Study Notes on Sampling Distributions and Central Limit Theorem

Sampling Distribution

  • Definition: Probability distribution for a statistic calculated from repeated samples of size n.

Sample Mean as a Random Variable

  • Sample mean () functions as a random variable.

  • For a random sample from population with mean (µ) and standard deviation (σ):

    • Mean of sampling distribution:  = µ

    • Standard deviation (SEM):  = σ/√n

Independence Assumption

  • Assume independence between observations for sampling characteristics to hold.

  • Random samples; if sampling without replacement: n < 0.05(N).

  • Question: Wouldn’t selecting only < 5% cause bias or scew the data? If you look at a chart of data, and you only focus in on a certain section with results that do not accurately represent your population, wouldn’t that lead you to a misleading conclusion about your data set?

  • Question: Can you show me how you pulled up the “1000 times” things on SC?

Shape of Sampling Distribution

  • Shape, center, and spread must be discussed:

    • Center = 

    • Spread = σ/√n

  • Shape influenced by parent distribution.

Normal Distribution Sampling

  • If X is normally distributed, then  is also normally distributed with:

    • Mean = µ

    • Standard deviation = σ/√n

Application of Sampling Distribution

  • Use probabilities for means:

    • P( > k) = P(Z > (k - µ)/(σ/√n)).

Central Limit Theorem (CLT)

  • CLT states sampling distribution of  approaches normality as sample size increases (n > 30).

Non-Normal Distributions

  • Even with non-normal population, mean and standard deviation of sampling distribution () held true regardless of n.

Practical Examples

  • SAT example with 100 students (mean=1500, SD=250).

  • Caloric intake examples illustrating P( > k) with given means and standard deviations.