Sampling and Sampling Distributions

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Sampling and Sampling Distributions

• Aims of Sampling
• Probability Distributions
• Sampling Distributions
• The Central Limit Theorem
• Types of Samples

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Aims of sampling

• Reduces cost of research (e.g. political polls)
• Generalize about a larger population (e.g.,
  benefits of sampling city r/t neighborhood)
• In some cases (e.g. industrial production)
  analysis may be destructive, so sampling is needed

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Probability

• Probability what is the chance that a given
  event will occur?
• Probability is expressed in numbers between 0 and
  1. Probability 0 means the event never
  happens probability 1 means it always happens.
• The total probability of all possible event
  always sums to 1.

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Probability distributions Permutations

• What is the probability distribution of number of
  girls in families with two children?
• 2 GG
• 1 BG
• 1 GB
• 0 BB

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How about family of three?
Num. Girls child 1 child 2 child 3
0 B B B
1 B B G
1 B G B
1 G B B
2 B G G
2 G B G
2 G G B
3 G G G
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Probability distribution of number of girls
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How about a family of 10?
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As family size increases, the binomial
distribution looks more and more normal.
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Normal distribution

• Same shape, if you adjusted the scales

B
C
A
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Coin toss

• Toss a coin 30 times
• Tabulate results

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Coin toss

• Suppose this were 12 randomly selected families,
  and heads were girls
• If you did it enough times distribution would
  approximate Normal distribution
• Think of the coin tosses as samples of all
  possible coin tosses

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Sampling distribution

• 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.
  

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Central Limit Theorem

• No matter what we are measuring, the distribution
  of any measure across all possible samples we
  could take approximates a normal distribution, as
  long as the number of cases in each sample is
  about 30 or larger.

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Central Limit Theorem

• If we repeatedly drew samples from a
  population and calculated the mean of a variable
  or a percentage or, those sample means or
  percentages would be normally distributed.

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Most empirical distributions are not normal
U.S. Income distribution 1992
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But the sampling distribution of mean income over
many samples is normal
Number of samples
Number of samples
18 19 20 21 22
23 24 25 26
Sampling Distribution of Income, 1992 (thousands)
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Standard Deviation

• Measures how spread out a distribution is.
• Square root of the sum of the squared deviations
  of each case from the mean over the number of
  cases, or

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Example of Standard Deviation
2
2
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Standard Deviation and Normal Distribution
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Distribution of Sample Means with 21 Samples
10 8 6 4 2 0
S.D. 2.02 Mean of means 41.0 Number of Means
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Frequency
37 38 39 40 41 42 43 44 45 46
Sample Means
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Distribution of Sample Means with 96 Samples
14 12 10 8 6 4 2 0
S.D. 1.80 Mean of Means 41.12 Number of Means
96
Frequency
37 38 39 40 41 42 43 44 45 46
Sample Means
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Distribution of Sample Means with 170 Samples
30 20 10 0
S.D. 1.71 Mean of Means 41.12 Number of Means
170
Frequency
37 38 39 40 41 42 43 44 45 46
Sample Means
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• The standard deviation of the sampling
  distribution is called the standard error

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The Central Limit Theorem
Standard error can be estimated from a single
sample
Where s is the sample standard deviation (i.e.,
the sample based estimate of the standard
deviation of the population), and n is the
size (number of observations) of the sample.
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Confidence intervals
Because we know that the sampling distribution
is normal, we know that 95.45 of samples will
fall within two standard errors. 95 of
samples fall within 1.96 standard errors.
99 of samples fall within 2.58 standard
errors.
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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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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

• Proportionate stratified sample The size of the
  sample selected from each subgroup is
  proportional to the size of that subgroup in the
  entire population. (Self weighting)
• Disproportionate stratified sample The size of
  the sample selected from each subgroup is
  disproportional to the size of that subgroup in
  the population. (needs weights)

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Disproportionate Stratified Sample
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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