Student's t test

Biostatistics BI45, Saint Anselm College

 

Background Information

Developed by Gossett to handle small sample sizes

Why

Z scores and Normal Distribution can be used to test differences between means but there is a problem

Calculation of Z scores requires knowledge of population parameters

population mean, population variance, population standard deviation to calculate the population standard error of the mean

Z score formula for review

Small samples do not provide reliable enough estimates of population parameters

Gossett developed the t distribution which he published under an assumed name - Student

Characteristics of the Student tdistribution

1) leptokurtic

2) as n and v (v=n-1) increase the t distribution begins to approach a normal distribution

Types of t tests

One-sample Student's t test

used to compare a population mean inferred from a sample with a hypothetical population mean (derived from the literature or otherwise set by the investigator)

Two Independent (Unpaired) Samples Student's t test

used to compare two independent population means inferred from two samples (independent indicates that the values from both samples are numerically independent of each - there is no correlation between corresponding values)

Two Dependent (Paired) Samples Student's t test

used to compare two dependent population means inferred from two samples (dependent indicates that the values from both samples are numerically dependent upon each other- there is a correlation between corresponding values)

Common Examples

same subject is exposed to 2 different experimental treatments

matched pairs - subjects in 2 different experimental treatments are different but have been matched based on their characteristics (gender, age, height, weight, behavior, etc...)

Means Testing Tree - Flow Chart of Statistical Tests for Means

Two-tailed and One-tailed versions of these tests

Two-tailed test - evaluates whether a difference exists between 2 samples, not the direction of the difference

One-tailed test - evaluates whether a difference exists between 2 samples, and specifically evaluates the direction of the difference (whether one sample is larger or smaller than the other)

Application and Usage of different Student's t tests

One Sample t test

One Sample t test - overview and mechanics

Example of a two-tailed One Sample t test

Example of a one-tailed One Sample t test

Two Independent (Unpaired) Samples t test

Two Independent (Unpaired) Samples t Test - overview and mechanics

Example of a two-tailed Two Independent Samples t test

Example of a one-tailed Two Independent Samples t test

F test for equality of variances and Welch's Approximate t

Example of F test of equality of variances

Two Dependent (Paired) Samples t test

Two Dependent (Paired) Samples t Test - overview and mechanics

Example of a two-tailed Two Dependent Samples t test

95% Confidence Intervals for all t Tests and Examples

Mechanics and Examples of 95% Confidence Intervals

 

 

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Copyright © 2001 Jay Pitocchelli. All rights reserved. The contents of this page are the intellectual property of Dr. Jay Pitocchelli for distribution to students enrolled in Biostatistics BI 45 at Saint Anselm College. These pages may not be copied, photocopied, reproduced, translated, or published in any electronic or machine-readable form in whole or in part without prior written approval of Jay Pitocchelli. Students enrolled in Biostatistics BI 45 at Saint Anselm College have permission to print this material for their lecture notes.