T-Tests
Overview of T-Tests
Definition of T-Tests and Statistical Principles
The t-test is a statistical test used to determine if there is a significant difference between the means of two groups.
The t-test is also known as the "Student's t-test," devised by William Sealy Gosset in 1908.
The purpose of the t-test is to help infer whether differences between sample means reflect true differences between populations.
The Student's Test
Origin and Purpose
Developed by William Sealy Gosset, a chemist at Guinness Brewery.
Designed to monitor the quality of beer batches by sampling.
Published under the pseudonym "Student" due to the company's restrictions on revealing statistical methods.
T-Tests
Significance of T-Tests
The first substantive statistical test encountered in statistical analysis.
Used for purposes such as:
Determining the likelihood of a sample belonging to a specific population.
Evaluating whether two samples originate from the same population (significantly different means).
Testing the degrees of difference between group means.
T-Distribution
Concept and Properties
When repeatedly sampling from a population and calculating means, the result approaches a normal distribution of means.
Small sample sizes often do not yield a normal distribution, complicating the determination of statistical significance.
The t-distribution addresses these issues with:
A lower peak and fatter tails, indicating greater variability of small sample estimates.
It converges toward a normal distribution as sample size increases.
One Tail vs. Two Tail Tests
Defining Test Types
Tests can be one-tailed or two-tailed:
A two-tailed test is default; it does not specify the direction of the difference and considers both extremes of the distribution.
A one-tailed test focuses on one direction (either higher or lower), effectively narrowing the scope of evaluation.
Two-Tailed Tests Explained
Significance Allocation
Half of the alpha level (risk of Type I error) is allocated to testing each direction in a two-tailed test:
For example, a two-tailed test with 0.05 allocates 0.025 to each tail of the distribution.
Determines if a mean is significantly different from a value based on both directions in the distribution.
Types of T-Tests
One Sample T-test
Compares the sample mean to a known population mean.
Requires knowledge of the population parameter.
Example: Assessing if COMG 102 students differ from the overall UH Manoa population in numerical understanding.
Independent Samples T-test
Compares means between two separate groups.
Example: Investigating differences in IQ scores between genders or political opinions between educated and non-educated groups.
Paired Samples T-test
Analyzes same subjects at two different times or under two conditions.
Example: Evaluating the impact of a treatment (e.g., blood pressure medication) before and after administration.
T-Test Formulas
General Formula for Two Independent Samples
Note on T-Scores
T-scores serve a similar role as z-scores, representing the number of standard deviations a measurement is from the expected mean.
T-scores are then used to determine significance levels.
P-Value
Definition and Implications
Represents the probability of observing a result at least as extreme as the test statistic under the null hypothesis (H0).
Conventional threshold for significance: (5%).
The threshold, while widely accepted, is not inherently sacred or absolute and varies across fields.
P-Value Table and Degrees of Freedom
Use for Decision Making
To assess statistical significance, consult a p-value table for the t-distribution based on degrees of freedom (df).
For two-sample t-tests, degrees of freedom are calculated as , where is the total sample size.
Summary of Topics Covered
Covered the “Student’s” Test, T-tests, purpose of a t-test, T-distribution, one-tail vs. two-tail distinctions, types of t-tests, test formulas, and implications of p-values.