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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 obtained—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 .
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.
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Key Term Reference
 bias
 Appears in these related concepts: Defining the Sample and Collecting Data, Generate Alternatives, and Gender Bias
 confounding
 Appears in these related concepts: Confounding, Sex Bias in Graduate Admissions, and tTest for Two Samples: Paired
 control
 Appears in these related concepts: Managing to Prevent Fraud, Controlling for a Variable, and Using a Bank for Control
 control group
 Appears in these related concepts: Random Assignment of Subjects, The Salk Vaccine Field Trial, and The Scientific Method
 critical value
 Appears in these related concepts: Two Regression Lines, Estimating the Target Parameter: Interval Estimation, and Estimating a Population Variance
 datum
 Appears in these related concepts: Graphs of Qualitative Data, Misleading Graphs, and Change of Scale
 degrees of freedom
 Appears in these related concepts: Specific Heat and Heat Capacity, Structure of the ChiSquared Test, and Inelastic Collisions in One Dimension
 error
 Appears in these related concepts: Outliers, Estimation, and Precise Definition of a Limit
 factor
 Appears in these related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 independent
 Appears in these related concepts: Fundamentals of Probability, Conditional Probability, and Party Identification
 independent sample
 Appears in these related concepts: Comparing Two Independent Population Means, Comparing Two Independent Population Proportions, and Assumptions
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 observational study
 Appears in these related concepts: The Clofibrate Trial, What are Observational Studies?, and Making Inferences About the Slope
 population
 Appears in these related concepts: Random Sampling, Social Definition of Race, and What constitutes a biome?
 probability
 Appears in these related concepts: The Addition Rule, Theoretical Probability, and Rules of Probability for Mendelian Inheritance
 sample
 Appears in these related concepts: What Is a Confidence Interval?, Sampling, and Fundamentals of Statistics
 sample mean
 Appears in these related concepts: Mean: The Average, Which Average: Mean, Mode, or Median?, and Shapes of Sampling Distributions
 sampling
 Appears in these related concepts: Inferential Statistics, Applications of Statistics, and Collecting and Measuring Data
 standard error
 Appears in these related concepts: Which Standard Deviation (SE)?, Estimating the Accuracy of an Average, and Estimating a Population Proportion
 ttest
 Appears in these related concepts: The tTest, tTest for One Sample, and One, Two, or More Groups?
 variance
 Appears in these related concepts: Testing a Single Variance, Variance Estimates, and Variance
Sources
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Cite This Source
Source: Boundless. “tTest for Two Samples: Independent and Overlapping.” Boundless Statistics. Boundless, 01 Jul. 2015. Retrieved 03 Jul. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/otherhypothesistests13/thettest60/ttestfortwosamplesindependentandoverlapping2922749/