30 Applying the 1% Guideline

image

image The Gardasil vaccine

In the Section 5.2 exercises, we found the probability that a randomly chosen patient completed the treatment to be . Suppose we take a random sample of size patients.

  1. Calculate the probability that both patients complete the treatment, using sampling without replacement.
  2. Confirm that the 1% Guideline applies.
  3. Approximate the probability that both patients complete the treatment using the 1% Guideline.
  4. Find the approximation error by comparing your answers from (a) and (c).

Solution

Define the following events:

  1. We have . Then, given that the first patient completed the treatment, we have, by sampling without replacement, . Then the Multiplication Rule gives us .
  2. The sample of size 2 represents of the population. Thus, the 1% Guideline applies, and we may treat the successive draws as independent.
  3. Thus, we can use the Multiplication Rule for Independent Events to solve this problem.

  4. The approximation error, the difference between (a) and (c), is . For most applications, this small approximation error is acceptable.

The 1% Guideline is also helpful when we do not know the size of the population, but may presume that the population is very large compared to the sample size.

NOW YOU CAN DO

Exercises 75–78