Independent
Measures T-Test
Quantitative Methods in HPELS
440:210
Agenda
Introduction
 The t Statistic for Independent-Measures
 Hypothesis Tests with IndependentMeasures t-Test
 Instat
 Assumptions

Introduction

Recall  Single-Sample t-Test:
 Collect
data from one sample
 Compare to population with:
Known µ
 Unknown 


This scenario is rare:
 Often
researchers must collect data from two
samples
 There are two possible scenarios
Introduction

Scenario #1:


Data from 1st sample are INDEPENDENT from data
from 2nd
AKA:



Independent-measures design
Between-subjects design
Scenario #2:


Data from 1st sample are RELATED or
DEPENDENT on data from 2nd
AKA:


Correlated-samples design
Within-subjects design
Agenda
Introduction
 The t Statistic for Independent-Measures
 Hypothesis Tests with IndependentMeasures t-Test
 Instat
 Assumptions

Independent-Measures t-Test

Statistical Notation:





µ1 + µ2: Population means for group 1 and group 2
M1 + M2: Sample means for group 1 and group 2
n1 + n2: Sample size for group 1 and group 2
SS1 + SS2: Sum of squares for group 1 and group 2
df1 + df2: Degrees of freedom for group 1 and group 2


Note: Total df = (n1 – 1) + (n2 – 1)
s(M1-M2): Estimated SEM
Independent-Measures t-Test

Formula Considerations:


Recall  Estimated SEM (s(M1-M2)):



t = (M1-M2) – (µ1-µ2) / s(M1-M2)
Sample estimate of a population  always error
SEM measures ability to estimate the population
Independent-Measures t-test uses two
samples therefore:


Two sources of error
SEM estimation must consider both


Pooled variance (s2p)
SEM (s(M1-M2)):


s(M1-M2) = √s2p/n1 + s2p/n2 where:
s2p = SS1+SS2 / df1+df2
Independent-Measures Designs

Static-Group Comparison Design:

Administer treatment to one group and
perform posttest
 Perform posttest to control group
 Compare groups
X
O
O
Independent-Measures Designs

Quasi-Experimental Pretest Posttest
Control Group Design:

Perform pretest on both groups
 Administer treatment to treatment group
 Perform posttests on both groups
 Compare delta (Δ) scores
O
O
X
O

Δ
O

Δ
Independent-Measures Designs

Randomized Pretest Posttest Control Group
Design:





Randomly select subjects from two populations
Perform pretest on both groups
Administer treatment to treatment group
Perform posttests on both groups
Compare delta (Δ) scores
R
O
R
O
X
O

Δ
O

Δ
Agenda
Introduction
 The t Statistic for Independent-Measures
 Hypothesis Tests with IndependentMeasures t-Test
 Instat
 Assumptions

Hypothesis Test: IndependentMeasures t-Test

1.
Recall  General Process:
State hypotheses

State relative to the two samples
 No effect  samples will be equal
2.
3.
4.
Set criteria for decision making
Sample data and calculate statistic
Make decision
Hypothesis Test: Independent-Measures t-Test


Example 10.1 (p 317)
Overview:
 Researchers are interested in determining the
effect of mental images on memory
 The researcher prepares 40 pairs of nouns
(dog/bicycle, lamp/piano . . .)
 Two separate groups (n1=10, n2=10) of
people are obtained
 n1  Provided 5-minutes to memorize the list
with instructions to use mental images
 n2  Provided 5-minutes to memorize the list
Hypothesis Test: Independent-Measures t-Test
Researchers provide the first noun and
ask subjects to recall second noun
 Number of correct answers recorded
 Questions:

 What
is the experimental design?
 What is the independent variable?
 What is the dependent variable?
Step 1: State Hypotheses
Degrees of Freedom:
Non-Directional
df = (n1 – 1) + (n2 – 1)
H0: µ1 = µ2
df = (10 – 1) + (10 – 1) = 18
H1: µ1 ≠ µ2
Directional
H0: µ1 ≤ µ2
H1: µ1 > µ2
Critical Values:
Non-Directional  2.101
Directional  1.734
Step 2: Set Criteria
Alpha (a) = 0.05
1.734
Step 3: Collect Data and Calculate Statistic
Pooled Variance (s2p)
s2p = SS1 + SS2 / df1 + df2
s2p = 200 + 160 / 9 + 9
s2p = 360 / 18
s2p = 20
SEM (s(M1-M2))
s(M1-M2) = √s2p / n1 + s2p / n2
s(M1-M2) = √20 / 10 + 20 / 10
s(M1-M2) = √2 +2
s(M1-M2) = 2
t-test:
t = (M1-M2) – (µ1-µ2) / s(M1-M2)
t = (25-19) – (0-0) / 2
t=6/2=3
Step 4: Make Decision
Accept or Reject?
Agenda
Introduction
 The t Statistic for Independent-Measures
 Hypothesis Tests with IndependentMeasures t-Test
 Instat
 Assumptions

Instat

Type data from sample into a column.
 Label
column appropriately.
Choose “Manage”
 Choose “Column Properties”
 Choose “Name”


Choose “Statistics”
 Choose


“Simple Models”
Choose “Normal, Two Samples”
Layout Menu:

Choose “Two Data Columns”
Instat

Data Column Menu:
 Choose

Parameter Menu:
 Choose

variable of interest
“Mean (t-interval)”
Confidence Level:
 90%
= alpha 0.10
 95% = alpha 0.05
Instat

Check “Significance Test” box:
 Check
“Two-Sided” if using non-directional
hypothesis.
 Enter value from null hypothesis.

If variances are unequal, check appropriate box
 More


on this later
Click OK.
Interpret the p-value!!!
Reporting t-Test Results


How to report the results of a t-test:
Information to include:

Value of the t statistic
 Degrees of freedom (n – 1)
 p-value

Examples:

Girls scored significantly higher than boys
(t(25) = 2.34, p = 0.001).
 There was no significant difference between
boys and girls (t(25) = 0.45, p = 0.34).
Agenda
Introduction
 The t Statistic for Independent-Measures
 Hypothesis Tests with IndependentMeasures t-Test
 Instat
 Assumptions

Assumptions of Independent-Measures
t-Test
Independent Observations
 Normal Distribution
 Scale of Measurement

 Interval

or ratio
Equal variances (homogeneity):
 Violated
if one variance twice as large as the
other
 Can still use parametric  with penalty
Violation of Assumptions
Nonparametric Version  Mann-Whitney U
(Chapter 17)
 When to use the Mann-Whitney U Test:

 Independent-Measures
design
 Scale of measurement assumption violation:

Ordinal data
 Normality

assumption violation:
Regardless of scale of measurement
Textbook Assignment

Problems: 3, 11, 19