Today’s Agenda

  • Central Limit Theorem for means
  • Central Limit Theorem for sums

Central Limit Theorem for means

When we collect a sufficiently large sample of \(n\) independent observations from a population with mean \(\mu\) and standard deviation \(\sigma\), the sampling distribution will approach

\[\bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})\]

  • \(\mu\) is the mean of the population

  • \(\frac{\sigma}{\sqrt{n}}\) is the standard deviation of the sampling distribution.

  • we call this the standard error so that we don’t confuse it with the standard deviation of the population.

  • The central limit theorem applies:

    • even if the original population is not normal
    • the sample must be sufficiently large. This is called the normality condition
      • if \(n \geq 30\) the original distribution can have any shape
      • if \(n < 30\) the original distribution must be normal
    • the observations must be independent, this happens when the sample is from a simple random sample.

Central Limit Theorem for sums

When we collect a sufficiently large sample of \(n\) independent observations from a population with mean \(\mu\) and standard deviation \(\sigma\), the sampling distribution will approach

\[\Sigma X \sim N(n\mu, \sqrt{n}\sigma)\]

# 1. Suppose we draw samples of size 45 from an unknown distribution with population mean 15 and standard deviation 3. What is the probability of drawing a sample of size 45 with a mean less than 14?

# find standard error
se <- 3/sqrt(45)

# find the z-score for the sample mean under investigation
z <- (14 - 15)/se

# use pnorm
pnorm(z)
## [1] 0.01267366
# what is the probability of drawing a sample within $0.5$ of the population mean?


# Provide a range of values for which 90% of sample means will fall in that range