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Probability Sampling

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After examining this page your knowledge on probability sampling will be enhanced


Before proceeding, we have to define some general terms.
A probability sampling method is any method of sampling that utilizes some form of random selection.
What is random selection? It is a selection made so that each person or item has an equal chance of being chosen.

With those general terms defined you may proceed to examine the history of probability and the different probability sampling methods.

The History of Probability
Probability started from the study of games of chance. Tossing a dice, playing poker and spinning a roulette wheel are just some examples of random sampling. Games of chance were not studied by mathematicians until the sixteenth and seventeenth centuries. Probability theory as a branch of mathematics arose in the seventeenth century when French gamblers asked Blaise Pascal and Pierre de Fermat, both well known pioneers in mathematics, for help in their gambling. In the eighteenth and nineteenth centuries, careful measurements in astronomy and surveying led to further advances in probability.
In the twentieth century probability is used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor. In addition, it is used to find the genetic makeup of individuals or populations, figure out the energy states of subatomic particles, estimate the spread of rumors, and predict the rate of return in risky investments.

Probability and Chance
Chance is a part of our everyday lives. Everyday we make judgements based on probability:
There is a 90% chance the Detroit Red Wings will win the game tomorrow. There is a 60% chance of thunderstorm this afternoon. We have a 50-50 chance of winning the game. There is a 20% chance of showers today. Although we assign certain probabilities to certain events, others might assign different probabilities to those same events due to their difference of opinion. For example, not everyone agrees with the high chance of the Detroit Red Wings winning the game. They might say that there is a 20% chance the Detroit Red Wings will win the game tomorrow. It all depends on what the person believes. Chance may result from human design such as casino games and the lottery, or it may result from nature such as determining a personís sex and other human characteristics. Probability is defined as the branch of mathematics that describes the pattern of chance outcomes.

Probability Theory

Probability Theory is the mathematical study of randomness. This theory deals with the possible outcomes of an event. It must be possible to list every outcome that can occur, and we must be able to state the expected relative frequencies of these outcomes. It is the method of assigning relative frequencies to each of the possible outcomes. If the outcomes of an experiment are equally likely, then the probability of an event is the ratio of the number of outcomes favourable to the event to the total number of outcomes.

Personal Probability
We can have a personal opinion about the next outcome of an event such as a coin toss. I can say that my personal probability of a head in the next toss is 1/2. Your personal probability may be different from mine. Personal probability sets us free from figuring out the outcome from many repetitions. Therefore, personal probability allows us to assign a probability to one time events such as a golf tournament.

Simple Random Sampling
Simple random sampling is the simplest form of random sampling. It is the basic sampling technique where you select a group of subjects, a sample, for study from a larger group, a population. Each individual is chosen entirely by chance and each member of the population has an equal chance of being included in the sample. Every possible sample of a given size has the same chance of selection. As a result, each member of the population is equally likely to be chosen at any stage in the sampling process.
For example, the thingamajig at the top is an ideal model of simple random sampling. Press the "Start" button to start the random selection. You will notice that at every second the thingamabob will pick up one of the three numbers 1, 2, or 3. You can terminate the process anytime by pressing the "Stop" button.
Randomly picking clients from a list of clients is another example of simple random sampling.

Simple random sampling is simple to accomplish and is easy to explain to others because it is a fair way to select a sample, it is reasonable to generalize the results from the sample back to the population. However, it is not the most statistically efficient method of sampling. It does not get a good representation of subgroups in a population because of the luck of the draw. To deal with these issues, we have to turn to other sampling methods.

Stratified Random Sampling
A stratified random sample, also called proportional or quota random sample, is obtained by taking samples from each stratum or sub-group of a population. It involves dividing your population into homogeneous subgroups and then taking a simple random sample in each subgroup. Stratified sampling techniques are generally used when the population is heterogeneous, or dissimilar, where certain homogeneous, or similar, sub-populations can be isolated. Simple random sampling is most appropriate when the entire population from which the sample is taken is homogeneous. There are several reasons why you would prefer stratified sampling over simple random sampling. Firstly, it assures that you will be able to represent not only the overall population, but also key subgroups of the population, especially small minority groups. Secondly, the cost per observation in the survey may be reduced and lastly, it provides each sub-population estimates of the population parameters.
Splitting clients into three different groups and picking from them is another example of stratified random sampling.

Take a farmer for example. Suppose he wishes to work out the average milk yield of each cow type in his herd which consists of Ayrshire, Friesian, Galloway and Jersey cows. He could divide up his herd into the four sub-groups and take samples from these.

Cluster Random Sampling
Cluster sampling is a sampling technique where the entire population is divided into groups, or clusters, and a random sample of these clusters are selected. All observations in the selected clusters are included in the sample. It is typically used when the researcher cannot get a complete list of the members of a population they wish to study but can get a complete list of groups or clusters of the population. It is also used when a random sample would produce a list of subjects so widely scattered that surveying them would prove to be far too expensive, for example, people who live in different postal districts in the UK. This sampling technique is more practical and economical than simple random sampling or stratified sampling. The problem with random sampling methods when we have to sample a population that's disbursed across a wide geographic region is that you will have to cover a lot of ground geographically in order to get to each of the units you sampled. Imagine taking a simple random sample of all the residents of New York State in order to conduct personal interviews. By the luck of the draw you will wind up with respondents who come from all over the state. Your interviewers are going to have a lot of traveling to do.
For instance, in the figure we see a map of the counties in New York State. Let's say that we have to do a survey of town governments that will require us going to the towns personally. If we do a simple random sample state-wide we'll have to cover the entire state geographically. Instead, we decide to do a cluster sampling of five counties, marked in red in the figure. Once these are selected, we go to every town government in the five areas. Clearly this strategy will help us to economize on our mileage. Cluster or area sampling, then, is useful in situations like this, and is done primarily for efficiency of administration.

Take this as another example, suppose that the Department of Agriculture wishes to investigate the use of pesticides by farmers in England. A cluster sample could be taken by identifying the different counties in England as clusters. A sample of these counties, clusters, would then be chosen at random, so all farmers in those counties selected would be included in the sample. It can be seen here then that it is easier to visit several farmers in the same county than it is to travel to each farm in a random sample to observe the use of pesticides.


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