Independent Measures T-Test

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Independent Measures T-Test. Quantitative Methods in HPELS 440:210. Agenda. Introduction The t Statistic for Independent-Measures Hypothesis Tests with Independent-Measures t-Test Instat Assumptions. Introduction. Recall  Single-Sample t-Test: Collect data from one sample

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### Independent Measures T-Test

Quantitative Methods in HPELS

440:210

Agenda
• Introduction
• The t Statistic for Independent-Measures
• Hypothesis Tests with Independent-Measures 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 Independent-Measures 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:
• t = (M1-M2) – (µ1-µ2) / s(M1-M2)
• Recall  Estimated SEM (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 X O  Δ

O 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 X O  Δ

R O O  Δ

Agenda
• Introduction
• The t Statistic for Independent-Measures
• Hypothesis Tests with Independent-Measures t-Test
• Instat
• Assumptions
Hypothesis Test: Independent-Measures t-Test
• Recall  General Process:
• State hypotheses
• State relative to the two samples
• No effect  samples will be equal
• 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

Non-Directional

H0: µ1 = µ2

H1: µ1≠ µ2

Directional

H0: µ1≤ µ2

H1: µ1 > µ2

Degrees of Freedom:

df = (n1 – 1) + (n2 – 1)

df = (10 – 1) + (10 – 1) = 18

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 Independent-Measures 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”
• Choose “Two Data Columns”
Instat
• Choose variable of interest
• Choose “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 Independent-Measures 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