---
title: "Lab 5"
subtitle: "36-315: Statistical Graphics and Visualization, Summer 2026"
format:
  pdf:
    colorlinks: true
    toc: true
    fontsize: 10pt
    geometry: margin=0.9in
execute:
  warning: false
  message: false
---

\newpage

# Problem 1: Transformations and changing smoothed densities (30 points)

In this lab, we will work with a dataset about food around the word, obtained and modified from Kaggle (see [here](https://www.kaggle.com/openfoodfacts/world-food-facts) for more information).

```{r}
library(tidyverse)
food <- read_csv("https://raw.githubusercontent.com/qntkhvn/36-315-summer26/refs/heads/master/data/food.csv")
```


## A (10pts) 

* First, create a histogram of the `energy_100g` variable on the **density scale**. There's no need to specify a `binwidth` for this part, but fill the bars with a non-gray color. Also, add a smoothed density curve to the plot using `geom_density()`.

```{r}
# YOUR CODE HERE
```

* Now create a histogram of the *square root* of `energy_100g`, with an overlayed density. Luckily, it's really easy to make transformations of variables in `ggplot`: simply replace `energy_100` with `sqrt(energy_100g)` within `ggplot()`.

```{r}
# YOUR CODE HERE
```


## B (10pts) 

It can be helpful to understand what bandwidth is being estimated in the density curve from Part A.

To check what the bandwidth is, write a line of code that applies the `density()` function to `sqrt(food$energy_100g)`.

```{r}
# YOUR CODE HERE
```

What bandwidth is being estimated? State the bandwidth value.

**YOUR ANSWER HERE**

Next, use the histogram code from Part A for `sqrt(energy_100)` and set `binwidth` equal to the above bandwidth value within `geom_histogram()`. (This is another way to automatically pick the `binwidth` for histograms!)

```{r}
# YOUR CODE HERE
```

## C (10pts) 

Finally, use the code from Part B and make the following changes:

* Change the `kernel` argument within `geom_density()`. Try a few different values, such as `kernel = "rectangular"`, `kernel = "triangular"`, etc. To see different options for `kernel`, see `kernel` within `help(density)`. Pick a non-Gaussian kernel for the plot.

+ Within `geom_density()`, specify a `color`, and set `linewidth = 2`. This will make the density curve better.

```{r}
# YOUR CODE HERE
```

\newpage

# Problem 2: Comparing conditional distributions (25 points) 

## A (5pts) 

Make a conditional smoothed density plot of `sqrt(energy_100g)` given `nutrition_grade_fr` using `geom_density()` with `color = nutrition_grade_fr`.

(There should be a separate density curve of `sqrt(energy_100g)` for each level of `nutrition_grade_fr`, each with its own color. There should NOT be any histograms on your plot, just smoothed density curves.)

```{r}
# YOUR CODE HERE
```

## B (10pts) 

Perform a Bartlett's test (using `bartlett.test()`) to test whether `sqrt(energy_100g)` has the same variance across different values of `nutrition_grade_fr`.

```{r}
# PUT YOUR CODE HERE
```

State your formal conclusion (in context) from this test, and explain how you arrived at your answer. Also, be sure to state the null hypothesis for this test in your answer.

**YOUR ANSWER HERE**

## C (10pts)

Perform a one-way ANOVA (using `oneway.test()`) to test whether `sqrt(energy_100g)` has the same mean across different values of `nutrition_grade_fr`. Within `oneway.test()`, specify either `var.equal = TRUE` or `var.equal = FALSE` to indicate whether the variances are equal (using the conclusions from the Bartlett's test above).

```{r}
# YOUR CODE HERE
```

State your formal conclusion (in context) from this test, and explain how you arrived at your answer. Also, be sure to state the null hypothesis for this test in your answer.

**YOUR ANSWER HERE**

\newpage

# Problem 3: Kolmogorov–Smirnov test (45 points)

## A (10pts) 

First, define a dataset, `food_d`, which contains only the Nutrition Grade D foods.

```{r}
# YOUR CODE HERE
```

Then, perform a Kolmogorov-Smirnov (KS) test to assess whether `energy_100g` for Grade D foods follows a normal distribution.

```{r}
# YOUR CODE HERE
```

State the p-value and your formal conclusion (in context) from this test.

**YOUR ANSWER HERE**

## B (20pts) 

The KS test relies on the empirical cumulative distribution functions (ECDFs).

First, make a plot to visualize the ECDF for `energy_100g` with the `food_d` dataset.

```{r}
# YOUR CODE HERE
```

Next, update the previous plot to include a comparison with the normal CDF using `stat_function()`.

```{r}
# YOUR CODE HERE
```

In 1--2 sentences: How would you use the plot above to report the median of `energy_100g` for Grade D foods? To help answer this question, draw a horizontal line on your ECDF plot by adding `geom_hline()` to your code, with an appropriate value specified.

```{r}
# YOUR CODE HERE
```

**YOUR ANSWER HERE**

## C (10pts) 

Test whether `energy_100g` has a different distribution for Grade B and Grade D foods. To do so, first define a dataset called `food_b` (which contains only the Nutrition Grade B foods), then perform a two-sample KS test.

```{r}
# YOUR CODE HERE
```

State your formal conclusion (in context) from this test. Be sure to mention what the null hypothesis is for this test.

**YOUR ANSWER HERE**

## D (5pts) 

Visualize the two-sample KS test results as follows.

* First, use the code from Part B, but remove the `geom_hline()` layer, and update the input dataset to be `food` instead of `food_d`.

* Then, update the ECDF layer so that it is color coded by `nutrition_grade_fr`.

```{r}
# YOUR CODE HERE
```

* Focusing on Grade B and Grade D foods, explain whether the graph supports the KS test conclusion in 1--2 sentences?

**YOUR ANSWER HERE**