Calculating Variances

The Problem Statement

Let X be a random variable following a Hypergeometric distribution. All Hypergeometric distributions have three parameters: sample size, population size, and number of successes in the population. For this problem, let X be a sample of size 19 taken from a population of size 31, in which there are 24 successes. An example of where such a distribution may arise is the following:

Statistics, Inc., wants to determine how well people like its candies. To determine this, they randomly select 19 people from its workforce of 31. (In this workforce, only 24 of those 31 actually like the candies.) Let X be the number of employees surveyed who like the candies.

From the description, we can tell that X has a Hypergeometric distribution. We know this because X is the number of successes in a finite population where subjects are not be measured more than once (i.e., “without replacement”).

For those who like pictures, here is a graphic of the probability mass function of X. Note that it has a positive skew (right skew), that its sample space is integers between 12 and 19, inclusive. In other words, S = {12, 13, 14, …, 19}.

The Probability Graphic

Here is the probability function of the Hypergeometric distribution described in the example:

To Calculate

Continuing the candy example, we know that the expected number of people in the sample who like the candies is . However, it is important to determine how variable X is about this mean. So, to that end, what is the variance of X? In symbols, calculate V[X].

Your Answer

Another Hypergeometric Distribution

Would you like to continue working on this topic? If so, click here for another Hypergeometric distribution.

Assistance