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library(tidyverse)

What is the Central Limit Theorem?

The central limit theorem is a statement about the distribution of the sample mean, ˉx

The central limit theorem guarantees that, when certain criteria are satisfied, the sample mean (ˉx) is normally distributed.

Specifically,

If

  1. Observations in the sample are independent. Two rules of thumb to check this:

    • completely random sampling
    • if sampling without replacement, sample should be less than 10% of the population size

and

  1. The sample is large enough. The required size varies in different contexts, but some good rules of thumb are:

    • if the population itself is normal, sample size does not matter.
    • if numerical require, >30 observations
    • if binary outcome, at least 10 successes and 10 failures.

then

ˉx∼N(μ,σ/√n)

i.e. ˉx is normally distributed (unimodal and symmetric with bell shape) with mean μ and standard deviation σ/√n.

Note the standard deviation depends on the number of samples, n.

Practice using CLT & Normal distribution

Suppose the bone density for 65-year-old women is normally distributed with mean 809mg/cm3 and standard deviation of 140mg/cm3.

Let x be the bone density of 65-year-old women. We can write this distribution of x in mathematical notation as

x∼N(809,140)

Visualize the population distribution

ggplot(data = data.frame(x = c(809 - 140*3, 809 + 140*3)), aes(x = x)) +
  stat_function(fun = dnorm, args = list(mean = 809, sd = 140),
                color = "black") +
  stat_function(fun = dnorm, args = list(mean = 809, sd = 140/sqrt(10)),
                color = "red",lty = 2) + theme_bw() +
  labs(title = "Black solid line = population dist., Red dotted line = sampling dist.")

Exercise 1

Before typing any code, based on what you know about the normal distribution, what do you expect the median bone density to be?

What bone densities correspond to Q1 (25th percentile), Q2 (50th percentile), and Q3 (the 75th percentile) of this distribution? Use the qnorm() function to calculate these values.

Exercise 2

The densities of three woods are below:

  • Plywood: 540 mg/cubic centimeter

  • Pine: 600 mg/cubic centimeter

  • Mahogany: 710 mg/cubic centimeter

  • What is the probability that a randomly selected 65-year-old woman has bones less dense than Pine?

  • Would you be surprised if a randomly selected 65-year-old woman had bone density less than Mahogany? What if she had bone density less than Plywood? Use the respective probabilities to support your response.

Exercise 3

Suppose you want to analyze the mean bone density for a group of 10 randomly selected 65-year-old women.

  • Are the conditions met use the Central Limit Theorem to define the distribution of ˉx, the mean density of 10 randomly selected 65-year-old women?

    • Independence?
    • Sample size/distribution?

  • What is the shape, center, and spread of the distribution of ˉx, the mean bone density for a group of 10 randomly selected 65-year-old women?

  • Write the distribution of ˉx using mathematical notation.

Exercise 4

  • What is the probability that the mean bone density for the group of 10 randomly-selected 65-year-old women is less dense than Pine?

  • Would you be surprised if a group of 10 randomly-selected 65-year old women had a mean bone density less than Mahogany? What the group had a mean bone density less than Plywood? Use the respective probabilities to support your response.

Exercise 5

Explain how your answers differ in Exercises 2 and 4.