PA 8.2: 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.

Remember

Four Types of Calculations

p – stands for probability (left tail area for a given value of $x$).
r – stands for random (for generating data).
q – stands for quantile (for a given probability).
d – stands for density (of the distribution at a given value of $x$).

Distributions

The normal distribution functions end with norm.
The uniform distribution functions end with unif.
The Chi-square distribution functions end with chisq.

Warm-up

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

# Q1 code

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

# Q2 code

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

# Q3 code

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

# Q4 code

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

# Q5 code

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 by finishing the code below.

Caution

Do not change the seed in the code provided below!

set.seed(1957)

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

Error: '...' used in an incorrect context

Canvas submission

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?

# Canvas submission code

--- title: "PA 8.2: Instrument Con" subtitle: "Distributions + Simulation" format: html: code-tools: true toc: true editor: source execute: error: true echo: true message: false warning: false --- ```{r setup} #| include: false library(purrr) ``` [Download starter .qmd template](../student-versions/pas/pa8-2.qmd) 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. ::: callout-tip # Remember **Four Types of Calculations** + `p` -- stands for probability (left tail area for a given value of $x$). + `r` -- stands for random (for generating data). + `q` -- stands for quantile (for a given probability). + `d` -- stands for density (of the distribution at a given value of $x$). **Distributions** + The normal distribution functions end with `norm`. + The uniform distribution functions end with `unif`. + The Chi-square distribution functions end with `chisq`. ::: ## Warm-up **1. What is the 95th percentile for trumpet weight?** ```{r} # Q1 code ``` **2. What is the 10th percentile for trombone weight?** ```{r} # Q2 code ``` **3. About what percent of trombones do you expect to be more than 5 pounds?** ```{r} # Q3 code ``` **4. About what percent of reed instruments do you expect to be more than 5 pounds?** ```{r} # Q4 code ``` **5. Simulate 100 random trombone weights. How many of them were below 4 pounds?** ```{r} # Q5 code ``` ## 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} 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 by finishing the code below.** ::: callout-caution **Do not** change the seed in the code provided below! ::: ```{r} set.seed(1957) my_weights <- map_dbl(.x = ... , .f = ~ music_man(n_tromb = 76, ...)) ``` ## Canvas submission > 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} # Canvas submission code ```