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**Chapter 10: Sampling and Sampling Distributions**

Aims of Sampling Basic Principles of Probability Types of Random Samples Sampling Distributions Sampling Distribution of the Mean Standard Error of the Mean The Central Limit Theorem

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Sampling Population – A group that includes all the cases (individuals, objects, or groups) in which the researcher is interested. Sample – A relatively small subset from a population.

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Notation

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Sampling Parameter – A measure (for example, mean or standard deviation) used to describe a population distribution. Statistic – A measure (for example, mean or standard deviation) used to describe a sample distribution.

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**Sampling: Parameter & Statistic**

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Probability Sampling Probability sampling – A method of sampling that enables the researcher to specify for each case in the population the probability of its inclusion in the sample.

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Random Sampling Simple Random Sample – A sample designed in such a way as to ensure that (1) every member of the population has an equal chance of being chosen and (2) every combination of N members has an equal chance of being chosen. This can be done using a computer, calculator, or a table of random numbers

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**Population inferences can be made...**

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**...by selecting a representative sample from the population**

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Random Sampling Systematic random sampling – A method of sampling in which every Kth member (K is a ration obtained by dividing the population size by the desired sample size) in the total population is chosen for inclusion in the sample after the first member of the sample is selected at random from among the first K members of the population.

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**Systematic Random Sampling**

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**Stratified Random Sampling**

Stratified random sample – A method of sampling obtained by (1) dividing the population into subgroups based on one or more variables central to our analysis and (2) then drawing a simple random sample from each of the subgroups

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**Stratified Random Sampling**

Proportionate stratified sample – The size of the sample selected from each subgroup is proportional to the size of that subgroup in the entire population. Disproportionate stratified sample – The size of the sample selected from each subgroup is disproportional to the size of that subgroup in the population.

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**Disproportionate Stratified Sample**

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**Sampling Distributions**

Sampling error – The discrepancy between a sample estimate of a population parameter and the real population parameter. Sampling distribution – A theoretical distribution of all possible sample values for the statistic in which we are interested.

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**Sampling Distributions**

Sampling distribution of the mean – A theoretical probability distribution of sample means that would be obtained by drawing from the population all possible samples of the same size. If we repeatedly drew samples from a population and calculated the sample means, those sample means would be normally distributed (as the number of samples drawn increases.) The next several slides demonstrate this. Standard error of the mean – The standard deviation of the sampling distribution of the mean. It describes how much dispersion there is in the sampling distribution of the mean. 50

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**Sampling Distributions**

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**Distribution of Sample Means with 21 Samples**

10 8 6 4 2 S.D. = 2.02 Mean of means = 41.0 Number of Means = 21 Frequency Sample Means 47

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**Distribution of Sample Means with 96 Samples**

14 12 10 8 6 4 2 S.D. = 1.80 Mean of Means = 41.12 Number of Means = 96 Frequency Sample Means

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**Distribution of Sample Means with 170 Samples**

30 20 10 S.D. = 1.71 Mean of Means= 41.12 Number of Means= 170 Frequency Sample Means

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**The Central Limit Theorem**

If all possible random samples of size N are drawn from a population with mean y and a standard deviation , then as N becomes larger, the sampling distribution of sample means becomes approximately normal, with mean y and standard deviation

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