Section 5.2 The Sampling Distribution of a Sample Mean:

Population Distribution:

• The population distribution of a variable is the distribution of values of the variable among all individuals in the population

• The population distribution is also the probability distribution of the variable when we choose one individual at random from the population

• Important Note: Sometimes the population of interest does not actually exist
  

Mean and Standard Deviation of a Sample Mean:

• Mean of a sampling distribution of a sample mean:

  • There is no tendency for a sample mean to fall systematically above or below μ, even if the distribution of the raw data is skewed.

  • Thus, the mean of the sampling distribution is an unbiased estimate of the population mean μ

• Standard deviation of a sampling distribution of a sample mean:

  • The standard deviation of the sampling distribution measures how much the sample statistic varies from sample to sample.

  • It is smaller than the standard deviation of the population by a factor of square root of n. (This is also sometimes called the standard error or SE)

  
  
  

Rules for Means:

• Rule 1: If X is a random variable and a and b are fixed numbers, then

  • μa+bX = a + bμX

• Rule 2: If X and Y are random variables, then

  • μX+Y = μX + μY

Rules for Variances and Standard Deviations:

• Rule 1: If X is a random variable and a and b are fixed numbers, then

  • σ2a+bX = b2σ2X

• Rule 2: If X and Y are independent random variables, then

  • σ2X+Y = σ2X + σ2Y

  • σ2X–Y = σ2X + σ2Y

The Sampling Distribution of a Sample Mean:

• Critical Idea: Averages are less variable than individual observations

The Central Limit Theorem:

• Remarkably, as the sample size increases, the distribution of sample means begins to look more and more like a Normal distribution!

• When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation

• Draw an SRS of size n from any population with mean μ and finite standard deviation σ. The central limit theorem (CLT) says that when n is large, the sampling distribution of the sample mean x is approximately Normal:

• x is approximately N(upside down h, o with hat / square root of n