Repeated Sampling
9.5 - 9.6

The Reality


We are never going to take thousands of samples and construct a sampling distribution. The concept of random sampling is entirely theoretical and lays the foundations for the derivations of the standard errors in Table 9.6.

Central Limit Theorem

  • A sampling distribution of a mean (\(\bar{x})\) will get closer and closer to the normal curve as the number of repetitions increases. This is true no matter the shape of the original data!

  • The mean of your sampling distribution is the same as the population mean. \(mean(\bar{x}) = \mu\).

  • The standard error of your sampling distribution is less than the spread of the population. \(SE(\bar{x}) = \frac{\sigma}{\sqrt{n} }\)

  • Statistical representation: \[\bar{x} \sim N\left(\mu_{\bar{x}} = \mu_x, SE(\bar{x})= \frac{\sigma_x}{\sqrt{n}}\right)\]

What makes a good sample?

  • representative of the population
  • picked at random (avoids bias)
  • big enough size to draw conclusions from (\(n \geq 30\))
  • Exception: If the population is already normal, the results hold for samples of any size \(n\)

Requirement for CLT to hold true

  • independent samples - includes less than 10% of the population (too big of a sample \(n\) will create dependence between samples)