Lecture 8: Paired-Samples t Test

Review of the Single-Sample t Test

  • Differences Between the z Test and the Single-Sample t Test

    • The primary difference in parameter knowledge requirements between a $z$ test and a single-sample $t$ test is the population standard deviation (σ\sigma).

    • In a $z$ test, both the population mean (μ\mu) and the population standard deviation (σ\sigma) are known.

    • In a single-sample $t$ test, the population standard deviation is unknown, requiring the use of a sample-based estimate.

  • Estimation of Population Standard Deviation

    • For a single-sample $t$ test, the statistic used to estimate the population standard deviation is the corrected sample standard deviation (ss).

    • The formula for the corrected sample standard deviation is:         s=∑i=1N(X−M)2N−1s = \sqrt{\frac{\sum_{i=1}^{N} (X - M)^2}{N - 1}}

  • Degrees of Freedom (dfdf)

    • In a single-sample $t$ test, the value of the degrees of freedom is calculated as N−1N - 1.

    • Knowing the degrees of freedom is essential because there are multiple $t$ distributions. The $df$ identifies which specific $t$ distribution must be used for a given sample size.

  • Critical Value Calculation Example

    • Scenario: A one-tailed test with an alpha (α\alpha) level of 0.100.10 and a sample size (NN) of 3232.

    • Calculation: df=32−1=31df = 32 - 1 = 31.

    • Result: The critical value (tcritt_{crit}) is 1.3111.311. When reporting this value, it should be rounded to two decimal places (1.311.31).

Introduction to the Paired-Samples t Test

  • Core Concepts and Applicability

    • Paired-samples $t$ tests are appropriate when observations in two sample groups are paired, such as when each participant is observed twice (repeated measures).

    • This test is used for within-groups designs, also known as within-subjects designs.

    • Example: Comparing empathy levels twice in the same participants—once after taking a control sugar pill and once after taking Tylenol.

  • Nature of the Comparison Distribution

    • Because each observation is paired, researchers calculate a difference score for each participant.

    • The relevant comparison distribution is the distribution of mean differences, rather than the distribution of means.

    • Performing this test requires all raw observations; it cannot be completed using only provided summary statistics and parameters.

  • Hypothesis Testing Framework Comparison

    • z Test: Known population mean (μ\mu) and population standard deviation (σ\sigma).

    • Single-Sample t Test: Compares a sample mean to a population mean (μ\mu) or a set value when σ\sigma is unknown.

    • Paired-Samples t Test: Compares paired observations from the same sample of participants.

    • Independent-Samples t Test: Compares means across different samples of participants.

    • ANOVA (One-Way Between-Subjects): Used when comparing across more than two samples.

    • Two-Way (or higher) ANOVA: Used if the test includes two or more variables.

    • Within-Subjects ANOVA (One-Way or higher): Used for more than two sets of observations within the same participants.

Step-by-Step Paired-Samples t Test: Caffeine and Speech Example

  • Experimental Scenario

    • Objective: Test if caffeine makes people talk more.

    • Design: Two conditions (caffeine treatment vs. sugar pill control) are counterbalanced.

    • Sample: N=5N = 5 (noted as unrealistically low for demonstration).

    • Dependent Variable: Number of words spoken in a 10 minute10\,minute group conversation.

  • Step 1: Identify Populations, Distribution, and Assumptions

    • Population 1: People who have taken caffeine.

    • Population 2: People who have taken a sugar pill.

    • Distribution: Distribution of mean difference scores.

    • Mean of Comparison Distribution (μM\mu_M): Under the null hypothesis, μM=0\mu_M = 0.

    • Assumptions:

      1. The dependent variable is continuous (Met).

      2. Participants were randomly selected (Met).

      3. The population and the distribution of mean difference scores are normally distributed (Uncertain due to small sample size; noted as a potential problem).

  • Step 2: State the Null and Research Hypotheses

    • Parameters: Two-tailed test, α=.05\alpha = .05.

    • Null Hypothesis (H0H_0): People who take caffeine will not speak a significantly different number of words compared to those who take a sugar pill (H0:μ1=μ2H_0: \mu_1 = \mu_2).

    • Research Hypothesis (H1H_1): People who take caffeine will speak a significantly different number of words compared to those who take a sugar pill (H1:μ1≠μ2H_1: \mu_1 \neq \mu_2).

  • Step 3: Determine Characteristics of the Comparison Distribution

    • The comparison distribution is the distribution of mean difference scores.

    • μM=0\mu_M = 0.

    • To find the standard error (sMs_M), the corrected sample standard deviation (ss) of the difference scores must first be calculated.

    • Raw Data and Difference Calculation:

      • Participant 1: Treatment 16111611, Control 14321432, Difference 179179.

      • Participant 2: Treatment 17501750, Control 13891389, Difference 361361.

      • Participant 3: Treatment 15701570, Control 14711471, Difference 9999.

      • Participant 4: Treatment 17241724, Control 13251325, Difference 399399.

      • Participant 5: Treatment 15571557, Control 12401240, Difference 317317.

    • Mean of Differences (MdifferenceM_{difference}):         Mdifference=179+361+99+399+3175=13555=271M_{difference} = \frac{179 + 361 + 99 + 399 + 317}{5} = \frac{1355}{5} = 271

    • Sum of Squares (SSSS) for Differences:

      • Deviations from Mean (Difference−MdifferenceDifference - M_{difference}):

        • 179−271=−92179 - 271 = -92

        • 361−271=90361 - 271 = 90

        • 99−271=−17299 - 271 = -172

        • 399−271=128399 - 271 = 128

        • 317−271=46317 - 271 = 46

      • Squared Deviations:

        • (−92)2=8,464(-92)^2 = 8,464

        • (90)2=8,100(90)^2 = 8,100

        • (−172)2=29,584(-172)^2 = 29,584

        • (128)2=16,384(128)^2 = 16,384

        • (46)2=2,116(46)^2 = 2,116

      • Sum of Squares (SSSS): 8,464+8,100+29,584+16,384+2,116=64,6488,464 + 8,100 + 29,584 + 16,384 + 2,116 = 64,648

    • Calculating ss and sMs_M:

      • Corrected Sample Standard Deviation (ss):             s=64,6485−1=64,6484=16,162≈127.12985s = \sqrt{\frac{64,648}{5 - 1}} = \sqrt{\frac{64,648}{4}} = \sqrt{16,162} \approx 127.12985

      • Standard Error (sMs_M):             sM=sN=127.129855=127.129852.23607≈56.85415s_M = \frac{s}{\sqrt{N}} = \frac{127.12985}{\sqrt{5}} = \frac{127.12985}{2.23607} \approx 56.85415

  • Step 4: Determine Critical Values

    • Degrees of Freedom (dfdf): 5−1=45 - 1 = 4.

    • For a two-tailed test at α=.05\alpha = .05 with df=4df = 4, the critical value from the $t$ table is:         tcritical=±2.776t_{critical} = \pm 2.776

  • Step 5: Calculate the Test Statistic

    • Formula:         t=Mdifference−μMsMt = \frac{M_{difference} - \mu_M}{s_M}         t=271−056.85415=4.77t = \frac{271 - 0}{56.85415} = 4.77

  • Step 6: Make a Decision

    • Since the obtained $t$ value (4.774.77) is more extreme than the critical $t$ value (±2.776\pm 2.776), the null hypothesis is rejected.

    • Statistical Reporting: t(4) = 4.77, p < .05.

Confidence Intervals and Effect Size

  • 95% Confidence Intervals for Paired Samples

    • Formula:         Lower bound=Mdifference−tcritical(sM)\text{Lower bound} = M_{difference} - t_{critical}(s_M)         Upper bound=Mdifference+tcritical(sM)\text{Upper bound} = M_{difference} + t_{critical}(s_M)

    • Calculation using the Caffeine Example:

      • Lower bound: 271−(2.776×56.85415)=271−157.83=113.17271 - (2.776 \times 56.85415) = 271 - 157.83 = 113.17

      • Upper bound: 271+(2.776×56.85415)=271+157.83=428.83271 + (2.776 \times 56.85415) = 271 + 157.83 = 428.83

    • APA Reporting style: 95% CI [113.17,428.83]95\%\,CI\,[113.17, 428.83].

  • Cohen’s d for Paired Samples t Test

    • Definition: Cohen's $d$ measures the magnitude of the effect in terms of difference scores.

    • Formula:         Cohen’s d=Mdifference−μs\text{Cohen's } d = \frac{M_{difference} - \mu}{s}

    • Calculation for the Caffeine Example:         Cohen’s d=271−0127.12985=2.13\text{Cohen's } d = \frac{271 - 0}{127.12985} = 2.13

    • Interpretation: An effect size of 2.132.13 is considered large. Reporting: "The observed effect size was large, d=2.13d = 2.13."

Key Principles Summary

  • Comparison Distribution Characteristics

    • The mean of the comparison distribution for a paired-samples $t$ test is 00. This is because the null hypothesis assumes there is no difference between the means of the compared groups.

    • The comparison distribution is specifically a distribution of mean differences.

  • Experimental Design

    • The paired-samples $t$ test is uniquely suited for within-groups or within-subjects designs where data points are linked across conditions.

Course Information and Midterm Details

  • Midterm 2 Specifications

    • Date: Tuesday.

    • Time: 12:00 noon12:00\,noon (punctuality required).

    • Duration: 40 minutes40\,minutes writing time.

    • Materials: A scientific calculator is required.

    • Format: Approximately half short answer and half multiple choice.

    • Topics: Includes Paired-Samples $t$ Test, and upcoming lecture on Independent-Samples $t$ Test.

  • Preparatory Tasks

    • Read textbook Chapter 11.

    • Complete Learning Curve Chapter 10.

    • SPSS Assignment 3 is pending.