AE 18: Sampling distributions & the Central Limit Theorem

Bulletin

• Due today:
  • quiz 06 on sakai (now)
  • lab 07 by 11:59pm
• Misc
  • Hw 04 due Saturday by 11:59pm
  • Proposal feedback on gradescope, be sure to read the notes
  • If you haven’t, check out AE17 solutions and additional example on sakai

Learning goals

• Recap simulation-based methods and infer
• Use Central Limit Theorem to define distribution of sample means
• Calculate probabilities from the normal distribution

library(tidyverse)

Recap of simulation-based methods and infer

Practice using CLT & Normal distribution

Suppose the bone density for 65-year-old women is normally distributed with mean \(809 mg/cm^3\) and standard deviation of \(140 mg/cm^3\).

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

\[x \sim N(809, 140)\] ## 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 \(Q_1\) (25th percentile), \(Q_2\) (50th percentile), and \(Q_3\) (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 \(\bar{x}\), 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 \(\bar{x}\), the mean bone density for a group of 10 randomly selected 65-year-old women?

• Write the distribution of \(\bar{x}\) 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.