Central Limit Theorem: Properties of Sampling Distribution

Central Limit Theorem (CLT)

• The CLT states that the frequency distribution of the sample mean (X̄) has the following properties:
  • Mean of the sample mean is equal to the mean of the population (𝝁)
  • Variance of the sampling distribution of means is equal to the variance of the population divided by the sample size (n)
  • Standard deviation of the sampling distribution of means is equal to the standard deviation of the population divided by the square root of the sample size (√n)
  • The sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough

Examples and Exercises

• Example 1:
  • Given a population with mean (𝜇) = 4.6 and variance (𝝈²) = 4, and samples of size n = 4,
  • The mean of the sampling distribution of the means is equal to the population mean (𝝁), and the standard deviation is equal to the population standard deviation divided by √n
• Example 2:
  • Given a population with mean (𝜇) = 3 and standard deviation (𝝈) = 1.8, and samples of size n = 2,
  • The mean of the sampling distribution of the means is equal to the population mean (𝝁), and the variance of the sampling distribution of the means is equal to the population variance divided by n
• Exercise 1:
  • Given a population with mean (𝜇) = 5 and standard deviation (𝝈) = 3.2, and samples of size n = 6,
  • Find the mean and variance of the population where the sampling distribution is derived
• Activity 1:
  • Given a population with mean (𝜇) = 7 and variance (𝝈²) = 4, and samples of size n = 9,
  • Find the mean and standard deviation of the sampling distribution of the means

Central Limit Theorem (with Replacement)

• The term "with replacement" implies that the samples can be selected repeatedly in random sampling
• To find the mean, variance, and standard deviation of the sampling distribution of the means:
  • Compute the mean, variance, and standard deviation of the sampling distribution of the means
  • Construct the sampling distribution
  • Draw the probability histogram of the sampling distribution of the means

Example (with Replacement)

• Consider all possible samples of size n = 2 that can be drawn with replacement from the population 1, 2, 3, and 4
• The sampling distribution of the mean is approximately normally distributed

Central Limit Theorem (without Replacement)

• The term "without replacement" implies that no samples can be selected repeatedly in random sampling