lesson 7

• The mean of the sampling distribution of the sample means is equal to the population mean
• The standard deviation of the sampling distribution of the sample means is equal to σ/√n
• Central Limit Theorem


  The Central Limit Theorem states that as the sample size n increases, the distribution of the sample means taken approaches a normal distribution
• If n is sufficiently large, then the distribution can be treated as a normal distribution
• Generally accepted "sufficiently large" sample size: n ≥ 30
• Galton board


  A device created to demonstrate the CLT
• If the sample size of a distribution is sufficiently large (n ≥ 30), then the z-score formula can be used
• The CLT can be applied if: the sample size is at least 30; or the sample size is less than 30, but the distribution is assumed to be normal; or the sample size is less than 30, but the population standard deviation is known
• 1. distribution


  The Student's t-distribution represents the standardized distances of sample means to the population mean wherein: the distribution is normal; the sample size is small (n < 30); and the population standard deviation is unknown
• The t-distribution is named after William S. Gosset, who published his works under the pen name Student
• 1. distribution characteristics

  • The mean, median, and mode are all equal to 0, and are located at the center of the distribution
  • The distribution is bell-shaped and is symmetric about the mean
  • The distribution is asymptotic to the x-axis
  • The variance is greater than 1