Ch. 10 - The t Test For Two Independent Samples
Page 1: Recap - Z-Test
Calculate Z-Test
Components needed:
Mean (M)
Population Mean (µ)
Population Standard Deviation (σ) or Sample Size (n1)
Page 2: Sample Variance
Sample Variance (s²)
Represents the mean squared distance from the mean.
Obtained by dividing the sum of squares by (n - 1).
Estimated Standard Error of the Mean (sM)
Used when σ is unknown.
Computed from the sample variance or standard deviation.
Provides an estimate of the difference between a sample mean (M) and the population mean (µ).
Page 3: Recap - Single Sample T-Test
Single Sample T-Test
Formula: t = (M - μ) / SM
M = Sample mean
μ = Population mean (hypothesized from H0)
SM = Estimated standard error (computed from the sample data)
Page 4: Recap - Single Sample T-Test Calculation
Calculate T-Test Components
We need:
Sample Mean (M)
Population Mean (µ)
Sum of Squares (SS) or Standard Deviation (s)
Sample Size (n)
Page 5: Recap - Variables
Independent Variable (IV)
Manipulated by the researcher.
The variable that is tested or manipulated.
Dependent Variable (DV)
Observed to assess the effect of the treatment.
The outcome variable; referred to as the ‘Dependent Measure’.
Page 6: The T-Test for Two Independent Samples
Introduction to Chapter 10.
Page 7: Research Designs for Two Sets of Data
Research studies often require the comparison of two or more sets of data.
Two General Research Designs:
Separate groups of participants.
Same group of participants.
Independent-Measures Design
Utilizes completely separate groups of participants for each treatment condition.
Page 8: Two Basic Designs
Between Groups (Independent Measures)
IV manipulated between different groups of subjects.
Different subjects experience a single condition.
Best design when carryover effects are expected.
Within Subjects (Repeated Measures)
IV manipulated within a single group of subjects.
Each participant is exposed to all levels of the IV; use when carryover effects are unlikely.
Page 9: Nomenclature for Independent Samples
Independent-Measures Terms
Between-Subjects Design
Between-Groups Design
Independent (sample) t-test
Values/Levels of an IV are Groups or Treatments.
IV can also be referred to as “Factor.”
Page 10: Levels of an IV
Differences in experiments:
Between Groups Experiment: Groups or Treatments.
Within Subjects Experiment: Conditions.
Page 11: Basic Between-Groups Design
Single IV manipulated at 2 levels
e.g., Placebo vs 5 mg of drug, Men vs Women.
Page 12: Testing the Basic Between-Groups Design
Use independent t-test to test the difference between means.
Group A and Group B scores are independent of each other.
Page 13: Independent-Measures Research Study
Involves two separate samples to obtain information about two unknown populations or treatment conditions.
Page 14: Goal of Independent-Measures Study
Evaluate the mean difference between two populations or treatment conditions.
Means represented as µ1 for the first population and µ2 for the second.
Null hypothesis states no difference: µ1 - µ2 = 0.
Alternative hypothesis states mean difference exists: µ1 - µ2 ≠ 0.
Page 15: Null Hypothesis for Independent-Measures Test
Null Hypothesis (H0): µ1 - µ2 = 0 or µ1 = µ2.
Alternative Hypothesis (H1): The means are different.
Non-directional hypothesis (two-tailed).
Page 16: One-Tailed Predictions for Population 1 and 2
Predictions for Population 1:
H0: μ1 ≤ μ2
H1: μ1 > μ2
Predictions for Population 2:
H0: µ1 ≥ µ2
H1: μ1 < μ2
Page 17: Independent-Measures T-Statistic
Evaluates hypothesis about differences between two population means using sample means difference (M1 − M2).
Page 18: Standard Error in T-Statistic
Measures error expected using sample means to represent population means.
Represented by symbols sM1-M2.
Page 19: Estimated Standard Error of M1 − M2
Interpreted as the average distance between a sample statistic and the population parameter.
Page 20: Formula Development for s(M1-M2)
Total error in using two sample means to approximate two population means calculated by adding errors from each sample.
Page 21: Error Measurement in Independent-Measures
Error from each sample contributes to total error in using sample means.
Page 22: Total Error Calculation for Sample Means
Calculate the total error for two sample means.
Page 23: Sample Variance with One Sample
Sample variance calculated and combined into pooled variance corrects bias in the standard error.
Page 24: Independent T-Test – Pooled Variance
Pooled Variance combines the sample variances to correct bias in standard error.
Allows larger sample to contribute more.
Page 25: Independent T-Test - Estimated Standard Error of the Mean
Relevant formula updates with pooled variance considerations.
Page 26: Final Formula for Independent-Measures T Statistic
Complete formula incorporating components and degrees of freedom.
Page 27: Degrees of Freedom Calculation for T Statistic
Degrees of freedom (df) calculated as: df = (n1 – 1) + (n2 – 1) = n1 + n2 - 2.
Page 28: Degrees of Freedom for Entire Data Set
Necessary for determining critical t-value.
Page 29: Basic T Statistic Calculation
Formula example: t = (M1 - M2) / SM.
Page 30: Elements of a T Statistic
Each component defined:
Hypothesized parameter, estimated error, standard variance.
Page 31: Independent-Measures T Statistic Overview
Uses two samples to evaluate mean difference between two treatment conditions.
Page 32: Result Write Up for Significant Results
Guide for writing results when significant:
"The analysis showed a significant effect of IV on DV ..."
Page 33: Result Write Up for Non-Significant Results
Guide for writing results when not significant:
"Even though mean measure was ... this difference was not significant."
Page 34: Hypothesis Tests Steps with Independent-Measures T Statistic
Steps involve stating hypotheses, computing degrees of freedom, obtaining data, and making decisions.
Page 35: The Critical Region in Hypothesis Tests
Overview of critical region and decision criteria for rejecting null hypothesis.
Page 36: Example Study: Sleepytime
Sample size analysis with control and experimental groups detailed.
Page 37: Step 0: Identification of DV and IV
Clarification of dependent and independent variables.
Page 38: Step 1: Hypothesis and Alpha Level
Null hypothesis defines equal means; alternative defines different means.
Page 39: Step 2: Locate T Critical Value
Critical value based on total score number and degrees of freedom.
Page 40: Step 3: Calculate Test Statistics
Preliminary calculations shown with example data.
Page 41: Formulas to Pool Variances
Summarizes formulas for combining sample variances.
Page 42: Step 4: Make a Decision
Decision-making process based on comparison of t-obtained vs t-critical.
Page 43: T Distribution Table Overview
Value entries for t corresponding to proportions in tails.
Page 44: Reporting the Result - Example Statement
Guide for structuring result interpretation statements in research.
Page 45: Assumptions of Independent-Measures T Formula
Key assumptions stated; independent observations and population normality required.
Page 46: Homogeneity of Variance Assumption
Importance of variance equality when sample sizes differ.
Page 47: Hartley’s F-Max Test Overview
Test for homogeneity of variance through sample variances comparison.
Page 48: Computing Hartley’s F-Max Test Steps
Steps to compute sample variance and F-max.
Page 49: Critical Value Comparison in Hartley's Test
Explain how to locate critical values in F-max table.
Page 50: F-Max Critical Values Table
Display of critical values by sample sizes and alpha levels.
Page 51: Hartley’s F-Max Test Decision Criteria
Outcome implications based on F-max values.
Page 52: Handling Violations of Homogeneity Assumption
Post-hoc analysis adjustments when homogeneity is violated.
Page 53: Effect Size Overview
Defines effect size and relevance of measuring treatment effects.
Page 54: Effect Size Decision-Making Criteria
Scenarios for determining when to reject the null hypothesis based on t statistics.
Page 55: Measuring Effect Size with Cohen’s d
Calculation of mean differences standardized through Cohen's d.
Page 56: Percentage of Variance Accounted for (r2)
Discusses effect size in terms of variance explained by treatment.
Page 57: Percentage of Variance Calculation Format
Formal method to express variance accounted for by treatment effect.
Page 58: Confidence Intervals for Mean Differences
Step for estimating the unknown population difference based on sample data.
Page 59: Estimated t Value Implications
Use of t distribution for estimating population mean difference through confidence intervals.
Page 60: Confidence Interval Interpretation
How to determine population mean score range based on sample analysis.
Page 61: Impact of Variability and Sample Size on Tests
Discussion on how both factors impact hypothesis test outcomes.
Page 62: Standard Error Influences on T Statistic
Exploration of relationships between variance, sample size, and outcomes.
Page 63: Independent-Measures T-Test Key Features
Review of key components needed for independent measures t-test construction.