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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

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.

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.
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.
Key Term Reference
 bias
 Appears in this related concepts: Determine a Course, Generate Alternatives, and Culture Bias
 confounding
 Appears in this related concepts: Applications of Statistics, Random Assignment of Subjects, and tTest for Two Samples: Paired
 control
 Appears in this related concepts: Experiments, Using a Bank for Control, and Internal and External
 control group
 Appears in this related concepts: The Salk Vaccine Field Trial, Statistical Controls, and The Scientific Method
 critical value
 Appears in this related concepts: Two Regression Lines, Estimating the Target Parameter: Interval Estimation, and 95% Critical Values of the Sample Correlation Coefficient Table
 datum
 Appears in this related concepts: Change of Scale, Comparing Nested Models, and Controlling for a Variable
 degrees of freedom
 Appears in this related concepts: Specific Heat and Heat Capacity, Structure of the ChiSquared Test, and Inelastic Collisions in One Dimension
 error
 Appears in this related concepts: Estimating the Accuracy of an Average, Estimation, and Precise Definition of a Limit
 factor
 Appears in this related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 independent
 Appears in this related concepts: Probability Histograms, Conditional Probability, and Party Identification
 independent sample
 Appears in this related concepts: Comparing Two Independent Population Means, Comparing Two Independent Population Proportions, and Assumptions
 mean
 Appears in this 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 this related concepts: The Clofibrate Trial, What are Observational Studies?, and Observational studies and sampling strategies exercises
 population
 Appears in this related concepts: Fundamentals of Statistics, Organismal Ecology and Population Ecology, and What constitutes a biome?
 probability
 Appears in this related concepts: Particle in a Box, The Addition Rule, and Theoretical Probability
 sample
 Appears in this related concepts: What Is a Confidence Interval?, Sampling, and Identify Product Benefits
 sample mean
 Appears in this related concepts: Which Standard Deviation (SE)?, Standard Error, and Degrees of Freedom
 sampling
 Appears in this related concepts: Wilcoxon tTest, Collecting and Measuring Data, and Continuous Sampling Distributions
 standard error
 Appears in this related concepts: Estimating a Population Proportion, Calculations for the tTest: One Sample, and Chance Error and Bias
 ttest
 Appears in this related concepts: The tTest, tTest for One Sample, and One, Two, or More Groups?
 variance
 Appears in this related concepts: Testing a Single Variance, Variance Estimates, and Variance
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Cite This Source
Source: Boundless. “tTest for Two Samples: Independent and Overlapping.” Boundless Statistics. Boundless, 02 Jul. 2014. Retrieved 24 Apr. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/otherhypothesistests13/thettest60/ttestfortwosamplesindependentandoverlapping2922749/