Lab 4 Data 606

library(tidyverse)
library(openintro)

Exercise 1

Observing the two distributions we can see the both center in the 150-200 calorie range. In terms of shape Mcdonalds looks more right-skewed and Dairy Queen has more of a symmetric shape (slightly right-skewed). The tail of Mcdonalds extends further than Dairyqueen, which tells me the spread is larger.

data("fastfood", package='openintro')
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")

hist(mcdonalds$cal_fat, breaks=20)

hist(dairy_queen$cal_fat, breaks=20)

Exercise 2

Plotting density plots over the frequency plots shows that Dairyqueen follows a symmetric normal distribution. Mcdonalds is right-skewed. It was harder to tell in the frequency plots, but the density plots make it clear.

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

mcmean <- mean(mcdonalds$cal_fat)
mcsd <- mean(mcdonalds$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`.

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

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

Exercise 3

I plotted the simulated data over the real QQ plot to see how they line up. It lines up pretty closely until the tail end. The 2 line plots diverage around x = 1.5. But, interestingly the slope in the simiulation changes similarly to the real data.

sim_norm <- data.frame(rnorm(n = nrow(dairy_queen), mean = dqmean, sd = dqsd))
colnames(sim_norm) <- c('data')

ggplot(data = sim_norm, aes(sample = data)) + geom_line(stat = "qq") + 
  geom_blank() +
  geom_line(data = dairy_queen, aes(sample = cal_fat),stat = "qq",  col = "tomato") 

Exercise 4

Just looking at the real data plot compared to the other 8 simulations I would say the it’s very close to a normal distribution.

qqnormsim(sample = cal_fat, data = dairy_queen)

Exercise 5

The Mcdonalds QQ plot isn’t really linear and does not match as closely with the simulation as the Dairyqueen dataset did. My first instinct is telling me the distribution is not normal, but let’s compare against multiple simiulations to be sure.

sim_norm <- data.frame(rnorm(n = nrow(mcdonalds), mean = mcmean, sd = mcsd))
colnames(sim_norm) <- c('data')

ggplot(data = sim_norm, aes(sample = data)) + geom_line(stat = "qq") + 
  geom_blank() +
  geom_line(data = mcdonalds, aes(sample = cal_fat),stat = "qq",  col = "tomato") 

Out of the 8 simulations there isn’t really one that matches up very closely with the real data set. I would say this data set is not normal.

qqnormsim(sample = cal_fat, data = mcdonalds)

Exercise 6

What is the probability of selecting a food item under 700 calories from Burger King? We don’t know if Burger King has a normal distribution, but calculating the probability using pnorm gives us a 62% probability.

burger_king <- fastfood %>%
  filter(restaurant == "Burger King")

bkmean <- mean(burger_king$calories)
bksd <- sd(burger_king$calories)

pnorm(q=700, mean=bkmean, sd=bksd)

## [1] 0.6235496

The probability we calculated empirically gives us a 64% probability, which is very close to the value we got calculating the Z score. Maybe burger king has a normal distribution.

burger_king %>% 
  filter(calories < 700) %>%
  summarise(percent = n() / nrow(burger_king))

## # A tibble: 1 x 1
##   percent
##     <dbl>
## 1   0.643

What is the probability of grabbing a food item over 300 calories from Taco Bell? The Z score calculation gives us a probability of 78%.

taco_bell <- fastfood %>%
  filter(restaurant == "Taco Bell")

tbmean <- mean(taco_bell$calories)
tbsd <- sd(taco_bell$calories)

1 - pnorm(q=300, mean=tbmean, sd=tbsd)

## [1] 0.782086

Empirically the probability is 76%. This is fairly close to what we calculated before.

taco_bell %>% 
  filter(calories > 300) %>%
  summarise(percent = n() / nrow(taco_bell))

## # A tibble: 1 x 1
##   percent
##     <dbl>
## 1   0.757

Exercise 7

Look at all the distributions and qq plots it looks like Arby’s, Burger King, and Taco bell have the closet to normal distributions. If I had to pick one it would be Arby’s - it has the closest to linear qq plot.

ggplot(data = fastfood, aes(sample = sodium)) + geom_line(stat = "qq") +  facet_wrap(~restaurant)

ggplot(data = fastfood) + geom_histogram(aes(x = sodium)) +  facet_wrap(~restaurant)

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

Exercise 8

Sodium in this data set is measured in mg and is a discrete variable.

Exercise 9

I chose subway and the distribution looks pretty sparse. Plotting density plot over it helped me understand the shape a little better. I would say it is slightly right-skewed - not normal.

subway <- fastfood %>%
  filter(restaurant == "Subway")

sbmean <- mean(subway$total_carb)
sbsd <- sd(subway$total_carb)

ggplot(data=subway, aes(x=total_carb)) +
  geom_blank() +
  geom_histogram(aes(y = ..density..)) +
  stat_function(fun = dnorm, args = c(mean = sbmean, sd = sbsd), col = "tomato")

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

sim_norm <- data.frame(rnorm(n = nrow(subway), mean = sbmean, sd = sbsd))
colnames(sim_norm) <- c('data')

ggplot(data = sim_norm, aes(sample = data)) + geom_line(stat = "qq") + 
  geom_blank() +
  geom_line(data = subway, aes(sample = total_carb),stat = "qq",  col = "tomato")