Random Sampling

• This chapter connects (bridges) the previous knowledge and the understanding of statistical inference.
• Outcome of a statistical experiment:
  • Numerical value: total value of a pair of dice tossed.
  • Descriptive representation: blood types in blood test.
• We focus on
  • sampling from distributions or populations
  • study such important quantities as the sample mean and sample variance.
• We extend the concept of probability distribution to that of a sample statistic.
• For instance, the distribution of a sample mean $\bar{X}$, which is a random variable because the different samples may result in different values of sample mean $\bar{x}$.
• The use of high speed computer enhances the use of formal statistical inference with graphical techniques.

• Definition 8.1:
  
• The number of observations in the population is defined to be the size of the population.
  • Finite size: 600 students are classified according to blood type: a population of size 600.
  • Infinite size: measuring the atmospheric pressure; some finite populations are so large.

• Each observation in a population is a value of a random variable $X$ having some probability distribution $f(x)$.
• If one is inspecting items coming off an assembly line for defects, then each observation in population might be a value 0 or 1 of the binomial random variable $X$ with probability distribution
  $$b(x;1,p) = p^xq^{1-x},  x = 0,1$$
  where 0 indicates a non-defective item and 1 indicates a defective item.
• Definition 8.2:
  
• Sometimes, it is impossible or impractical to observe the entire set of observations that make up the population.

• Obtain representative samples to have a valid inference.
• Biased sampling procedure produces inference that consistently overestimate/underestimate some characteristics of the population.
• Random sample: selected independently and at random.
• Definition 8.3:
  
• If we assume the population of battery lives to be normal, the possible values of any $X_i,~i=1,2,\ldots,8$, will be precisely the same as those in the original population, and hence $X_i$ has the same identical normal distribution as $X$.