Midterm Exam 2

Week 6: Single-Sample t-Test

Purpose of the Test:

Comparing sampling distribution of the sample mean to the sampling distribution of the hypothetical mean.

Example: Weight or height

Components of the Formula:

The t-statistic is calculated using the formula t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} where:

  • x^- :Mean of Sample

  • (\mu) : Population mean

  • (s): Sample standard deviation

  • (n): Sample size

Degrees of Freedom for Single-Sample t-Test

  • Computed as df = n - 1 where (n) is the sample size.

Null and Research Hypotheses for Single-Sample t-Test

  • Null Hypothesis (H0): The sample mean equals the population mean (keep null)

  • Research Hypothesis (H1): The sample mean does not equal the population mean (rejected null)

Week 7: Independent-Samples t-Test

Purpose of the Test:

Used to compare the means of two independent groups means.

Example: Mean test scores in between 2 different classes, girls vs. boys

  • Components of the Formula: The t-statistic is calculated with the formula

    • x^-: Means of group 1 and group 2

    • (s_{p}) : Pooled standard deviation

    • (n1), (n2): Sample sizes of group 1 and group 2

Degrees of Freedom for Independent-Samples t-Test

  • Degrees of Freedom (df): Calculated as df = n1 + n2 - 2.

Running and Interpreting in Jamovi

  • Similar steps as the single-sample t-test.

Pros and Cons of Independent-Samples t-Test

  • Pros: Simple to use, can be applied to any independent groups.

  • Cons: Assumes equal variance between groups, potential for type I error if assumptions are violated.

Assumptions of Independent-Samples t-Test

  1. Normal distribution of the outcome variable in both groups.

  2. Homogeneity of variances.

  3. Independent observations.

Identifying Variables

  • Independent Variable (IV): Known as the "grouping variable" in Jamovi.

  • Dependent Variable (DV): The outcome measurement being compared.

Between-Subjects Design

  • The independent-samples t-test is categorized as a between-subjects design because different subjects are measured under different conditions.

Week 7: Paired-Samples t-Test

  • Purpose of the Test: Used to compare means from the same group at different times or under different conditions.

  • Components of the Formula: The t-statistic is calculated using the formula t = \frac{\bar{D}}{s_{D}/\sqrt{n}} where:

    • (\bar{D}): Mean difference across pairs

    • (s_{D}): Standard deviation of the differences

    • (n): Number of pairs

Degrees of Freedom for Paired-Samples t-Test

  • Degrees of Freedom (df): Computed as df = n - 1 (where (n) is the number of paired differences).

Running and Interpreting in Jamovi

  • Only one box is used for variable input in this test, differing from the independent-samples t-test.

Pros and Cons of Paired-Samples t-Test

  • Pros: Reduces variability and increases power by controlling for individual differences.

  • Cons: Requires pairs, limiting the sample size.

Assumptions of Paired-Samples t-Test

  1. The differences are normally distributed.

Within-Subjects Design

  • The paired-samples t-test is classified as a within-subjects design as it measures the same subjects across conditions.

Null and Research Hypotheses for Paired-Samples t-Test

  • Null Hypothesis (H0): The mean of the differences equals zero (H0: (\bar{D} = 0)).

  • Research Hypothesis (H1): The mean of the differences does not equal zero (H1: (\bar{D} \neq 0)).

Week 8: Steps for Hypothesis Testing

  • Overview of Steps:

    1. Define Hypotheses: Clearly articulate both null and alternative hypotheses.

    2. Set Decision Rules: Determine the alpha level (usually 0.05).

    3. Calculate Test Statistic: Use the appropriate formula for t-tests or ANOVA.

    4. Evaluate Significance: Compare p-value to alpha level.

    5. Interpret Results: Relate findings back to research hypotheses.

Possible Outcomes of Research Studies

  • Type I Error: Rejecting a true null hypothesis (false positive).

  • Type II Error: Failing to reject a false null hypothesis (false negative).

Statistical Power

  • Definition: The probability of correctly rejecting a false null hypothesis.

  • Ways to Increase Power:

    1. Increase sample size.

    2. Reduce variability within groups.

    3. Use a more precise measurement.

    4. Increase the effect size.

Effect Size (Cohen’s d)

  • Formula: d = \frac{\bar{x}1 - \bar{x}2}{s} where:

    • (\bar{x}1), (\bar{x}2): Means of two groups

    • (s): Pooled standard deviation.

  • Interpretation of Effect Sizes:

    • Small: 0.2

    • Medium: 0.5

    • Large: 0.8

Confidence Interval (CI)

  • Definition: A range of values that likely contain the population parameter.

  • Factors in CI Calculation:

    1. Sample mean.

    2. Critical value from t-distribution.

    3. Standard error.

  • Key Points:

    • Sample CI should contain the parameter 95% of the time.

    • If CI for the mean difference includes 0, the difference is NOT significant.

    • If CI does not include 0, the difference IS significant.

Thought Questions

  • Significance of CI containing zero: If the confidence interval for mean difference contains zero, it implies that a true difference in population means might not exist, hence we fail to reject H0.

Introduction to ANOVA

  • Reason for ANOVA: ANOVA is used to test for differences across multiple groups simultaneously, reducing the risk of Type I error associated with multiple t-tests.

  • One-Way ANOVA: Compares means across three or more groups based on a single independent variable.

  • Between vs. Within Groups Variance:

    • Between Groups Variance: Variation due to differences between group means — larger is preferred for significant findings.

    • Within Groups Variance: Variability within each group — smaller is preferred as it represents error variance.

Degrees of Freedom for ANOVA

  • Two components of degrees of freedom:

    1. Between-groups df: k - 1 (where (k) is the number of groups).

    2. Within-groups df: N - k (where (N) is the total number of observations).

Pros and Cons of ANOVA

  • Pros: Controls Type I error rate, evaluates all groups simultaneously.

  • Cons: Assumes homogeneity of variances and normally distributed populations.

Running and Interpreting ANOVA in Jamovi

  • Familiarity with running ANOVA similar to t-tests.

Important ANOVA Concepts

  • Grand Mean: The mean of the entire sample.

  • Group Means: The means specific to each group.

  • ANOVA Table: Summarizes the results of the ANOVA procedure.

  • Assumptions of ANOVA: Normality, independence, and homogeneity of variances.

  • Effect Size for ANOVA: Eta squared (η²) which quantifies effect size similar to Cohen's d, defined as:

    • Small: 0.01

    • Medium: 0.06

    • Large: 0.14

Post Hoc Analyses

  • Definition: Conducted after ANOVA if results are significant to determine which group means differ.

  • Popular Tests: Tukey and Scheffe tests, each having its pros and cons.

Running Post Hoc Analyses in Jamovi

  • Steps and APA reporting similar to ANOVA.

Week 10: Review and Preparation

  • No New Material: Focus on revisiting all content in preparation for the exam.

Overarching Concepts

  • Signal vs. Noise: Key to understanding the validity of statistical findings. Signal refers to the actual effect whereas noise refers to random variability.

Steps for Hypothesis Testing

  • Go through hypothesis testing steps logically for each type of test covered (t-tests and ANOVA) with contextual thought questions for clarity:

    1. Develop a specific research hypothesis.

    2. Distinguish between research and statistical hypotheses.

    3. Understand the differences between null and alternative hypotheses.

    4. Clearly outline decision rules before analysis.

    5. Calculate the test statistic properly.

    6. Significance evaluation against predefined thresholds.

    7. Result interpretation in light of research hypothesis.

APA-Style Sentence Writing

  • Be prepared to articulate findings from research scenarios in a standardized manner adherent to APA formatting rules.

Statistical Tests Summary

  • Types of Tests and Their Characteristics:

    • One Sample t-test: Paired, related group, focuses on one mean.

    • Independent Samples t-test: Between two groups, assessing mean differences.

    • Paired Samples t-test: Within subjects, comparing two conditions of the same group.

    • ANOVA: Involves multiple group means comparison to uncover significant differences.