Central Limit Theorem and Sampling Methods in Statistics for Business, Lecture notes of Statistics

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STATISTICS FOR BUSINESS

II

Doç. Dr. Yüksel Akay ÜNVAN

CENTRAL LIMIT THEOREM Central limit theorem is one of the most important theorems in statistics. If you take independent samples one by one from a population with a known arithmetic mean and standard deviation and calculate the means of these samples, central limit theorem states that the shape of the distribution of the sample means will be approximately Normal distribution. Central Limit Theorem also indicates that without the regard of the shape of a population parameter’s distribution (symmetric, asymmetric, left skewed or right skewed), the sampling distribution of the sample mean approximates to the normal distribution as the sample size increases.

POPULATION AND SAMPLE

Sampling is a technique of selecting items from a group called a

population. A population is the universe to be sampled, such as all

Turkish people, all universities in Turkey, all pregnant women over 40

years of age, all adult males who smoke 20 cigarettes or more daily,

or all the buildings in Eskişehir, Turkey. Population can be finite or

infinite. The boundaries of the population should be clearly declared.

Populations can be homogenous or heterogenous. It could be

consisting of any type of object. In statistics the word population is

used in a broader sense.

SAMPLING METHODS The main purpose of the inferential statistics is to generalize the information obtained from sample to the population of interest. A sample is a part of the population selected so that inferences can be drawn from it about the population. In most cases, studying on sampling is more feasible than the all elements of the population. In this section, we discuss basic methods for selecting a sample from a population. Before discussing the sampling methods, we distinguish between two types of samples, probability samples and non-probability samples. The following examples demonstrates the differences between probability sample and non- probability sample. Consider a sample of five firms from a population of 30 firms. The selected firms are to be used to determine the production quality of a new

SAMPLING METHODS The selection procedure has the fact that 20! / [5! × (20–5)! ] = 15,504 different combinations of five firms that can be drawn from a population of 20 firms. If the sampling procedure gives each combination an equal probability (1/15,504) of being chosen, then this sampling procedure is called probability sample because a probability mechanism is used, and the probabilities of the selection is known. A television program was previewed in a studio. The director of the program selected 30 persons she judged to be representative of the audience of 700 persons for detailed interview about their impression of the program. This selection is a judgement selection because the director judged it to be representative sample. Sampling methods are usually categorized into two types. These are probability sampling and

PROBABILITY SAMPLING METHODS efficient than simple random sampling but the sample size for each stratum must be calculated. In cluster sampling, naturally occurring clusters are sampled then the members of each selected group subsampled afterward. It is convenient for existing units such as cities, schools, hospitals but each cluster should be composed of units that is like one another. In systematic sampling, every kth element after a random starting point is selected. Recurring patterns must be watched within the sampling frame.

NONPROBABILITY SAMPLING METHODS

Researchers sometimes use nonprobability sampling methods because the members appear to be representative or because they can be assembled conveniently. The subsections below describe commonly used nonprobability sampling methods. In convenience sampling, a group of individuals is ready and available for the sample. It is a practical sampling method because it uses available members, In snowball sampling, previously identified members identify other members of the population. It is useful if there is no any listing of the population, but an unbiased sample may not be obtained because of recommendations. In quota sampling, population is divided into subgroups being studied. It is practical if reliable records or data exist to estimate proportions, but records must be up-to-date to get accurate proportions.

SAMPLING DISTRIBUTION

The most commonly used estimators for the population parameter are sample mean and sample proportion. The sample mean x provides information about the population mean μ. Let’s consider the following situation. For example, you want to determine the mean height of the Open Education Faculty students who are registered to a programme in spring term for 2018-2019. For this purpose, you may take a simple random sample, (the sample is the students taken from the population of students who registered to a programme in the given time period), with a sample size of 100, then calculate the mean of the heights of the students in the

SAMPLING DISTRIBUTION

simple random samples with a size of 100 and calculate the sample mean x for each sample, you would obtain a different value of x for each sample. The reason is that each sample has different students. The sampling distribution of the sample mean is the probability distribution of all possible sample means, that can be created from a population with a fixed sample size of n.