Practice problems for these concepts can be found at:

- Sampling Distributions Solved Problems for Beginning Statistics
- Sampling Distributions Supplementary Problems for Beginning Statistics

In order to obtain information about some population, either a *census* of the whole population is taken or a *sample* is chosen from the population and the information is inferred from the sample. The second approach is usually taken, since it is much cheaper to obtain a sample than to conduct a census. In choosing a sample, it is desirable to obtain one that is representative of the population. The average weight of the football players at a college would not be a representative estimate of the average weight of students attending the college, for example. A *simple random sample* of size n from a population of size N is one selected in such a way that every sample of size n has the same chance of occurring. In *simple random sampling with replacement*, a member of the population can be selected more than once. In *simple random sampling without replacement*, a member of the population can be selected at most once. Simple random sampling without replacement is the most common type of simple random sampling.

**EXAMPLE 7.1** Consider the population consisting of the world's five busiest airports. This population consists of the following: A: Chicago O'Hare, B: Atlanta Hartsfield, C: London Heathrow, D: Dallas–Fort Worth, and E: Los Angeles Intl. The number of possible samples of size 2 from this population of size 5 is given by the combination of 5 items selected two at a time, that is, = 10. In simple random sampling, each possible pair would have probability 0.1 of being the pair selected. That is, Chicago O'Hare and Atlanta Hartsfield would have probability 0.1 of being chosen, Chicago O'Hare and London Heathrow would have probability 0.1 of being chosen, etc. One way of ensuring that each pair would have an equal chance of being selected would be to write the names of the five airports on separate sheets of paper and select two of the sheets randomly from a box.

### Using Random Number Tables

The technique of writing names on slips of paper and selecting them from a box is not practical for most real world situations. Tables of random numbers are available in a variety of sources. The digits 0 through 9 occur randomly throughout a random number table with each digit having an equal chance of occurring. Table 7.1 is an example of a random number table. This particular table has 50 columns and 20 rows. To use a random number table, first randomly select a starting position and then move in any direction to select the numbers.

**EXAMPLE 7.2** The money section of *USA Today* gives the 1900 most active New York Stock Exchange issues. The random numbers in Table 7.1 can be used to randomly select 10 of these issues. Imagine that the issues are numbered from 0001 to 1900. Suppose we randomly decide to start in row 1 and columns 21 through 24. The four-digit number located here is 0345. Reading down these four columns and discarding any number exceeding 1900, we obtain the following eight random numbers between 0001 and 1900: 0345, 1304, 0990, 1580, 1461, 1064, 0676, and 0347. To obtain our other two numbers, we proceed to row 1 and columns 26 through 29. Reading down this column, we find 1149 and 1074. To obtain the 10 stock issues, we read down the columns and select the ones located in positions 345, 347, 676, 990, 1064, 1074, 1149, 1304, 1461, and 1580.

### Using the Computer to Obtain a Simple Random Sample

Most computer statistical software packages can be used to select random numbers and to some extent have replaced random number tables. As the capability and availability of computers continue to increase, many of the statistical tables are becoming obsolete.

**EXAMPLE 7.3** MINITAB can be used to select the random sample of stock issues in Example 7.2. The pull down **Calc** **Random Data** **Integer** is used. The Integer Distribution dialog box is filled as follows: Generate 10 rows of data, store in column C1, minimum value = 1 and maximum value = 1900. The following 10 numbers are printed when column C1 is printed out.

MTB > print c1

Data Display

C1

830 166 918 1028 383 1136 1256 1658 731 19

These are the ten numbers of stocks to be selected.

Practice problems for these concepts can be found at:

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