Sampling

Definition Statistics- A branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data

Terminologies Population is the complete set of all items that we are interested in studying. The number of items in a population is called the population size, usually denoted by N. Sample is a subset of the population. Usually n denotes sample size (the number of observations in a sample).

Terminologies Variable is a characteristic or property of an item on which we take measurements. The items or the individuals from whom/which the data are collected are often called cases.

Terminologies Data are the observed values of the variable. Study of a whole population is called census, and that of sample is known as sample survey. Sampling frame is a list of all the elements in the population from which the sample is drawn. e.g.- A list of all low birth weight infants admitted to the neonatal ICUs in St. Louis city & county in 1998

Terminologies Randomization -each individual in the population has an equal opportunity to be selected for the sample. Parameter- a numerical value or measure of a characteristic of the population Statistic - numerical value or measure of a characteristic of the sample  

Why Sample? Sometimes "measuring" or "testing" something destroys it. The government requires automakers who want to sell cars in the U.S. to demonstrate that their cars can survive certain crash tests. Obviously, the company can't be expected to crash every car, to see if it survives! So the company crashes only a sample of cars.

Why Sample? Another reason for sampling is that not all units in the population can be identified, such as all the air molecules in the LA basin. So to measure air pollution, you take a sample of air molecules. Also, even if all those air molecules could be identified, it would be too expensive and too time consuming to measure them all. 

Why Sample? Save time, money etc. Blood testing-a drop is enough!

Why Sample? Representativeness - sample must be as much like the population in as many ways as possible Sample reflects the characteristics of the population, so those sample findings can be generalized to the population Most effective way to achieve representativeness is through randomization; random selection or random assignment

Types of samples Non Random or non probability Samples: Random or probability Samples:

Non Random or non probability Samples These samples focus on volunteers, easily available units, or those that just happen to be present when the research is done. Non-probability samples are useful for quick and cheap studies, for case studies, for qualitative research, for pilot studies, and for developing hypotheses for future research. Not every element of the population has the opportunity for selection in the sample Non-random selection restricts generalization

Random or probability Samples Everyone in the population has equal opportunity for selection as a subject Increases sample's representativeness of the population Decreases sampling error and sampling bias

Types of probability sampling Simple random sample: Each unit in the population is identified, and each unit has an equal chance of being in the sample. The selection of each unit is independent of the selection of every other unit. Selection of one unit does not affect the chances of any other unit. There are two types of simple random samples: a) Simple Random Sampling with Replacement(SRSWR) b) Simple random sampling without replacement(SRSWOR).

Types of simple random sampling SRSWR: An object is selected from the population and is replaced back in the population and is available for selection again. Here an object can be selected twice. (SRSWOR): An object once selected cannot be selected again.

Types of random sampling Stratified random sample: Sometimes the population is first sliced into homogeneous groups called strata and simple random sampling is used within each stratum. Finally, these subsamples are combined into a sample. This sampling scheme is known as “stratified random sampling” or simply “stratified sampling”.

Types of random sampling When to use stratified random sampling? Suppose we would like to know how students feel about funding for the football team in a large university and the student population consists of 30% men and 70% women. Suppose we feel that men and women would have different views on the funding. In this case, a simple random sample won’t do a good job. Instead, it is better to divide the student population in two strata: male and female students. We can then choose a stratified sample consisting of 40 male students and 60 female students. This will be a better representative of the population.  

Types of random sampling Cluster Sampling: Splitting the population into representative clusters can make sampling more practical. Then we could simply select one or a few clusters at random and perform a census within each of them. This sampling scheme is called cluster sampling

Types of random sampling When to use Cluster Sampling? Suppose I am trying to find out what MSU freshmen think about the dining service on campus and I know that freshmen at MSU are all housed in 10 freshman dorms. In this case, I shall select two or three of these 10 dorms at random and contact all the residents of these selected dorms.

Stratified vs. Cluster Sampling Strata are homogeneous but different from one another while clusters are heterogeneous and resemble the overall population. We perform simple random sampling in ALL strata where as we only choose a few clusters at random and perform a census in those clusters.

Types of random sampling Multistage Sampling Sampling schemes that combine several methods are called multistage sampling. Most surveys conducted by professional organizations use multistage sampling. The exact scheme depends on the nature of the populations and the nature of the survey

Types of random sampling Systematic sampling- Samples are collected in a predetermined order.

Example To represent the population of MSU students: Simple Random Sample (SRS) - randomly generate a subset of the PIDs of all students or put all the names in a hat, shake it up and draw some out. Cluster - a set of large lecture classes of different disciplines. Stratified Random - randomly generate a set of PIDs for each class: freshmen, sophomores, juniors and seniors.

Example Multistage - randomly choose 3 dorms on campus, then randomly choose 2 floors of each dorm and sample from each of the floors using SRS. non random- our STT 200 class. Systematic - every 5thstudent I meet in the food-court.

Question An elementary school teacher with 25 students plans to have each of them make a poster about 2 different states. The teacher 1st numbers the states in alphabetical order from 1-Alabama to 50-Wyoming, then uses a random number table to decide which states each kid gets. 45921 01710 22892 37076 Which 2 states does the 1st student get? Which 2 states does the 2nd student get?