4 assignments in Business Statistics due dates listed within description.

Sampling Methods and Estimation

Sampling Methods

We sample because it is often impossible to study every item, or individual, in some populations.  It would be too expensive and time-consuming, for example, to contact and record the annual incomes of all U.S. bank officers.  Many times, it is also not feasible.

A sample is a part of the population.  To estimate a population parameter, therefore, we sample a population.  Care must be taken to ensure that every member of our population has a chance of being selected; otherwise, the conclusions might be biased.  A number of probability-type sampling methods can be used, including simple random, systematic, stratified, and cluster sampling.  Regardless of the sampling method selected, a sample statistic is seldom equal to the corresponding population parameter.  The difference between this sample statistic and the population parameter is the sampling error.

Sampling is necessary because we want to make statements about a population, but we do not want to or cannot examine all the items in that population.  Recall from Chapter 1 that a population refers to the entire group of objects or persons of interest.  The population of interest might be all the persons in the city receiving welfare payments or all the computer chips produced during the last hour.  A sample is a portion, a part, or a subset of the population.  Fifty welfare recipients out of 4,000 receiving payments might constitute the sample, or 20 computer chips might be sampled out of 1,500 produced last hour.

Please study this  Populations and Samples Website  for more information on what is a population and sample.

Reasons for Sampling

Why is it necessary to sample?  Why can't we just inspect all the items?  There are several reasons:

1. To contact the whole population would often be very time consuming.  To ask every eligible voter if they plan to vote for the current senator in the forthcoming election would take months.  The election would probably be over before the survey was completed.

2. The cost of studying all the items in the population is often prohibitive.  Some television program ratings are established by analyzing the viewing habits of about 1,200 viewers.  The cost of studying all the homes having television would be exorbi¬tant.

3. The physical impossibility of checking all the items in the population.  The South Dakota Game Commission, for example, cannot check all the deer, grouse, and other wild game because they are always moving.

4. The destructive nature of certain tests.  The manufacturer of fuses cannot test all of them because in the testing the fuse is destroyed and none would be available for sale.

5. The adequacy of sample results.  If the sample results of the viewing habits of 1,200 homes revealed that only 1.1 percent of the homes watched “60 Minutes,” there is no doubt that the program would be replaced by another show.  Checking the viewing habits of all the homes regarding "60 Minutes" probably would not change the percent significantly.

Probability Sampling Methods

Four basic types of probability sampling are commonly used: simple random sampling, systematic random sampling, stratified random sampling, and cluster sampling.  The most widely used type of sampling is a simple random sample.

Figure 1

Several ways of selecting a simple random sample are:

1. The name or identifying number of each item in the population is recorded on a slip of paper and placed in a box.  The slips of paper are shuffled and the required sample size is chosen from the box.

2. Each item is numbered and a table of random numbers, such as the one in Appendix B.6, is used to select the members of the sample.

3. There are many software programs, such as MINITAB and Excel, which have routines that will randomly select a given number of items from the population.

Another type of sampling is a systematic random sample.

Figure 2

In a systematic random sample, the items or individuals of the population are arranged in some way — alphabetically, in a file drawer by date received, or by some other method.  A random starting point is selected, and then every kth member of the population is selected for the sample.  In a systematic random sample, you might take all the items in the population and number them 1, 2, 3…. Next, a random starting point is selected, for example, 39.  Every kth item thereafter, such as every 100th, is selected for the sample.  This means that 39, 139, 239, 339, and so on would be a part of the sample.  Systematic random sampling is desirable not only because it reduces the necessary sample size compared to a simple random sample, but it is also flexible while maintaining its simplicity.  Some potential traps exist that a research designer needs to keep in mind.  For example, studies of retail traffic would not want to use the number 7 as the skip interval when studying sales by day.  The general guideline here is not to use a skip interval that might mimic a naturally occurring cycle of one kind or another.

Another type of probability sample is referred to as stratified random sampling.

Figure 3

For example, if our study involved Army personnel, we might decide to stratify the population of all Army personnel into generals, other officers, and enlisted personnel.  The number selected from each of the three strata could be proportional to the total number in the population for the corresponding strata.  Each member of the population can belong to only one of the strata.  That is, one individual cannot be a general and a private at the same time.  In this particular case, the mutual exclusivity is obvious.  In other cases, make sure to maintain mutual exclusivity when designing your sample format.  For example, when designing a study to test your advertising, setting up a category of those who saw your ads on television versus in the newspaper versus a billboard would not necessarily be mutually exclusive.  Yes, they can only see one ad in one place at a time, but that is not how the category is defined.

One additional consideration is whether you wish to maintain a proportionate sampling or a disproportionate design.  The proportionate model has greater statistical efficiency in most cases, is easier to carry out, and is self-weighting (Cooper & Schindler, 2008).  One obvious case for using disproportionate sampling is a sampling classification is based on gender in the United States.  Since females comprise a slightly higher proportion of the total population, using a stratified design to mirror that disparity makes sense.

Another common type of sampling is cluster sampling.

Figure 4

Cluster sampling is often used to reduce the cost of sampling when the population is scattered over a large geographic area.  Suppose the objective is to study household waste collection in a large city.

1. Step 1: Divide the city into smaller units, perhaps precincts.

2. Step 2: The precincts are numbered and several are randomly selected.

3. Step 3: Households within each of these precincts are randomly selected and interviewed.

There are several questions that need to be answered when designing a clustered sample.  Homogeneity within the cluster is important.  Several aspects of the cluster size also need to be addressed.  How large do the clusters need to be and are they going to be of equal size?  Once you find homogenous clusters, do some need to be combined or others split to counter the effects of unequal size?  Unfortunately, you have no general rule to go by here.  The determination is based solely on the impact on ease of conducting the study.

There are further considerations for cluster sampling.  One consideration that should be emphasized is the difference between multistage and multiphase sampling.  They are different and should not be confused.  Multistage sampling could be thought of as subsampling.  In cases where you have large, naturally occurring clusters, going through and manually creating artificial smaller clusters to conduct one-stage cluster sampling may not be as economical as subsampling.

There is another form of subsampling, in which the initial sample is then approached a second time.  A subsample of that initial sample is then the subject of further study.  This can be referred to as double, sequential, or multi-phase sampling (Cooper & Schindler, 2008).

Please visit the  Probability Sampling Website  for an excellent resource for probability samples.

Point Estimates and Confidence Intervals

Known s or a Large Sample

In many situations, the population is large or it is difficult to identify all the members, so we need to rely on sample information.  A single number used to estimate a population parameter is called a  point estimate .

Figure 5

· The sample mean,  , is a point estimate of the population mean, µ.

· The sample standard deviation, s, is a point estimate of the population standard deviation, σ.

For example, a sample of 100 recent accounting graduates reveals a mean starting salary of $30,000.  The $30,000 is a point estimate.  The sample mean is a point estimate of the mean starting salary of all (population) accounting graduates.

We expect the point estimate to be close to the population parameter, but we would like to measure how close it really is.  We need a measure that gives us a range of values into which our point estimate will fit.  We use a confidence interval for this purpose.

Confidence Interval

The range of values, within which a population parameter is expected to lie, is usually referred to as the confidence interval .

Figure 6

Figure 7

The  95 percent confidence interval  means that ninety-five percent of the sample means selected from a population will be within 1.96 standard deviations of the population mean µ.

The  99 percent confidence interval  means that ninety-nine percent of the sample means selected from a population will be within 2.58 standard deviations of the population mean µ.

The central limit theorem allows us to state or specify a range of values within which a population parameter – such as the population mean – can be expected to occur.

When the sample size (n) is at least 30, it is generally accepted that the central limit theorem will ensure a normal distribution of the sample means.  This important consideration allows us to use the standard normal distribution, that is, z in our calculation of the confidence interval.  In general, the confidence interval for the mean of a sample is computed by text formula [9-1].

Where:

· z depends on the level of confidence.

· s is the sample standard deviation.

· n is the size of the sample.

Referring back to the example of 100 recent accounting graduates, construct a 95% confidence interval estimate of the population mean starting salary.  Assume that s = 2000.  In this case, the sample mean is 30,000, z = 1.96, and n = 100.

To calculate the confidence interval:

30,000 ± 1.96(2000)/√100

30,000 ± 1.96(200)

30,000 ± 392

We are 95% confident that recent accounting graduates will have an average salary of 29,608 to 30,392.

Watch this  Confidence Interval for the Population Mean Video  to better understand how to compute the confidence interval for a population mean.