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Chapter 11 – 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
Chapter 11 – 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.
Chapter 11 – 3
Notation
Chapter 11 – 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.
Chapter 11 – 5
Sampling: Parameter & Statistic
Chapter 11 – 6
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. Simple Random Method, Systematic Sampling, Stratified Sampling and Cluster or Multistage Sampling
• Non-probability sampling – Accidental Sampling, Quota Sampling and Purposive Sampling.
Chapter 11 – 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
Chapter 11 – 8
Population inferences can be made...
Chapter 11 – 9
...by selecting a representative sample from the population
Chapter 11 – 10
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.
Chapter 11 – 11
Systematic Random Sampling
Chapter 11 – 12
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
Chapter 11 – 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.
Chapter 11 – 14
Disproportionate Stratified Sample
Chapter 11 – 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.
Chapter 11 – 16
• 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.
Sampling Distributions
Chapter 11 – 17
Sam
plin
g Distrib
ution
s
Chapter 11 – 18
• 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 .
y
Ny /
The Central Limit Theorem
Chapter 11 – 19
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SRS w/o replacement SRS w/ replacement
Chapter 11 – 20
The Central Limit Theorem
Chapter 11 – 21
Sampling distribution of 100 sample means
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3. 46 3. 47 3. 48 3. 49 3. 5 3. 51 3. 52 3. 53 3. 54 3. 55 3. 56 3. 57 3. 58 3. 59 3. 6 3. 62 3. 64
Chapter 11 – 22
Sampling distribution of 1000 sample means
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3. 44 3. 45 3. 46 3. 47 3. 48 3. 49 3. 5 3. 51 3. 52 3. 53 3. 54 3. 55 3. 56 3. 57 3. 58 3. 59 3. 6 3. 61 3. 62 3. 63 3. 64 3. 65
Chapter 11 – 23
Sampling distribution of 10,000 sample means
Chapter 11 – 24
Sampling distribution of 100,000 sample means
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3. 44 3. 45 3. 46 3. 47 3. 48 3. 49 3. 5 3. 51 3. 52 3. 53 3. 54 3. 55 3. 56 3. 57 3. 58 3. 59 3. 6 3. 61 3. 62 3. 63 3. 64 3. 65 3. 66
Chapter 11 – 25
Example: The Central Limit Theorem
• Example
• Simulation
• http://onlinestatbook.com/stat_sim/sampling_dist/index.html
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Is CLT useful?
Chapter 11 – 27
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Chapter 11 – 28
Law of large numbers
• Law of large numbers
A law that states that if the size of the sample, n, is sufficiently large, then the central limit theorem will apply even if the population is not normally distributed along variable Y.
• Example: Simulation
Chapter 11 – 29
The Central Limit Theorem Revisited
