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  • A probability sample is a sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample
  • The sampling distribution of the sample mean is a probability distribution consisting of all possible sample means of a given sample size selected from a population
  • Need for Sampling:
    • The physical impossibility of checking all items in the population
    • The cost of studying all the items in a population
    • The sample results are usually adequate
    • Contacting the whole population would often be time-consuming

The destructive nature of certain tests


Methods of probability sampling

  • Methods of Probability Sampling:
    • Simple Random Sampling: A sample formulated so that each item or person in the population has the same chance of being included
    • Systematic Random Sampling: The items or individuals of the population are arranged in some order.  A random starting point is selected and then every kth member of the population is selected for the sample
    • Stratified Random Sampling: A population is first divided into subgroups called strata, and a sample is selected from each stratum
  • Independently and identically distributed (iid) random variables: If a sample contains random variables such that each of them have the same probability distribution function then such random variables are called as i.i.d
  • The sampling error is the difference between a sample statistic and its corresponding
  • population parameter

Developing sampling distributions

  • Suppose there’s a population of 4 oldest scientists in a university: Jack, Andrew, Michelle and Tom
  • Random variable, X is the ages of the individuals
    • Values of X: 78, 76, 72, 74



  • Summary Measure for Population Distribution



All Possible samples of size n = 2

Summary measures for the sampling distribution

  • The mean of the sample



  • The standard deviation of the sample:


  • Two important points worth noting in population and sampling distributions:
    • Population mean and the sample mean is same which is equal to 75
    • Variance of the population = 2.2362=5 and Variance of the sample = 1.582=2.5 which is lower than the population variance
  • Also the: 

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