PA 9: Instrument Con

Distributions + Simulation

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 sub-par materials to scam the townsfolk.

You do some research on the weights of properly crafted brass instruments, and you learn the following facts:

Trombone weights are normally distributed, with a mean weight of 4.6 lbs and a standard deviation of 0.8 lbs.
Trumpet weights are uniformly distributed between 1.5 and 3.5 lbs.
Reed instruments (like clarinets, bassoons, and saxophones) have weights that are Chi-square distributed with 4 degrees of freedom.

Warm-up

1. What is the 95th percentile for trumpet weight?

2. What is the 10th percentile for trombone weight?

3. About what percent of trombones do you expect to be more than 5 pounds?

4. About what percent of reed instruments do you expect to be more than 5 pounds?

5. Simulate 100 random trombone weights. How many of them were below 4 pounds?

Catching a Con

You manage to intercept a shipping notice for a 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!

6. Write a function that samples the correct number of trombones, cornets (trumpets), and reed instruments (saxophones, clarinets, bassoons), and then returns the total weight of the shipment.

music_man <- function(n_tromb, n_cor, n_reed){
  
  trombones <- rnorm(n_tromb, ...)
  cornets   <- ...
  reeds     <- ...
  
  ...
  
  return()
  
}

7. Use the function you just wrote to create random samples of the total weight of 1000 possible shipments. Do not change the seed in the code provided below!

set.seed(1957)

my_weights <- map_dbl(.x = 1:1000, 
                      .f = ~ music_man(n_tromb = 76, ...)
                      )

Error in `map_dbl()`:
ℹ In index: 1.
Caused by error in `music_man()`:
! '...' used in an incorrect context

Canvas submission

You and your partner together should address the following questions:

How many of these samples had a weight less than or equal to Professor Hill’s shipment?

Do you beleive Professor Hill ordered genuine instruments?

--- title: "PA 9: Instrument Con" subtitle: "Distributions + Simulation" format: html: embed-resources: true code-tools: true toc: true editor: source execute: error: true echo: true message: false warning: false --- ```{r setup} #| include: false library(purrr) set.seed(1957) ``` 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 sub-par materials to scam the townsfolk. You do some research on the weights of properly crafted brass instruments, and you learn the following facts: + Trombone weights are normally distributed, with a mean weight of 4.6 lbs and a standard deviation of 0.8 lbs. + Trumpet weights are uniformly distributed between 1.5 and 3.5 lbs. + Reed instruments (like clarinets, bassoons, and saxophones) have weights that are Chi-square distributed with 4 degrees of freedom. ## Warm-up  **1. What is the 95th percentile for trumpet weight?** ```{r} #| label: q1 ``` **2. What is the 10th percentile for trombone weight?** ```{r} #| label: q2 ``` **3. About what percent of trombones do you expect to be more than 5 pounds?** ```{r} #| label: q3 ```  **4. About what percent of reed instruments do you expect to be more than 5 pounds?** ```{r} #| label: q4 ``` **5. Simulate 100 random trombone weights. How many of them were below 4 pounds?** ```{r} #| label: q5 ``` ## Catching a Con You manage to intercept a shipping notice for a 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!  **6. Write a function that samples the correct number of trombones, cornets (trumpets), and reed instruments (saxophones, clarinets, bassoons), and then returns the total weight of the shipment.** ```{r} #| label: function-for-instrument-weights music_man <- function(n_tromb, n_cor, n_reed){ trombones <- rnorm(n_tromb, ...) cornets <- ... reeds <- ... ... return() } ```  **7. Use the function you just wrote to create random samples of the total weight of 1000 possible shipments.** *Do not* change the seed in the code provided below! ```{r} #| label: simulating-instrument-weights set.seed(1957) my_weights <- map_dbl(.x = 1:1000, .f = ~ music_man(n_tromb = 76, ...) ) ``` ## Canvas submission You and your partner together should address the following questions: > How many of these samples had a weight less than or equal to Professor Hill's > shipment? > > Do you beleive Professor Hill ordered genuine instruments? ```{r} #| label: catching-a-con ```