Download Inferential Statistics

Data Analysis:
Inferential Statistics
Dr. Mark Baron
University of South Dakota
Inferential Statistics
• that part of statistical procedures that deals
with making inferences about population
parameters from sample statistics
• Statistics - sample measures
• Parameters - population measures
Dr. Mark Baron
University of South Dakota
Uses of Inferential Stats
• Testing hypotheses about population
parameters
• Estimating population parameters
• Point estimate
• Interval estimate
Dr. Mark Baron
University of South Dakota
Sampling Distribution
• The distribution (usually theoretical) of all
possible values of the statistic from all
possible samples of a given size selected
from the population
Dr. Mark Baron
University of South Dakota
Central Limit Theorum
• Given any population with mean μ and
finite variance σ2, as the sample size
increases without limit, the distribution of
the sample mean approaches a normal
distribution with mean μ and variance σ2/n,
where n is the sample size
Dr. Mark Baron
University of South Dakota
Standard Normal Distribution
Dr. Mark Baron
University of South Dakota
Standard Normal Distribution
• The normal distribution with a mean of 0 and a
standard deviation of 1.0
• Z-score:
Dr. Mark Baron
x-x
z=
s
University of South Dakota
Significance Level (α)
• Maximum probability that a sample statistic
would differ significantly from the
corresponding population parameter (or
from another statistic) due to chance alone.
• A criterion used in making the decision of
whether or not to reject a hypothesis.
Dr. Mark Baron
University of South Dakota
Errors in Hypothesis Testing
• Null Hypothesis (Ho) - a hypothesis stated such
that no difference or no relationship exists.
• Ho: μ1 = μ2 OR Ho: μ1 - μ2 = 0
• Type I Error (α) - rejection of a true null
hypothesis (i.e. finding a significant difference
when it does not actually exist)
• Type II Error (β) - failure to reject a false null
hypothesis (i.e. failure to find a significant
difference when it does exist)
Dr. Mark Baron
University of South Dakota
Possible Testing Outcomes
Reject
Fail to
Reject
True Ho
False Ho
Type I
Error
(α)
Correct
Correct
Type II
Error
(β)
Dr. Mark Baron
University of South Dakota
Parametric Analyses
• A set of statistical analyses that make the
following parametric assumptions:
• measurement of the dependent variable is on the
interval scale
• individuals' scores are independent
• the scores are selected from a normally distributed
population
• when two or more populations are being studied, they
have homogeneous variance (i.e. both have similar
dispersions in their distributions)
Dr. Mark Baron
University of South Dakota
Parametric Analyses
• t test
• independent means
• dependent means / correlated means
• Analysis of Variance (ANOVA)
•
•
•
•
one-way ANOVA
Independent variable(s) / dependent variable
Post-hoc tests (Tukey / Scheffé / etc.)
two-way ANOVA (main effects / interaction)
Dr. Mark Baron
University of South Dakota
Parametric Analyses
• Analysis of covariance (ANCOVA)
• covariate
• Multivariate ANOVA (MANOVA)
• Regression (linear / multiple)
• predictors / outcomes
• regression coefficient (R)
• Pearson product-moment
• Canonical correlation
Dr. Mark Baron
University of South Dakota
Nonparametric Analyses
• Inferential statistics used to test null
hypotheses when the parametric
assumptions cannot be met
• require few (if any) assumptions about the
population
• usually nominal and ordinal data
• involve statistics other than means
Dr. Mark Baron
University of South Dakota
Nonparametric Analyses
• Chi Square (Χ2)
• Goodness of Fit
• Test of Independence
• Correlation
• Phi / Contingency Coefficient
• Spearman Rho
• Biserial / Point Biserial / Tetrachoric / Gamma
Dr. Mark Baron
University of South Dakota
Nonparametric Analyses
•
•
•
•
•
Sign Test
Rank-Sums Test
Mann-Whitney U Test
Kruskal-Wallis Test
Friedman’s Test
Dr. Mark Baron
University of South Dakota
Meta-analyses
• a statistical procedure used to synthesize the
results across numerous independently
conducted research studies
X exper - X control
EFFECT SIZE =
SD
Dr. Mark Baron
University of South Dakota