tTest for Two Samples: Independent and Overlapping
Twosample ttests for a difference in mean involve independent samples, paired samples, and overlapping samples.
Learning Objective

Contrast paired and unpaired samples in a twosample ttest
Key Points
 For the null hypothesis, the observed tstatistic is equal to the difference between the two sample means divided by the standard error of the difference between the sample means.
 The independent samples ttest is used when two separate sets of independent and identically distributed samples are obtainedâ€”one from each of the two populations being compared.
 An overlapping samples ttest is used when there are paired samples with data missing in one or the other samples.
Terms

blocking
A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental results due to a known change in raw materials, operators, machines, etc., become concentrated in the levels of the blocking variable.

null hypothesis
A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical evidence in the form of a hypothesis test indicates otherwise.
Full Text
The two sample ttest is used to compare the means of two independent samples. For the null hypothesis, the observed tstatistic is equal to the difference between the two sample means divided by the standard error of the difference between the sample means. If the two population variances can be assumed equal, the standard error of the difference is estimated from the weighted variance about the means. If the variances cannot be assumed equal, then the standard error of the difference between means is taken as the square root of the sum of the individual variances divided by their sample size. In the latter case the estimated tstatistic must either be tested with modified degrees of freedom, or it can be tested against different critical values. A weighted ttest must be used if the unit of analysis comprises percentages or means based on different sample sizes.
The twosample ttest is probably the most widely used (and misused) statistical test. Comparing means based on convenience sampling or nonrandom allocation is meaningless. If, for any reason, one is forced to use haphazard rather than probability sampling, then every effort must be made to minimize selection bias.
Unpaired and Overlapping TwoSample TTests
Twosample ttests for a difference in mean involve independent samples, paired samples and overlapping samples. Paired ttests are a form of blocking, and have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared. In a different context, paired ttests can be used to reduce the effects of confounding factors in an observational study.
Independent Samples
The independent samples ttest is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomize 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the ttest .
Medical Treatment Research
Medical experimentation may utilize any two independent samples ttest.
Overlapping Samples
An overlapping samples ttest is used when there are paired samples with data missing in one or the other samples (e.g., due to selection of "I don't know" options in questionnaires, or because respondents are randomly assigned to a subset question). These tests are widely used in commercial survey research (e.g., by polling companies) and are available in many standard crosstab software packages.
Key Term Reference
 bias
 Appears in these related concepts: Interpreting Distributions Constructed by Others, Culture Bias, and Chance Error and Bias
 confounding
 Appears in these related concepts: Random Assignment of Subjects, tTest for Two Samples: Paired, and Line fitting, residuals, and correlation exercises
 control
 Appears in these related concepts: Experiments, Using a Bank for Control, and Internal and External
 control group
 Appears in these related concepts: The Salk Vaccine Field Trial, Statistical Controls, and The Scientific Method
 critical value
 Appears in these related concepts: Two Regression Lines, Estimating the Target Parameter: Interval Estimation, and Calculations of Probabilities
 datum
 Appears in these related concepts: Change of Scale, Controlling for a Variable, and Summary for inference of the difference of two means
 degrees of freedom
 Appears in these related concepts: tTest for One Sample, Structure of the ChiSquared Test, and Specific Heat and Heat Capacity
 error
 Appears in these related concepts: Estimation, Precise Definition of a Limit, and Basic properties of point estimates
 factor
 Appears in these related concepts: Rational Algebraic Expressions, Factors, and Finding Factors of Polynomials
 independent
 Appears in these related concepts: Fundamentals of Probability, Unions and Intersections, and Party Identification
 independent sample
 Appears in these related concepts: Comparing Two Independent Population Proportions, Comparing Two Independent Population Proportions, and Wilcoxon tTest
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, Averages, and Understanding Statistics
 observational study
 Appears in these related concepts: What are Observational Studies?, Case study: gender discrimination exercises, and One sample t tests
 population
 Appears in these related concepts: Applications of Statistics, The Functionalist Perspective on Deviance, and Quorum Sensing
 probability
 Appears in these related concepts: Theoretical Probability, Rules of Probability for Mendelian Inheritance, and The Addition Rule
 sample
 Appears in these related concepts: Identifying Product Benefits, Surveys, and Basic Inferential Statistics
 sample mean
 Appears in these related concepts: Lab 1: Confidence Interval (Home Costs), Type I and II Errors, and Introduction to confidence intervals
 sampling
 Appears in these related concepts: Collecting and Measuring Data, Continuous Sampling Distributions, and Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student'st
 standard error
 Appears in these related concepts: Which Standard Deviation (SE)?, Estimating the Accuracy of an Average, and Calculations for the tTest: One Sample
 ttest
 Appears in these related concepts: The tTest, One, Two, or More Groups?, and Assumptions
 variance
 Appears in these related concepts: Testing a Single Variance, Variance, and Variance Estimates
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources: