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Independent Measures T-Test Quantitative Methods in HPELS 440:210

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Agenda Introduction The t Statistic for Independent-Measures Hypothesis Tests with Independent- Measures t-Test Instat Assumptions

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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

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Introduction Scenario #1: Data from 1 st sample are INDEPENDENT from data from 2 nd AKA: Independent-measures design Between-subjects design Scenario #2: Data from 1 st sample are RELATED or DEPENDENT on data from 2 nd AKA: Correlated-samples design Within-subjects design

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Agenda Introduction The t Statistic for Independent-Measures Hypothesis Tests with Independent- Measures t-Test Instat Assumptions

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Independent-Measures t-Test Statistical Notation: µ 1 + µ 2 : Population means for group 1 and group 2 M 1 + M 2 : Sample means for group 1 and group 2 n 1 + n 2 : Sample size for group 1 and group 2 SS 1 + SS 2 : Sum of squares for group 1 and group 2 df 1 + df 2 : Degrees of freedom for group 1 and group 2 Note: Total df = (n 1 – 1) + (n 2 – 1) s (M1-M2) : Estimated SEM

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Independent-Measures t-Test Formula Considerations: t = (M 1 -M 2 ) – (µ 1 -µ 2 ) / 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 (s 2 p ) SEM (s (M1-M2) ): s (M1-M2) = s 2 p /n 1 + s 2 p /n 2 where: s 2 p = SS 1 +SS 2 / df 1 +df 2

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Independent-Measures Designs Static-Group Comparison Design: Administer treatment to one group and perform posttest Perform posttest to control group Compare groups XOOXOO

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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 OXO Δ OO Δ

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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 ROXO Δ ROO Δ

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Agenda Introduction The t Statistic for Independent-Measures Hypothesis Tests with Independent- Measures t-Test Instat Assumptions

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Recall General Process: 1. State hypotheses State relative to the two samples No effect samples will be equal 2. Set criteria for decision making 3. Sample data and calculate statistic 4. Make decision Hypothesis Test: Independent- Measures t-Test

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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 (n 1 =10, n 2 =10) of people are obtained n 1 Provided 5-minutes to memorize the list with instructions to use mental images n 2 Provided 5-minutes to memorize the list

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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?

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1.734 Step 1: State Hypotheses Non-Directional H 0 : µ 1 = µ 2 H 1 : µ 1 µ 2 Directional H 0 : µ 1 µ 2 H 1 : µ 1 > µ 2 Step 2: Set Criteria Alpha ( ) = 0.05 Degrees of Freedom: df = (n 1 – 1) + (n 2 – 1) df = (10 – 1) + (10 – 1) = 18 Critical Values: Non-Directional 2.101 Directional 1.734

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Step 4: Make Decision Accept or Reject? Step 3: Collect Data and Calculate Statistic Pooled Variance (s 2 p ) s 2 p = SS 1 + SS 2 / df 1 + df 2 s 2 p = 200 + 160 / 9 + 9 s 2 p = 360 / 18 s 2 p = 20 SEM (s (M1-M2) ) s (M1-M2) = s 2 p / n 1 + s 2 p / n 2 s (M1-M2) = 20 / 10 + 20 / 10 s (M1-M2) = 2 +2 s (M1-M2) = 2 t-test: t = (M 1 -M 2 ) – (µ 1 -µ 2 ) / s (M1-M2) t = (25-19) – (0-0) / 2 t = 6 / 2 = 3

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Agenda Introduction The t Statistic for Independent-Measures Hypothesis Tests with Independent- Measures t-Test Instat Assumptions

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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

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Instat Data Column Menu: Choose variable of interest Parameter Menu: Choose Mean (t-interval) Confidence Level: 90% = alpha 0.10 95% = alpha 0.05

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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!!!

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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).

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Agenda Introduction The t Statistic for Independent-Measures Hypothesis Tests with Independent- Measures t-Test Instat Assumptions

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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

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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

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Textbook Assignment Problems: 3, 11, 19

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