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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

Independent Measures T-Test

Quantitative Methods in HPELS

440:210


Agenda
Agenda

  • Introduction

  • The t Statistic for Independent-Measures

  • Hypothesis Tests with Independent-Measures t-Test

  • Instat

  • Assumptions


Introduction
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


Introduction1
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


Agenda1
Agenda

  • Introduction

  • The t Statistic for Independent-Measures

  • Hypothesis Tests with Independent-Measures t-Test

  • Instat

  • Assumptions


Independent measures t test1
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 test2
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
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 designs1
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 designs2
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


Agenda2
Agenda

  • Introduction

  • The t Statistic for Independent-Measures

  • Hypothesis Tests with Independent-Measures t-Test

  • Instat

  • Assumptions


Hypothesis test independent measures t test
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 test1
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 test2
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?


Agenda3
Agenda

  • Introduction

  • The t Statistic for Independent-Measures

  • Hypothesis Tests with Independent-Measures t-Test

  • Instat

  • Assumptions


Instat
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”


Instat1
Instat

  • Data Column Menu:

    • Choose variable of interest

  • Parameter Menu:

    • Choose “Mean (t-interval)”

  • Confidence Level:

    • 90% = alpha 0.10

    • 95% = alpha 0.05


Instat2
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
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).


Agenda4
Agenda

  • Introduction

  • The t Statistic for Independent-Measures

  • Hypothesis Tests with Independent-Measures t-Test

  • Instat

  • Assumptions


Assumptions of independent measures t test
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
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
Textbook Assignment

  • Problems: 3, 11, 19


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