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Part of a research study of identical twins who had been separated at birth involves a random sample of 9 pairs, in which one of the twin has been raised by the natural parent and the other by adoptive parents. The IQ scores of these twins were measured.

It maybe assumed that the difference in IQ has a normal distribution. The mean IQ scores of separated twins raised by natural parents and adoptive parent are denoted by $u_n$ and $u_a$

Obtain the confidence interval for $u_n-u_a$

The answer for this question treats the two samples as paired samples. How is this a paired sample instead of a separate two samples? My understanding for a paired sample is that if a SAME sample undergoes a transformation and provide different results then the different data are considered as a paired sample.

But in this case it's obvious that the twins are two different human beings and hence the result obtained using them constitutes to be two different samples.

Can somebody please explain the fallacy in my proposed statements?

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Paired samples can be paired for any reason you want. We often use them to check whether that reason influences the outcome. For example, maybe I have a factory producing boards that are nominally eight feet long. We can study the distribution in lengths of my output by taking a sample an measuring. Now maybe someone has a theory that some of the variation is explained by a setup at the start of each day. We could test that theory by taking paired samples of two boards cut on the same day and asking whether the variation between the elements of the pairs is less than the variation over the whole sample taken over a long time. In your example we are trying to separate the genetic and environmental influences on IQ. The twins of a pair have the same genes, so we assume that any difference in IQ is caused by environment. If genetics has a (large enough) impact, we would expect that the standard deviation of the difference in IQs between the twins is less than the standard deviation of the difference of two unrelated people.

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  • $\begingroup$ Can you please also provide some distinction between two sample t-tests and paired sample t-tests and how to know which to use ? $\endgroup$ – mathnoob123 Jun 1 '17 at 20:20
  • $\begingroup$ No, because I haven't studied t-tests for years. $\endgroup$ – Ross Millikan Jun 1 '17 at 20:21
  • $\begingroup$ Not necessary t-tests, just how to know when two samples are separate and when they are paired? $\endgroup$ – mathnoob123 Jun 1 '17 at 20:22
  • $\begingroup$ I thought I answered that. They are paired when you select them in a way that they share some characteristic. You are usually testing for the effect of that characteristic. $\endgroup$ – Ross Millikan Jun 1 '17 at 20:28
  • $\begingroup$ So in statistical terms, for paired samples we know the difference in populations means should be 0 and test that. Whereas for separate sample we are aware that population means could differ and test according to the required task. Right? $\endgroup$ – mathnoob123 Jun 1 '17 at 20:30

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