Lecture Notes on t Tests

Chapter 7: t Tests

Outline

  • Estimating Population Variance from a Sample
    • How to do it
    • Degrees of Freedom
    • The t Distribution
    • t Scores
    • Miscellaneous Related Points
    • Effect Size with Estimated Variance
  • t Tests in Psychological Research
    • Types of t Tests
    • Single Sample Tests
    • Paired-Samples Tests
    • Assumptions

Hypothesis Testing without Population Parameters

In psychology, relevant population parameters are often unknown.

  • For example, when measuring scores on a memory or attention test, previous studies might not exist for comparison.
  • t Test: A hypothesis testing method where the population variance is not known. It compares sample t scores to the t distribution.
    • Single Sample t Test: A procedure applied when the sample population mean is known.

Estimating Population Variance from Sample Scores

  • Populations with high variability tend to yield highly variable samples, while populations with low variability yield samples with low variability.
  • Sample variance is generally less than true population variance, leading to biased estimates.
    • Biased Estimate: An estimate that consistently over or underestimates a parameter.
Finding an Unbiased Estimate
  • An unbiased estimate means it is equally likely to overestimate or underestimate a parameter.
  • The formula for estimating population variance:
    S² = Σ(X-M)² / (N-1)
  • Population standard deviation is estimated as: S = √S².

Degrees of Freedom

  • Degrees of Freedom (df): the number of scores that can vary freely (in this case, N-1).
  • If the mean is known, and we have N-1 scores, knowing all the other scores keeps the last score fixed.
    • Example: For three scores averaging to 10, if the scores are [5, 15, x], knowing the mean helps us compute x.

The t Distribution

  • The estimation of population variance assumes a different distribution than the true variance, often resulting in fatter tails compared to normal distributions.
  • The shape of the t distribution is affected by degrees of freedom; more degrees of freedom yield a shape closer to the normal distribution.
    • At around 30 df, the t distribution resembles the normal distribution closely.

Understanding t Scores

  • A t Score indicates how many standard deviations a sample mean is from the population mean on the t distribution, similar to a Z score but adjusted for estimation.
  • Critical Values are determined using a t table based on degrees of freedom and the significance level (alpha).
  • t Scores are calculated using the formula:
    t = (M - μM) / SM, where M is sample mean, μM is population mean, and SM is the unbiased estimator of standard deviation.

Conducting a Single Sample t Test

  • Similar to hypothesis tests in Chapter 5, but with adjustments:
    • Standard deviation is estimated using N-1.
    • Compute Standard Error of Mean (SEM) using sample standard deviation.
    • Retrieve critical value from t distribution chart and convert raw score to t score.
  • Example context: Comparing ratings for new song quality against a known average rating.

Paired-Samples t Test

  • Referred to as Dependent Means t Test when repeated measures for each participant are involved.
  • Tests for differences between two related conditions using difference scores (M₁ - M₂) with comparison distribution mean set to zero.
  • Each score pair must be linked meaningfully, such as in repeated measures or matched subjects.

Assumptions of Normality

  • For hypothesis testing, scores must be normally distributed; violating this can lead to incorrect conclusions.
  • However, t tests are robust to moderate violations of normal distribution premises, as distributions of means normalize with larger samples.
    • Robustness reflects the accuracy of results despite assumption violations.

Effect Size with Estimated Variance

  • Calculated similar to previous chapters, but using the unbiased estimator of standard deviation.
  • Cohen's d Formula:
    d = (μ - μ₂) / S (where μ₂ equals 0 in paired-samples cases).

Example of Cohen's d with Estimated Variance

  • Based on a previous example of athletes' performance, the difference in weight lifted qualifies d calculations.

t Tests in Psychological Research

  • Consideration of power in repeated measures to focus on difference score variance rather than overall levels.
  • Example reporting in research: t(28) = 7.36, Bayes factor = 450,000, suggesting strong evidence against the null hypothesis.