Lab 4: Normal Distribution

The Normal Distribution

Getting started

Load packages

library(tidyverse)

## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --

## v ggplot2 3.3.3     v purrr   0.3.4
## v tibble  3.0.6     v dplyr   1.0.3
## v tidyr   1.1.2     v stringr 1.4.0
## v readr   1.4.0     v forcats 0.5.1

## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(openintro)

## Loading required package: airports

## Loading required package: cherryblossom

## Loading required package: usdata

The Data

head(fastfood)

## # A tibble: 6 x 17
##   restaurant item  calories cal_fat total_fat sat_fat trans_fat cholesterol
##   <chr>      <chr>    <dbl>   <dbl>     <dbl>   <dbl>     <dbl>       <dbl>
## 1 Mcdonalds  Arti~      380      60         7       2       0            95
## 2 Mcdonalds  Sing~      840     410        45      17       1.5         130
## 3 Mcdonalds  Doub~     1130     600        67      27       3           220
## 4 Mcdonalds  Gril~      750     280        31      10       0.5         155
## 5 Mcdonalds  Cris~      920     410        45      12       0.5         120
## 6 Mcdonalds  Big ~      540     250        28      10       1            80
## # ... with 9 more variables: sodium <dbl>, total_carb <dbl>, fiber <dbl>,
## #   sugar <dbl>, protein <dbl>, vit_a <dbl>, vit_c <dbl>, calcium <dbl>,
## #   salad <chr>

mcdonalds <- fastfood %>%
  filter(restaurant == "Mcdonalds")
dairy_queen <- fastfood %>%
  filter(restaurant == "Dairy Queen")

Exercise 1

summary(mcdonalds)

##   restaurant            item              calories         cal_fat      
##  Length:57          Length:57          Min.   : 140.0   Min.   :  50.0  
##  Class :character   Class :character   1st Qu.: 380.0   1st Qu.: 160.0  
##  Mode  :character   Mode  :character   Median : 540.0   Median : 240.0  
##                                        Mean   : 640.4   Mean   : 285.6  
##                                        3rd Qu.: 740.0   3rd Qu.: 320.0  
##                                        Max.   :2430.0   Max.   :1270.0  
##    total_fat         sat_fat         trans_fat       cholesterol   
##  Min.   :  5.00   Min.   : 0.500   Min.   :0.0000   Min.   :  0.0  
##  1st Qu.: 18.00   1st Qu.: 4.500   1st Qu.:0.0000   1st Qu.: 70.0  
##  Median : 27.00   Median : 7.000   Median :0.0000   Median : 95.0  
##  Mean   : 31.81   Mean   : 8.289   Mean   :0.4649   Mean   :109.7  
##  3rd Qu.: 36.00   3rd Qu.:11.000   3rd Qu.:1.0000   3rd Qu.:125.0  
##  Max.   :141.00   Max.   :27.000   Max.   :3.0000   Max.   :475.0  
##      sodium       total_carb         fiber           sugar      
##  Min.   :  20   Min.   :  9.00   Min.   :0.000   Min.   : 0.00  
##  1st Qu.: 870   1st Qu.: 32.00   1st Qu.:2.000   1st Qu.: 4.00  
##  Median :1120   Median : 46.00   Median :3.000   Median : 9.00  
##  Mean   :1438   Mean   : 48.79   Mean   :3.228   Mean   :11.07  
##  3rd Qu.:1780   3rd Qu.: 62.00   3rd Qu.:4.000   3rd Qu.:13.00  
##  Max.   :6080   Max.   :156.00   Max.   :8.000   Max.   :87.00  
##     protein          vit_a            vit_c         calcium     
##  Min.   :  7.0   Min.   :  0.00   Min.   : 0.0   Min.   :  0.0  
##  1st Qu.: 25.0   1st Qu.:  2.00   1st Qu.: 2.0   1st Qu.:  6.0  
##  Median : 33.0   Median :  6.00   Median :15.0   Median : 15.0  
##  Mean   : 40.3   Mean   : 33.72   Mean   :18.3   Mean   : 20.6  
##  3rd Qu.: 46.0   3rd Qu.: 20.00   3rd Qu.:25.0   3rd Qu.: 20.0  
##  Max.   :186.0   Max.   :180.00   Max.   :70.0   Max.   :290.0  
##     salad          
##  Length:57         
##  Class :character  
##  Mode  :character  
##                    
##                    
## 

ggplot(data= mcdonalds, aes(x= cal_fat)) +
  geom_histogram(bins= 30, fill= "blue") +
  ggtitle("Distribution Calorie Fat from Mc Donalds")

summary(dairy_queen)

##   restaurant            item              calories         cal_fat     
##  Length:42          Length:42          Min.   :  20.0   Min.   :  0.0  
##  Class :character   Class :character   1st Qu.: 350.0   1st Qu.:160.0  
##  Mode  :character   Mode  :character   Median : 485.0   Median :220.0  
##                                        Mean   : 520.2   Mean   :260.5  
##                                        3rd Qu.: 630.0   3rd Qu.:310.0  
##                                        Max.   :1260.0   Max.   :670.0  
##                                                                        
##    total_fat        sat_fat        trans_fat       cholesterol    
##  Min.   : 0.00   Min.   : 0.00   Min.   :0.0000   Min.   :  0.00  
##  1st Qu.:18.00   1st Qu.: 5.00   1st Qu.:0.0000   1st Qu.: 41.25  
##  Median :24.50   Median : 9.00   Median :1.0000   Median : 60.00  
##  Mean   :28.86   Mean   :10.44   Mean   :0.6786   Mean   : 71.55  
##  3rd Qu.:34.75   3rd Qu.:12.50   3rd Qu.:1.0000   3rd Qu.:100.00  
##  Max.   :75.00   Max.   :43.00   Max.   :2.0000   Max.   :180.00  
##                                                                   
##      sodium         total_carb         fiber            sugar       
##  Min.   :  15.0   Min.   :  0.00   Min.   : 0.000   Min.   : 0.000  
##  1st Qu.: 847.5   1st Qu.: 25.25   1st Qu.: 1.000   1st Qu.: 3.000  
##  Median :1030.0   Median : 34.00   Median : 2.000   Median : 6.000  
##  Mean   :1181.8   Mean   : 38.69   Mean   : 2.833   Mean   : 6.357  
##  3rd Qu.:1362.5   3rd Qu.: 44.75   3rd Qu.: 3.000   3rd Qu.: 8.750  
##  Max.   :3500.0   Max.   :121.00   Max.   :12.000   Max.   :30.000  
##                                                                     
##     protein          vit_a        vit_c          calcium      
##  Min.   : 1.00   Min.   : 0   Min.   : 0.00   Min.   :  0.00  
##  1st Qu.:17.00   1st Qu.: 9   1st Qu.: 0.00   1st Qu.:  6.00  
##  Median :23.00   Median :10   Median : 4.00   Median : 10.00  
##  Mean   :24.83   Mean   :14   Mean   : 4.37   Mean   : 16.41  
##  3rd Qu.:34.00   3rd Qu.:20   3rd Qu.: 6.00   3rd Qu.: 20.00  
##  Max.   :49.00   Max.   :50   Max.   :30.00   Max.   :100.00  
##                  NA's   :15   NA's   :15      NA's   :15      
##     salad          
##  Length:42         
##  Class :character  
##  Mode  :character  
##                    
##                    
##                    
## 

ggplot(data = dairy_queen, aes(x = cal_fat)) +
  geom_histogram(fill = "red", binwidth = 30) + 
  ggtitle("Distribution Calorie Fat from Dairy Queen")

Answer: The distribution of the calories from fat of Dairy Queen’s items and Mc Donalds’ are close to normal. Each set is right skewed (the tail runs to the right). With that said, the frequency histogram(s) highlight a number of unique differences between the distributions. McDonald’s has a higher minimum, maximum (approximately 1250) and center (around 280) for fat calories. Whereas Dairy Queen’s curve is far less skewed, with a small skew to the right, the center of Dairy Queen’s curve is around 250 calories from fat and a max value around 675 calories from fat. Also, the McDonald’s x axis increases in increments of 200 cals while the Dairy Queen one increases in increments of 100 cals.

The Normal Distribution

dqmean <- mean(dairy_queen$cal_fat)
dqsd   <- sd(dairy_queen$cal_fat)

ggplot(data = dairy_queen, aes(x = cal_fat)) + geom_blank() + geom_histogram(aes(y = ..density..)) + stat_function(fun = dnorm, args = c(mean = dqmean, sd = dqsd), col = "tomato")

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Exercise 2

Answer: Yes, Dairy Queen’s calories from fat curve follows a nearly normal distribution.

Evaluating the normal distribution

ggplot(dairy_queen, aes(sample = cal_fat))+
   stat_qq()+stat_qq_line()

sim_norm <- rnorm(n = nrow(dairy_queen), mean = dqmean, sd = dqsd)

Exercice 3

ggplot(data= NULL, aes(sample= sim_norm)) +
  stat_qq()+stat_qq_line()

Answer: No, all the points on the line do not collapse. The probability graphs are identical but not the same for the actual data of the simulated. There is a smaller slope from x= -2 to x= -1 and a larger slope from x= 1 to x= 2.3 for the actual results. The plots are quite close, rather than the nuance.

qqnormsim(sample = cal_fat, data = dairy_queen)

Exercice 4

Answer: Yes, the Dairy Queen “cal_fat” normal probability plot is pretty closely aligned with all our simulated data probability plots, although it curves slightly below the qqline (diagonal line) while the simulations did not.

Exercice 5

qqnormsim(sample = cal_fat, data = mcdonalds)

Answer: The Mc Donalds “cal_fat” data is nearly normal. Although the slope is rather small near the beginning and rather larger later on, it does form a diagonal line near the qqline (diagonal line) and closely mimics a couple of the simulated plots up until these higher values.

Normal probabilities

1 - pnorm(q = 600, mean = dqmean, sd = dqsd)

## [1] 0.01501523

dairy_queen %>% 
  filter(cal_fat > 600) %>%
  summarise(percent = n() / nrow(dairy_queen))

## # A tibble: 1 x 1
##   percent
##     <dbl>
## 1  0.0476

Exercice 6

Question 1: What is the probability that a randomly chosen Chick Fil-A product has more than 400 calories from fat?

chick_fil_a <- fastfood %>%
  filter(restaurant == "Chick Fil-A")

a_mean <- mean(chick_fil_a$cal_fat)
a_sd <- sd(chick_fil_a$cal_fat)

1 - pnorm(q = 400, mean = a_mean, sd = a_sd)

## [1] 0.006429412

chick_fil_a %>% 
  filter(cal_fat > 400) %>%
  summarise(percent = n() / nrow(chick_fil_a))

## # A tibble: 1 x 1
##   percent
##     <dbl>
## 1  0.0741

Answer: There is around a 0.074%, around 0%, chance of randomly selecting a Chick Fil-A item above 400 calories from fat. Quit healthy! (sort of)

Question 2: What is the probability that a randomly chosen product from any of these fast food restaurants is less than 400 calories?

ff_mean <- mean(fastfood$calories)
ff_sd <- sd(fastfood$calories)

pnorm(q = 400, mean = ff_mean, sd = ff_sd)

## [1] 0.3214986

fastfood %>% 
  filter(calories < 400) %>%
  summarise(percent = n() / nrow(fastfood))

## # A tibble: 1 x 1
##   percent
##     <dbl>
## 1   0.359

Answer: There is approximately 0.36% chance of randomly selecting a product from any of these fast food restaurants is less than 400 calories.

Answer: Although the calculated probabilities varied slightly for both calculations, those for probability that a randomly chosen product from any of these fast food restaurants was less than 400 calories were in closer agreement.

More Practise

unique(fastfood$restaurant)

## [1] "Mcdonalds"   "Chick Fil-A" "Sonic"       "Arbys"       "Burger King"
## [6] "Dairy Queen" "Subway"      "Taco Bell"

Exercice 7

Arby’s Restaurant

arbys <- fastfood %>%
  filter(restaurant == "Arbys")
ggplot(arbys, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Burger King Restaurant

burgerking <- fastfood %>%
  filter(restaurant == "Burger King")
ggplot(burgerking, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Chick Fil-A Restaurant

chick_fil_a <- fastfood %>%
  filter(restaurant == "Chick Fil-A")
ggplot(chick_fil_a, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Dairy Queen Restaurant

dairy_queen <- fastfood %>%
  filter(restaurant == "Dairy Queen")
ggplot(dairy_queen, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Mc Donald’s Restaurant

mcdonalds <- fastfood %>%
  filter(restaurant == "Mcdonalds")
ggplot(mcdonalds, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Sonic Restaurant

sonic <- fastfood %>%
  filter(restaurant == "Sonic")
ggplot(sonic, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Subway Restaurant

subway <- fastfood %>%
  filter(restaurant == "Subway")
ggplot(subway, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Taco Bell Restaurant

tacobell <- fastfood %>%
  filter(restaurant == "Taco Bell")
ggplot(tacobell, aes(sample = sodium))+
   stat_qq()+stat_qq_line()

Answer: Burger King appear to have the closest to normal distributions for their sodium data.

Exercice 8

Answer: Some of the normal probability plots for sodium distributions seem to have a stepwise pattern. I think this might be the case because the data collected were rounding.

Exercice 9

ggplot(dairy_queen, aes(sample = total_carb))+
   stat_qq()+stat_qq_line()

ggplot(data= dairy_queen, aes(x= total_carb)) +
        geom_blank() +
        geom_histogram(aes(y= ..density..), bins= 7) +
        stat_function(fun= dnorm, args= c(mean= mean(dairy_queen$total_carb),
                      sd= sd(dairy_queen$total_carb)), col= "red")

Answer: Based on the normal probability plot, this variable (total carbohydrates) is right skewed. Also, the histogram confirms this with data being concentrated on the left with a tail running to the right.