Packages

In this lab we will work with the tidyverse, fivethirtyeight and broom packages. We start by loading the packages.

library(tidyverse)
library(fivethirtyeight)
library(broom)

⊕Note: Make sure you knit your HTML document frequently - this will also save your .Rmd file! This is good practice to make sure you are catching any bugs early (and saving your output regularly). Note that these packages are also loaded in your R Markdown document.

Load the data

The data we will be working with today is called candy_rankings and it’s in the fivethirtyeight package.

To find out more about the dataset, type the following in your Console: ?candy_rankings A question mark before the name of an object will always bring up its help file. This command must be ran in the Console.

candy_rankings is a tidy data frame, with each row representing an observation and each column representing a variable.

To view the data, run data(candy_rankings) and click on the name of the data frame in the Environment tab.

You can also take a quick peek at your data frame and view its dimensions with the glimpse function.

glimpse(candy_rankings)

The description of the variables, i.e. the codebook, is given below.

Exercises

1. How many observations are in this dataset? What does each observation represent?

2. How many variables are in this dataset? Describe the variable types - are they numeric / continuous / quantitative or are they categorical? If they are categorical, what are the categories? Are they a special type of categorical variable known as a binary variable, a categorical variable with just two levels?

3. Create a new dataset called single_candy_rankings that only contains observations where pluribus == 0. This new dataset contains only candies that come with one candy (instead of many in a bag or box). The rest of this lab will use this new dataset.

single_candy_rankings  <- candy_rankings %>%
  filter(---)

4. Let’s take a closer look at two variables, sugarpercent and winpercent. Summarize the center and spread of these two variables by calculating the mean and standard deviation using the code below and describe what you see. Create two histograms using the single_candy_rankings dataset, one of sugarpercent and one of winpercent. Be sure to select an appropriate binwidth.

single_candy_rankings %>%
  summarise(mean_sugarpct = mean(sugarpercent),
            mean_winpct = mean(winpercent),
            sd_sugarpct = sd(sugarpercent),
            sd_winpct = sd(winpercent))

5. Using the single_candy_rankings, create a visualization to describe the relationship between sugarpercent and winpercent (where winpercent is on the y-axis). Describe the relationship that you see.

6. Examine the correlation between sugarpercent and winpercent using the cor() function by modifying the code below. Comment on the correlation coefficient - how does it compare to what you saw in Exercise 5.

single_candy_rankings %>%
  summarise(r = cor(---))

7. In the special case of a simple linear regression, you can calculate \(\hat{\beta}_1\) without having to use any calculus! It turns out the following equation works:

\[\hat{\beta}_1 = \hat{r} \frac{\hat{\sigma}_y}{\hat{\sigma}_x}\]

Where \(\hat{r}\) is the estimated correlation between \(x\) and \(y\), \(\hat{\sigma}_x\) is the standard deviation of \(x\), and \(\hat{\sigma}_y\) is the standard deviation of \(y\). Let’s see if we can calculate this. Edit the code below to calculate:

• sd_x: the standard deviation of sugarpercent
  
• sd_y: the standard deviation of winpercent
  
• r: the correlation between sugarpercent and winpercent
• beta1: The slope of the simple linear regression between winpercent ~ sugarpercent (I have written the equation for beta for you, you do not have to edit this final part).

single_candy_rankings %>%
  summarise(sd_x = ---,
            sd_y = ---,
            r = ---,
            beta1 = r * (sd_y / sd_x))

8. We are interested in fitting a simple linear regression of the form:

\(y = \beta_0 + \beta_1x + \epsilon\)

where \(y\) is winpercent and \(x\) is sugarpercent. Use the lm() function to fit this model. Then use the tidy() function from the broom package to output a tidy table of the model output.

lm(---, data = single_candy_rankings) %>%
  ---()

9. What is \(\hat{\beta}_0\)? What is \(\hat{\beta}_1\)? Interpret these results in words. How does this \(\hat{\beta}_1\) compare to the one you calculated in Exercise 7?

10. Calculate the residuals for the model fit in Exercise 8. Create a scatterplot of the residuals by \(\hat{y}\), the predicted outcome, winpercent. Do the conditions of linearity and constant variance seem to hold?

y_hat <- lm(---, data = single_candy_rankings) %>%
  ---()

single_candy_rankings <- single_candy_rankings %>%
  mutate(---)

11. Plot the residuals using a histogram with binwidth = 9 to observe whether they follow a bell-shaped distribution. Comment on what you see.

12. Calculate the regression standard error for the model created in Exercise 8. How do you interpret this result?

Grading