1. THE CONCEPT OF STATISTICAL SIGNIFICANCE:
CHI-SQUARE AND THE NULL HYPOTHESIS

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

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

4. 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

5. 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 individual—more 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

6. 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.

7. 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

8. Problems in Sampling (II)
Ho for Sample
Accepted Rejected
Ho for Population
True Type I
False Type II
Where Ho = null hypothesis

10. 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 = Σ (fo – fe)2/fe

12. Calculating Expected Frequencies:
fe = col Σ (row Σ/total N)
for upper left-hand cell
= 802 (200/1,679)
= 95.5
fo = 44
fo – fe = 44 – 95.5 = (fo – fe)2 = 2,652.25
(fo – fe)2/fe =

13. 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?

14. Figuring Degrees of Freedom:
df = (r – 1)(c – 1)
Illustration: Given marginal values,
________X________
__Y__ L H Σ L H
Σ and df = 1

16. Characteristics of Chi-Square
Distribution for null hypothesis has a known distribution—skewed to the right
Specific distributions have corresponding degrees of freedom, defined as (r-1)(c-1)
For a 2x2 table, chi-square of or greater would occur no more than 5% of the time in event of null hypothesis (thus, “.05 level or better”)

19. 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 < .05)

20. 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

21. 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”