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 sampleCompare to population with:

Known µ Unknown

This scenario is rare:Often researchers must collect data from two

samplesThere 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) = √s2

p/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

Recall General Process:1. State hypotheses

State relative to the two samples No effect samples will be equal

2. Set criteria for decision making3. Sample data and calculate statistic4. Make decision

Hypothesis Test: Independent-Measures t-Test

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?

1.734

Step 1: State Hypotheses

Non-Directional

H0: µ1 = µ2

H1: µ1 ≠ µ2

Directional

H0: µ1 ≤ µ2

H1: µ1 > µ2

Step 2: Set Criteria

Alpha () = 0.05

Degrees of Freedom:

df = (n1 – 1) + (n2 – 1)

df = (10 – 1) + (10 – 1) = 18

Critical Values:

Non-Directional 2.101

Directional 1.734

Step 4: Make Decision

Accept or Reject?

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 + s2

p / 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

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”

Layout Menu: Choose “Two Data Columns”

Instat

Data Column Menu:Choose variable of interest

Parameter Menu:Choose “Mean (t-interval)”

Confidence Level:90% = alpha 0.1095% = 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 designScale of measurement assumption violation:

Ordinal data

Normality assumption violation: Regardless of scale of measurement

Textbook Assignment

Problems: 3, 11, 19