PA: Simulation
A man named Professor Harold Hill wanders into your small town. He claims to be selling top-quality instruments for a marching band. He begins selling his instruments to all the children in town.
You are suspicious of Professor Hill, and think that perhaps he is selling instruments made of subpar materials, to scam the townsfolk.
You do some research on the weights of properly crafted brass instruments, and you learn the following facts:
Trombones weights are Normally distributed, with a mean weight of 4.6 pounds and a standard deviation of 0.8 pounds.
Trumpet weights are Uniformly distributed between 1.5 and 3.5 pounds.
Reed instruments (like clarinets, bassoons, and saxophones) have weights that are Chi-Square distributed with 4 degrees of freedom.
Warmup
What is the 95th percentile for trumpet weight?
What is the 10th percentile for trombone weight?
About what percent of trombones do you expect to be more than 5 pounds?
About what percent of reed instruments do you expect to be more than 5 pounds?
Simulate 100 random trombone weights. How many of them were below 4 pounds?
Catching a con
You manage to intercept a shipping notice for delivery to Professor Hill. It says the following:
Wells Fargo Shipment 1957
To: Harold Hill, Mason City, Iowa
From: Music Factory, Gary, Indiana
Items included:
Trombones: 76
Cornets: 110
Saxophones: 542
Clarinets: 318
Bassoons: 175
Total Shipped Weight: 4532 lbs.
This is your chance to catch Professor Hill in his lie!
First, write a function that samples the correct number of weights of trombones, cornets (trumpets), and reed instruments, and then returns the total weight.
Then, use that function to create 1000 random samples of total weight.
(Some code is provided for you below. Do not change the seed!)
music_man <- function(n_tromb, n_cor, n_reed){
trombones <- rnorm(n_tromb, ...)
cornets <- ...
reeds <- ...
return()
}
set.seed(1957)
my_weights <- replicate(1000, music_man(76, ...))How many of these samples had a weight less than Prof. Hill’s shipment? Do you believe Professor Hill is lying, or not?