Bone density

Application exercise

Goals

Calculate probabilities under the normal curve
Understand Z scores
Build intuition for hypothesis testing and confidence interval building with the normal distribution

Packages and data

We’ll use the tidyverse and openintro packages.

library(tidyverse)
library(openintro)

Bone density - population

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

\[ x \sim N(\mu = 809, \sigma = 140) \]

Let’s set the parameters of this distribution as objects you can use later.

bone_density_mean <- 809
bone_density_sd <- 140

Exercise 1

Visualize the distribution of bone density of 65-year-old women.

# add code here

Exercise 2

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

Add response here.

Exercise 3

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.

# add code here

Exercise 4

The densities of three woods are below:

Plywood: 540 \(mg/cm^3\)
Pine: 600 \(mg/cm^3\)
Mahogany: 710 \(mg/cm^3\)

Let’s set these as variables we can use later:

plywood <- 540
pine <- 600
mahogany <- 710

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

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Exercise 5

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.

Add response here.

# add code here

Bone density - sampling

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

Exercise 6

Are the conditions for the Central Limit Theorem met?

Add response here.

Exercise 7

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

Add response here.

Exercise 8

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?

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Exercise 9

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.

# add code here

Exercise 10

Explain how your answers differ in similar sounding earlier vs. later exercises.

Add response here.

# add code here