1. Chapter 7: 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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2. 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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3. Notation
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4. 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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5. Sampling: Parameter & Statistic
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6. 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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7. 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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8. Population inferences can be made...
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9. ...by selecting a representative sample from the population
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10. Random Sampling
Systematic random sampling – A method of sampling in which every Kth member 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.
Where
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11. Systematic Random Sampling
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12. Stratified Random Sampling
Stratified random sample – A method of sampling obtained by
dividing the population into subgroups based on one or more variables central to our analysis and
then drawing a simple random sample from each of the subgroups
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13. 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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14. Disproportionate Stratified Sample
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15. 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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16. 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.
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17. Sampling Distributions
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18. 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 and standard deviation
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19. Distribution of Sample Means with 21 Samples10 8 6 4 2
Sample Means
S.D. = 2.02
Mean of means = 41.0
Number of Means = 21
Frequency
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20. Distribution of Sample Means with 96 Samples
Frequency 14 12 10 8 6 4 2
Sample Means
S.D. = 1.80
Mean of Means = 41.12
Number of Means = 96
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21. Distribution of Sample Means with 170 Samples
Frequency 30 20 10
Sample Means
S.D. = 1.71
Mean of Means= 41.12
Number of Means= 170
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