Sampling from a Small Population

When we sample observations from a population, usually we’re only sampling a small fraction of the possible individuals or cases. However, sometimes our sample size is large enough or the population is small enough that we sample more than 10% of a population44 without replacement (meaning we do not have a chance of sampling the same cases twice). Sampling such a notable fraction of a population can be important for how we analyze the sample.

EXAMPLE

Professors sometimes select a student at random to answer a question. If each student has an equal chance of being selected and there are 15 people in your class, what is the chance that she will pick you for the next question?

If there are 15 people to ask and none are skipping class, then the probability is 1/15, or about 0.067.

EXAMPLE

If the professor asks 3 questions, what is the probability that you will not be selected? Assume that she will not pick the same person twice in a given lecture.

For the first question, she will pick someone else with probability 14/15. When she asks the second question, she only has 14 people who have not yet been asked. Thus, if you were not picked on the first question, the probability you are again not picked is 13/14. Similarly, the probability you are again not picked on the third question is 12/13, and the probability of not being picked for any of the three questions is

P (not picked in 3 questions)

= P (Q1 = not picked, Q2 = not picked, Q3 = not picked.)

= 14

15 × 13

14 × 12

13 =

12

15 = 0.80

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