SAMPLING STRATEGIES

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THE CONCEPT OF STATISTICAL SIGNIFICANCE CHI-SQUAR
E AND THE NULL HYPOTHESIS
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READINGS

• Pollock, Essentials, ch. 5 and ch. 6, pp. 121-135
• Pollock, SPSS Companion, ch. 7

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OUTLINE

• Strategies for Sampling
• Establishing Confidence Intervals
• Chi-Square and the Null Hypothesis
• Critical Values of Chi-Square

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• Why Sample?
• Goal description of a population
• Advantages savings of time and money
• Basic paradox credibility of results from a
  sample
• depends on size and quality of the sample
  itself,
• and not on the size of the population

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Types of Samples Probability sampling Every
individual in the population has a known
probability of being included in the
sample Random sample (SRS) each individual has
an equal chance of being selected, and all
combinations are equally possible Systematic
sample every kth individualmore or
less equivalent to SRS if first selection is made
through random process Stratified sample
individuals separated into categories, and
independent (SRS) samples selected within the
categories Cluster sample population divided
into clusters, and random sample (SRS) then drawn
of the clusters
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• Parameters and Statistics
• A parameter is a number that describes the
  population. It is
• a fixed number, though we do not know its value.
• A statistic is a number that describes a sample.
  We use
• statistics to estimate unknown parameters.
• A goal of statistics To estimate the
  probability that the
• null hypothesis holds true for the population.
  Forms
• Parameter may not fall within a confidence band
  that can be placed around a sample statistic, or
• A relationship observed within a sample may not
  have a satisfactory probability of existing
  within the population.

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• Problems with Sampling (I)
• Bias
• A consistent, repeated deviation of the sample
  statistic
• from the population parameter
• Convenience sampling
• Voluntary response sampling
• Solution Use SRS
• Variation
• Signal large standard deviation within sample
• Range of sample statistics
• Solution Use larger N

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Problems in Sampling (II) Ho for
Sample Accepted Rejected Ho for Population
True Type I False Type II
Where Ho null hypothesis
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What is Chi-square? A measure of
significance for cross-tabular
relationships Where fo observed frequency
(or cell count) And fe expected frequency
(or cell count) X2 S (fo fe)2/fe
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Calculating Expected Frequencies fe col S
(row S/total N) for upper left-hand cell
802 (200/1,679) 95.5 fo 44 fo fe
44 95.5 -51.5 (fo fe)2 2,652.25 (fo
fe)2/fe 27.77
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Conceptualizing Chi-Square

• Expected frequencies represent the null
  hypothesis (no relationship)
• Observed frequencies present visible results
• Question 1 Are observed frequencies different
  from expected frequencies?
• Question 2 Are they sufficiently different to
  allow us to reject the possibility that the true
  relationship (within the universe of case) is
  null?

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Figuring Degrees of Freedom df (r 1)(c
1) Illustration Given marginal values,
________X________ __Y__ L H S
L 30 50 H 50 S
60 40 100 and df 1
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Characteristics of Chi-Square

• Distribution for null hypothesis has a known
  distributionskewed to the right
• Specific distributions have corresponding degrees
  of freedom, defined as (r-1)(c-1)
• For a 2x2 table, chi-square of 3.841 or greater
  would occur no more than 5 of the time in event
  of null hypothesis (thus, .05 level or better)

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POSTSCRIPT

• X2 f (strength of relationship, sample size)
• The stronger the observed relationship within the
  sample, the higher the X2
• The larger the sample (SRS), the higher the X2
• The higher the X2 (given degrees of freedom), the
  greater the probability that null hypothesis does
  not hold in the population (p lt .05)

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Limitations of Chi-Square

• No more than 20 of expected frequencies less
  than 5 and all individual expected frequencies
  are 1 or greater
• Directly proportional to N observations
• Rejection of null hypothesis does not directly
  confirm strength or direction of relationship

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Review Summary Measures for Cross-Tabulations

• Lambda-b PRE, ranges from zero to unity
  measures strength only
• Gamma Form and strength (-1 to 1), based on
  pairs of observations
• Chi-square Significance, based on deviation
  from null hypothesis