1 Sampling Techniques

2 Introduction I l A unit is the entity that is of interest to us. l The population consists of all units of interest. l The population size is the number of units in the population. l A frame is a list of all units in the population.

3 Introduction II l Census is the process of investigating all the units in a population. l A sample is a subset of units selected from the population. Sampling is the process of taking a sample. l A parameter is a numerical summary derived from the population. A statistic is that derived from a sample. l The main purpose of sampling is to derive statistics and use them to approximate the parameters.

4 Sampling problems l What is the appropriate sample size? This question will be answered in Chapter 4, Confidence interval analysis. l Which n units of the population should be selected? This question will be answered in this chapter.

5 Sampling error and bias l Sampling error, introduced when sample mean is used to approximate population mean, is always there. -Sampling error due to luck is always there. -But sampling bias, the systematic error, can be removed. -Other errors: non-response, false response, etc., will be there whether a sample or a census is used. l Simple random sample is used to remove bias.

6 Simple random sample I In a simple random sample, any n units will have equal chance to be selected before the selection. -Example 1: 10 balls are in a bag. Only one ball will be selected, but each ball should have the same chance to be selected before the the selection. This is a simple random sample. -Example 2: Lottery numbers. A group of six two digit numbers will be selected. Before the selection, each group of six two digit numbers should have the same chance to be selected.

7 Small University I -- a counter example l A pollster comes to a small university with only 8 students, and decides to take a simple random sample of two students (sample size = 2) l The university has only 4 courses, in each course two students are enrolled. Each student can only take one course. l The pollster decides to randomly select two courses, then in each course randomly pick up one student. Eventually two students will be selected. Does this approach provide a simple random sample?

8 Small University John Tina Mary Janet David James Ben Joan Course

9 Simple random sample II l Step 1: Get the frame. l Step 2: Determine the random digits needed. l Step 3: Select n random numbers from the random number table. l Step 4: Select the corresponding units.

10 Small University 1John 2David 3Tina 4James 5Mary 6Janet 7Ben 8 Joan

11 Remarks l For simple random sample, a frame is always needed. It may be difficult to obtain a frame. l We say “a simple random sample should ensure that any group of n units should have a equal chance to be selected”. The equal opportunity to be selected is before the selection, not after.

12 Other random sampling techniques l Stratified sampling -First divide the whole population into several non- overlapping sub-populations which are called strata. -A simple random sample will be taken from each stratum. l Cluster sampling -Divide the population into a number of sub-populations. -Only a random selection of sub-populations will be studied. l Systematic sampling -One unit from the list is randomly selected. -Every k-th (it can be the second, the 5-th, etc.) unit is then systematically selected.