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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 .
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.
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Key Term Reference
 bias
 Appears in these related concepts: Interpreting Distributions Constructed by Others, Defining the Sample and Collecting Data, and Social Psychology
 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: Managing to Prevent Fraud, Fulfilling the Controlling Function, and Controlling for a Variable
 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, Summary for inference of the difference of two means, and Lab 1: Confidence Interval (Home Costs)
 degrees of freedom
 Appears in these related concepts: Specific Heat and Heat Capacity, tTest for One Sample, and Structure of the ChiSquared Test
 error
 Appears in these related concepts: Estimation, Precise Definition of a Limit, and Basic properties of point estimates
 factor
 Appears in these related concepts: Randomized Design: SingleFactor, Finding Factors of Polynomials, and Solving Quadratic Equations by Factoring
 independent
 Appears in these related concepts: Probability Histograms, The Rise of Independents, and Unions and Intersections
 independent sample
 Appears in these related concepts: Comparing Two Independent Population Proportions, Wilcoxon tTest, and Comparing Two Independent Population Proportions
 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: What are Observational Studies?, Case study: gender discrimination exercises, and One sample t tests
 population
 Appears in these related concepts: Random Samples, Random Sampling, and Organismal Ecology and Population Ecology
 probability
 Appears in these related concepts: The Addition Rule, Theoretical Probability, and Rules of Probability for Mendelian Inheritance
 sample
 Appears in these related concepts: Identifying Product Benefits, Surveys, and Basic Inferential Statistics
 sample mean
 Appears in these related concepts: Which Standard Deviation (SE)?, Introduction to confidence intervals, and Type I and II Errors
 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: Estimating a Population Proportion, Calculations for the tTest: One Sample, and Chance Error and Bias
 ttest
 Appears in these related concepts: The tTest, Assumptions, 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, 21 Jul. 2015. Retrieved 30 Apr. 2016 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/otherhypothesistests13/thettest60/ttestfortwosamplesindependentandoverlapping2922749/