How do I ensure that the sample that I obtained is a random sample? Give an example, thx.

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If you are asking how to select a random sample:

Suppose you want a random sample of people. First you select the population you want to study. Then you assign each person in the population a number.

Use a table (book) of random numbers.(It is important to know that the table of random numbers is truly random. A random number generator on a computer does not create random numbers, but pseudo-random numbers. If you truly need random numbers they should be generated by a nondeterministic process like atomic decay or detection of cosmic rays, etc...) Open the book randomly and choose a number to start at randomly. (This is harder to do than it seems -- people will tend to choose pages in the middle and select a number in the middle of the page, so you will need a way to choose these randomly. Perhaps some nondeterministic event like rolling a set of colored die, etc...)

If you want to choose 25 people from a population, starting at your chosen number read off numbers two at a time. (02=2,etc...) If you get a repeat number you skip to the next 2 digit number until you have the 25 test subjects.

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If you already have a sample and want to know if it was a random sample it is a much harder question.

You can test to see that each digit is as likely to appear as any other digit. (For a large sample, this is necessary but not sufficient. It is easy to come up with a list like this that isn't random e.g. 12345678910111213141516...will fit the bill for a large sample.)

Probably in real life you would ask to see how the sample was created. If there was no nondeterministic mechanism, the list is not random.

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