Independent Samples T-Test

Introduction to Writing Up Test Results

• Clarification of expectations for writing up test results.
• Importance of following specified formats during lab reports and tests.

Writing Format for Reporting T-Test Results

• t value: Always represented as a lowercase 't'. Autocorrect issues can be ignored for reporting.

• Degrees of freedom (df):

  • Calculation: For independent samples t-test, use the formula:
    $df = n - 2$
    where $n$ is the sample size (total number of observations).

• p value estimate:

  • Typical format: "$p$ greater than or less than 0.05"
  • Example: If p < 0.05, provide the conclusion about the null hypothesis.

• Conclusion: State whether you reject or fail to reject the null hypothesis based on the p-value.

Example of Reporting Process

• Noting that if p < 0.05, one should reject the null hypothesis (p is low).

Use of Alpha Levels in Testing

• When calculating obtained values:
  • Always use the default alpha level of 0.05 for two-tailed tests.
  • Different alpha levels can be used for different critical comparisons:
  • Example of critical values based on alpha levels for sample size = 20:
    • For $ext{alpha} = 0.10$: Critical Value = 1.734
    • For $ext{alpha} = 0.05$: Critical Value = 2.101
    • For $ext{alpha} = 0.01$: Critical Value = 2.879

Hypothetical Analysis Example

• Suppose the obtained t value is 1.67 with a sample size of 20:
  • Calculate degrees of freedom: $df = 20 - 2 = 18$
  • Compare to critical values for each alpha level to determine whether to reject or fail to reject the null hypothesis.
  • At $ext{alpha} = 0.10$, 1.67 < 1.734 — Fail to reject.
  • At $ext{alpha} = 0.05$, 1.67 < 2.101 — Fail to reject.
  • At $ext{alpha} = 0.01$, 1.67 < 2.879 — Fail to reject.

Degrees of Freedom

• Concept of degrees of freedom: Refers to the number of independent values in a statistical calculation that can vary.

• For an independent samples t-test, calculate as:

  • $df = n - k$
    where $n$ is the total sample size and $k$ is the number of groups.

• Example Breakdown:

  • For a sample of 3 scores (n=3), if mean=5:
  • First two scores can vary (e.g., scores 2 and 9), but the third score is dependent to maintain the mean of 5 (total must equal 15).

Visual Representation of T-Distributions

• The shape of the t-distribution:
  • Bell-shaped, centered at zero; critical values change based on degrees of freedom.
  • As degrees of freedom increase, the distribution approaches a normal distribution.

Effect Size

• Effect size measures the degree of difference between groups and is not dependent on sample size, crucial for understanding the significance of test results.

• Cohen's d: Commonly used effect size measure in t-tests, calculated as:

  • $d = \frac{M<em>1 - M</em>2}{SD}$
    where $M<em>1$ and $M</em>2$ are the means of the two groups and $SD$ is the pooled standard deviation.

• Interpretation guidelines for effect sizes:

  • Small effect: $d ext{ approximately } 0.2$
  • Medium effect: 0.2 < d < 0.5
  • Large effect: d ext{ > } 0.5

Significance of Results

• Significant test results do not imply practical significance; hence, effect size offers more insight into the meaningfulness of results.

Understanding Power in Statistical Tests

• Definition of Power: The probability that a test correctly rejects a false null hypothesis.

  • High power (e.g., 80%) indicates a good likelihood of detecting an effect when one exists.

• Influencing factors of power:

  • Effect size: Larger effect sizes lead to higher power.
  • Sample size: Increasing sample size improves power (more data reduces variability).
  • Alpha level: Higher alpha levels (like changing from 0.05 to 0.10) increase power.
  • Type of test: One-tailed tests have more power than two-tailed tests because they focus on one potential direction of effect.

• Use of power in experimental design involves setting an estimate, typical goal being 80% power and utilizing it to determine necessary sample size for achieving reliable results.

• Example calculation considerations:

  • Example scenario: Determining sample size for a clinical trial based on expected dropout rates and previous effect sizes.

Conclusion

• Understanding these statistical concepts not only prepares students for the exam but also improves ability to critically assess research findings in psychology and related fields.