Sampling Distributions

Question 1

What does the CLT tell us about the standard deviation of sample means?

Answer 1

No matter what model a random sample is taken from, and for any sample size n, the standard deviation of the sample mean is the model standard deviation, σ (the true population standard deviation) divided by square root of n i.e. σ/√n

Question 2

What is the CLT?

Answer 2

• If X is a random variable with a mean µ and a variance σ2, then in general
• X̅ → N (µ, σ2/n)
  
  • The sample averages will approach a normal distribution, with centre µ and variance of σ2/n

Question 3

What is the standard error of the mean?

Answer 3

• SEMean
• (σ/√n)
• Standard error just means spread of the sample averages
• As we usually don’t know the theoretical value σ, we usually have to estimate it during the sample standard deviation, s, and s/√n is called the (sample) standard error of the mean - this is SEMean in Minitab

Question 4

How large does n have to be for the CLT?

Answer 4

• If X has a normal distribution, then the sample mean has a normal distribution for all sample sizes
• If X has a distribution that is close to normal, the approximation is good for small ample sizes (e.g. n=20)
• If X has a distribution that is far from normal, the approximation requires larger sample sizes (eg. n=50)