5.5 Hypergeometric distribution - university of calgary - base  (Page 2/9)

Notation for the hypergeometric: h = hypergeometric probability distribution function

X ~ H ( r , N - r , n )

Read this as " X is a random variable with a hypergeometric distribution." The parameters are r , N - r , and n ; r = the size of the group of interest (first group), N - r = the size of the second group ( N is the population size minus the group of interest), n = the size of the chosen sample.

Hypergeometric formula

P ( X = x )= $\frac{\left(\begin{array}{c}r\\ x\end{array}\right)\times \left(\begin{array}{c}N-r\\ n-x\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)}$

• $\left(\begin{array}{c}r\\ x\end{array}\right)$ is the number of ways of selecting x from our first group ( r )
• In $\left(\begin{array}{c}\mathrm{N-r}\\ \mathrm{n-x}\end{array}\right)$ , N - r represents the number in our second group (the remainder in the population). For example, if the population is 20 and our first has 14 members, then the second group has N - r = 20-14 = 6 members. ( n - x ) is the number in our sample that is selected from the second group. For example, if we are selecting a sample of four, if x = 2 (number selected from first group), then n - x = 5 - 2 = 3 is the number in the sample selected from the second group.
• Using the multiplicative rule, we get $\left(\begin{array}{c}r\\ x\end{array}\right)\times \left(\begin{array}{c}N-r\\ n-x\end{array}\right)$ for the number of ways of selecting x from the first group and n-x from the second group.
• $\left(\begin{array}{c}N\\ n\end{array}\right)$ is the number of ways of selecting a random sample of n from the population ( N ).

A school site committee is to be chosen randomly from six men and five women. If the committee consists of four members chosen randomly, what is the probability that two of them are men? How many men do you expect to be on the committee and what is the standard deviation?

Let X = the number of men on the committee of four. The men are the group of interest (first group).

X takes on the values 0, 1, 2, 3, 4, where r = 6 , N - r = 5 , and n = 4 . X ~ H (6, 5, 4)

P ( X = 2 )= $\frac{\left(\begin{array}{c}6\\ 2\end{array}\right)\times \left(\begin{array}{c}5\\ 2\end{array}\right)}{\left(\begin{array}{c}\mathrm{11}\\ 4\end{array}\right)}$ = 0.4545

Note

Currently, the TI-83+ and TI-84 do not have hypergeometric probability functions. There are a number of computer packages, including Microsoft Excel, that do.

The probability that there are two men on the committee is about 0.45.

The graph of X ~ H (6, 5, 4) is:

The y -axis contains the probability of X , where X = the number of men on the committee.

You would expect μ = 2.18 (about two) men on the committee with a standard deviation of σ = 0.8332

The formula for the mean is μ = $\frac{nr}{N}$ = $\frac{(4)(6)}{\mathrm{11}}$ = 2.18

The formula for the standard deviation is $\sigma =\sqrt{\frac{nr}{N}(1-\frac{r}{N})\frac{N-n}{N-1}}$ = $\sqrt{\frac{4\times 6}{\mathrm{11}}(1-\frac{6}{\mathrm{11}})\frac{\mathrm{11}-4}{\mathrm{11}-1}}$ = 0.8332

Chapter review

A hypergeometric experiment is a statistical experiment with the following properties:

1. You take samples from two groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups.
4. Each pick is not independent, since sampling is without replacement.
5. You are not dealing with Bernoulli Trials.

The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. The distribution of X is denoted X ~ H ( r , N-r , n ), where r = the size of the group of interest (first group), N-r = the size of the second group, and n = the size of the chosen sample. It follows that
n ≤ N . The mean of X is μ = $\frac{nr}{N}$ and the standard deviation is $\sigma =\sqrt{\frac{nr}{N}(1-\frac{r}{N})\frac{N-n}{N-1}}$ .

Formula review

X ~ H ( r , N-r , n ) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), N-r = the size of the second group, and n = the size of the chosen sample.

X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, ..., up to the size of the group of interest. (The minimum value for X may be larger than zero in some instances.)

n ≤ N . This means that the sample size cannot be more than the population size.

The mean of X is given by the formula μ = $\frac{nr}{N}$ and the standard deviation is given as $\sigma =\sqrt{\frac{nr}{N}(1-\frac{r}{N})\frac{N-n}{N-1}}$ .

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.