Chapter 8 - Sampling, Sampling Methods, and the Central Limit Theorem

Research and Sampling

Research is the process of studying a problem, question or hypothesis. Typically, data are collected on relevant variables by sampling from a clearly defined population. Then statistical and analytical techniques are applied to summarize and analyze the data. The analytical results are the basis for making inferences and stating conclusions about the population. The primary goal of sampling is that the sample is unbiased and hence a good representation of the population. Using unbiased sample information, valid inferences and conclusions about the research hypothesis can be made.

Sampling Methods

There are different sampling methods that can be used to select an unbiased sample.

Simple Random Sampling

The most widely used sampling method - a sample is selected so that each item or person in the population has the same probability or chance of being included.

Systematic Random Sampling

In Systematic Random Sampling a random starting point is selected, and then every kth member of the population is selected.

Before using systematic random sampling, we should carefully observe the physical order of the population. When the physical order is related to the population characteristic, then systematic random sampling should not be used because the sample could be biased.

For example, if we wanted to audit the invoices in a file drawer that were ordered in increasing dollar amounts, systematic random sampling would not guarantee an unbiased random sample. Other sampling methods should be used.

Stratified Random Sampling

When a population can be clearly divided into groups based on some characteristic, we may use stratified random sampling. It guarantees each group is represented in the sample. The groups are called strata.

For example, college students can be grouped as full time or part time, or as freshmen, sophomores, juniors, or seniors. Usually the strata are formed based on members’ shared attributes or characteristics. A random sample from each stratusm is taken in a number proportional to the stratum’s size when compared to the population. Once the strata are defined, we apply simple random sampling within each group or stratum to collect the sample.

Stratified Random Sample - a poulation is divided into subgroups, called strata, and a sample is randomly selected from each stratum.

Cluster Sampling

Another common type of sampling is cluster sampling. It is often employed to reduce the cost of sampling a population scattered over a large geographic area.

Cluster Sampling - A population is divided into clusters using naturally occuring geographic or other boundaries. Then, clusters are randomly selected and a sample is collected by randomly selecting from each cluster.

For example:

    Suppose you want to determine the views of residents in the greater Chicago, Illinois, metropolitan area about state and federal environmental protection policies. Selecting a random sample of residents in this region and personally contacting each one would be time consuming and very expensive., Instead, you could employ cluster sampling by subdividing the region into small units, perhaps by counties. These are often called primary units.

    There are 12 counties in the greater Chicago metropolitan area., Suppose you randomly select three counties. The three chosen are LaPorte, Cook, and Kenosha. Next, you select a random sample of the residents in each of these counties and interview them. This is also referred to as sampling through an intermediate unit. In this case, the intermediate unit is the county.

Sample Mean as a Random Variable

The result of sampling is a random selection of population objects or individuals. Data collected about the sampled objects are used to compute sample statistics, such as the mean and standard deviation. Because each sample is different, each sample will have a different mean and standard deviation. Therefore, sample statistics are random variables that can be described with distributions.

Sampling error - the difference between the sampling mean and the true population mean. Can be either positive (+) or negative (-).

Sampling Distribution of the Sample Mean

Fundamentally, the distribution of all possible sample means, calculated from all possible random samples, is normally distributed.

With this knowledge, probability statements regarding the size of a specified sample error can be made. For example, knowing that the distribution of sample means is normally distributed, we can apply the empirical rule to say that approximately 68% of all sample means are within one standard error of the mean. Or, 95% of all sample means are within two standard errors of the mean. The complement of this probability statement is that 5% of all sample means are more than two standard errors distrant from the mean. So, using sampling and knowing about the sampling distribution of the sample mean ensures that large sampling errors are unlikely. An earlier section in this chapter listed several practical reasons to prefer sampling over collecting information on a population. Here, we recognize that sampling results in a small probability, or risk, of a large sampling error.

Sampling Distribution of the Sample Mean - a probability distribution of all possible sample means of a given sample size.

Relationships between the population distribution and the sampling distribution of the sampling mean

1. The mean of the sample means is exactly equal to the population mean.
2. The dispersion of the sampling distribution of the sample mean is narrower than the population distribution.
3. The sampling distribution of the sample mean tends to become bell shaped and to approximate the normal probability distribution.

    Given a bell-shaped or normal probability distribution, we will be able to apply concepts from Chapter 7 to determine the probability of selecting a sample with a specified sample mean. In the next section, we will show the importance of sample size as it relates to the sampling distribution of the sample mean.

The Central Limit Theorem

Applying the central limit theorem to the sampling distribution of the sample mean allows us to use the normal probability distribution to create confidence intervals for the population mean and perform tests of hypothesis.

Central limit theorem - If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples.

Review - the difference between the sample mean (\(\bar{x}\)) and the population mean (µ) is called sampling error. In other words, the difference of 3.80 years between the sample mean of 8.60 and the population mean of 4.80 is the sampling error. It is due to chance.

Standard Error of the Mean

The central limit theorem does not say anything about the dispersion of the sampling distribution of the sample mean. However, in our example, we observed that there was less dispersion in the distribution of the sample mean than in the population distribution by noting the difference in the range in the population and the range of the sample means. We observe that the mean of the sample means is close to the mean of the population. It can be demonstrated that the mean of the sampling distribution is exactly equal to the population mean (i.e., \(\mu_{\bar{x}} = µ\)), and if the standard deviation in the population is \(\sigma\), the standard deviation of the sample means is \(\sigma/\sqrt{n}\) where n is the number of observations in each sample. We refer to \(\sigma/\sqrt{n}\) as the standard error of the mean. Its longer name is actually the standard deviation of the sampling distribution of the sampling mean.

Standard Error of the Mean -> \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)

In the formula, note the relationship between the standard error and sample size, n. As the sample size increases, the dispersion of the sampling distribution of the mean decreases and the shape of the distribution narrows around the population mean.

Using the Sampling Distribution of the Sample Mean

Conditions necessary for a normal distribution of the sampling distribution

1. When the samples are taken from populations known to follow the normal distribution, the size of the sample is not a factor.
2. When the shape of the population distribution is not known, sample size is important. In general, the sampling distribution will be normally distributed as the sample size approaches infinity. In practice, a sampling distribtuion will be close to a normal distribution with samples of at least 30 observations.

Find the z-value of \(\bar{x}\) when the Population Standard Deviation is known

\(z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}\)