Inferring Sample Findings
to the Population and
Testing for Differences
Jump to first page
Statistics versus
Parameters




Values computed from samples
are statistics!
Values computed from the
population are parameters!
Use Greek letters when referring to
parameters.
Use Roman letters for statistics.
Inference and Statistical
Inference



Inference - generalize about an
entire class based on what you
have observed about a small set of
members of that class.
Draw a conclusion from a small
amount of evidence.
Statistical Inference - Sample
size and sample statistics are used
to make estimates of population
parameters.


Hypothesis
Testing
Statistical procedure used to accept or reject the hypothesis
based on sample information.
Steps in hypothesis testing
 Begin with a statement about what you believe exists
in the population.




Draw a random sample and determine the sample
statistic.
Compare the statistic with the hypothesized parameter.
Decide whether or not the sample supports the original
hypothesis
If the sample does not support the hypothesis, revise
the hypothesis to be consistent with the sample's
statistic.
Test of the Hypothesized
Population Parameter Value
For example, we hypothesize that the average GPA for
business majors is not the same as Recreation majors.
z
x
µH
sx
The sample mean is compared to the hypothesized mean,
if z exceeds critical value of z (e.g., 1.96) then we reject the
hypothesis that the population mean is Mu.
z
p
H
sp
Directional Hypotheses



Indicates the direction in which you believe the
population parameter falls.
For example, the average GPA of business majors
is higher than the average GPA of Recreation
Majors.
Note that we are now interested in the volume of the
curve on only one side of the mean.
Interpretation


If the hypothesis about the population
parameter is correct or true,then a high
percentage of the sample means must fall
close to this value (i.e., within +/-1.96 sd.)
Failure to support the hypothesis tells the
hypothesizer that the assumptions about the
population are in error.
Testing for Differences
Between Two Means


z
Ho: There is no difference between
two means. (Mu1=Mu2)
Ha: There is a difference between
two means. (Mu1 does not equal
Mu2).
x1x2
s x x
1
2
sx
x2
1
2

s1
2
s2

n1 n2
Testing for Differences Between
Two Means: Example



Is there a statistically significant difference
between men and women on how many
movies they have seen in the last month?
Ho: There is no difference between two
means. (MuW=MuM)
Ha: There is a difference between two
means. (MuW/=MuM)
Example
Gender
N
Mean
St. Dev
male
19
2.3684
1.98
female
13
2.5385
2.18
t
TOTAL
df
-.229 30
Significance (2 tailed)
.820
F
Sig.
Levene’s test for equality of variance .004 .952
Testing for Differences Between
Two Means


Fail to reject the null hypothesis, that the means are equal
Why?
 Significance = .82
Reject any significance lower than .05
 .82 > ,05; therefore, fail to reject null
 there is no statistically significant difference between men
and women on how many movies seen in last month
 makes sense - look at means (2.36 & 2.53)

Small Sample Size - t-Test




Normal bell curve assumptions are
invalid when sample sizes are 30
or less.
Alternative choice is t-Test
Shape of t distribution is
determined by sample size (i.e.,
degrees of freedom).
df = n-1
ANOVA




ANOVA = Analysis of Variance
Compares means across multiple groups
ANOVA will tell you that one pair of means has a
statistically significant difference but not which
one
Assumptions:
 independence
 normality
 equality
of variance (Levene test)
Analysis of Variance
When researchers want to compare the means of three or
more groups.
ANOVA used to facilitate comparison!
Basic Analysis
Does a statistical significance difference exist between at
least two groups of means?
ANOVA does not communicate how many pairs of means are
statistically significant in their differences.
Hypothesis Testing
Ho: There is no difference among the
population means for the various groups.
Ha: At least two groups have different
population means..
When MSBetween is significantly greater than
MSWithin then we reject Ho.
F value
F = MSBetween/MSWithin
If F exceeds Critical F(df1, df2) then we reject Ho.
Visual Representation
Population 2
Population 1
Population 3
Population 5
Population 4
Appears that at least 2 populations have different
means.
Visual Representation
Population 3
Population 2
Population 1
Population 4
Population 5
Appears that populations do not have significantly
different means.
Tests of Differences

Chi-square goodness-of-fit
 Does
some observed pattern of frequencies
correspond to an expected pattern?

Z-test/T-test
 Is
there a significant difference between the
means of two groups?

ANOVA
 Is
there a significant difference between the
means of more than two groups?
When to Use each Test

Chi-square goodness-of-fit
 Both

variables are categorical/nominal.
T-test
 One
variable is continuous; the other is
categorical with two groups/categories.

ANOVA
 One
variable is continuous (i.e., interval or
ratio); the other is categorical with more than
two groups.
How to Interpret a Significant
p-value (p < .05)

Chi-square goodness-of-fit


T-test


“There is a significant difference in frequency of responses
among the different groups (or categories).”
“The means (averages) of the 2 population groups are
different on the characteristic being tested.”
ANOVA

“The means of the (multiple) population groups are different need post hoc test (e.g., Bonferroni) to determine exactly
which group means are different from one another.”
Measuring Association



Is there any association
(correlation) between two or more
variables?
If so, what is the strength and
direction of the correlation?
Can we predict one variable
(dependent variable) based on its
association with other variables
(independent variables)?
Correlation Analysis



estimate of correlation between two A statistical
technique used to measure the closeness of the
linear relationship between two or more variables.
Can offer evidence of causality, but is not enough to
establish causality by itself (must also have evidence
of knowledge/theory and correct sequence of
variables).
Scatterplots can give visual variables.
Regression Analysis

Simple Regression
 relate
a single criterion (dependent) variable to
a single predictor (independent) variable

Multiple Regression
 relate
a single criterion variable to multiple
predictor variables

All variables should be at least interval!
Correlation/Regression

Coefficient of Correlation (r)
 measure
of the strength of linear association
between two variables
 also called “Pearson’s r” or “product-moment”
 ranges from -1 to +1

Coefficient of Determination (r2)
 proportion
of variance in the criterion explained by
the fitted regression equation (of predictors)