Sampling Mathsfest 2014

Why Sample? Jan8, 2003 Air Midwest Flight 5481 from Douglas International Airport in North Carolina stalled after take off, crashed into a hangar and burst into flames. All 21 people on board perished. A subsequent investigation revealed that the weight of the passengers was a factor that contributed to the crash. This prompted the FAA to collect weight information from randomly selected flights so that old assumptions about passenger weights could be updated.

Sampling Distributions Suppose that we draw all possible samples of size n from a given population. Suppose further that we compute a statistic (e.g., a mean) for each sample. The probability distribution of this statistic is called a sampling distribution Population125 Sample(1, 1)(1, 2)(1, 5)(2, 1)(2, 2)(2, 5)(5, 1)(5, 2)(5, 5) Mean Probability

Sampling With Replacement 1.When selecting a relatively small sample from a large population, it makes no significant difference whether we sample with replacement or without replacement. 2.Sampling with replacement results in independent events that are unaffected by previous outcomes. Independent events are easier to analyse and result in simpler formulas.

Simple Random Sample

Advantages of Simple Random Sampling Every member of the population has an equal chance of being represented in the sample The simple random sample should be representative of the population. Theoretically the only thing that can compromise its representativeness is luck If the sample is not representative of the population, then the random variation is called sampling error

Disadvantages of Simple Random Sampling A complete and up to date list of all the population is required Such a list is usually not available for large populations

Estimators Population125 Mean (µ) Sample(1, 1)(1, 2)(1, 5)(2, 1)(2, 2)(2, 5)(5, 1)(5, 2)(5, 5) Mean of Sampling Distribution The sample statistic targets the population parameter

Estimators Population125 Standard Deviation (σ) Sample(1, 1)(1, 2)(1, 5)(2, 1)(2, 2)(2, 5)(5, 1)(5, 2)(5, 5) SD (s) Mean of Sample Standard Deviations The sample statistic does not target the population parameter

Stratified Random Sampling

Advantages of Stratified Random Sampling Provides greater precision than a simple random sample of the same size Smaller samples are required, thereby saving money Can guard against an unrepresentative sample

Disadvantages of Stratified Random Sampling May require more administrative effort than a simple random sample A complete and up to date list of the population is required

Uniform Population Distribution frequency raw score What is the mean of this population? 5 What is the standard deviation of this population?

Distribution of Sample Means: Samples of Size 2 12, 22 22,43 32,64 42,85 54,23 64,44 74,65 84,86 96,24 106,45 116,66 126,87 138,25 148,46 158,67 168,88 Sample Scores Mean ( )

Distribution of Sample Means from Samples of Size n = frequency sample mean

Distribution of Sample Means from Samples of Size n = frequency sample mean p( > 7) = ?

Distribution of Sample Means from Samples of Size n = frequency sample mean = 6 %

frequency raw score frequency sample mean Population Distribution Distribution of Sample Means P(X > 7) = 25% Distribution of Sample Means

Cluster Sampling

Advantages of Cluster Sampling Inexpensive Limited resources can be allocated to a few randomly selected clusters. Easy to implement Subjects are easily accessed

Disadvantages of Cluster Sampling From all the different probability sampling methods, this technique is the least representative of the population. There is a tendency for individuals within a cluster to have similar characteristics, therefore there is a chance that a researcher may have an over represented or under represented cluster.

Mean of Sample Means Means Mean of Population = 5

Skewed Population Distribution frequency raw score 7 8 9

Distribution of Sample Means Samples of Size frequency ` sample mean Spreadsheet

Systematic Random Sample

Advantages of Systematic Random Sampling Representative of the population Because the sample is random, we can make statistical conclusions that would be considered valid

Disadvantages of Systematic Random Sampling A complete and up to date list of all the population is required If the population is listed in some standardised pattern, then systematic sampling could pick out similar members rather than completely random members

Uniform Population Distribution 6 raw score frequency Mean = 5 Standard deviation =

Distribution of Sample Means Sample Size frequency sample mean Spreadsheet

Things to Notice 1.The sample means tend to pile up around the population mean. 2.The distribution of sample means is approximately normal in shape, even though the population distribution was not. 3.The distribution of sample means has less variability than does the population distribution. 4.Increasing sample size decreases the variability in the distribution of samples.

The Central Limit Theorem

Non Probabilistic Sampling Quota Sampling Convenience Sampling Snowball Sampling