Important

Go to the course GitHub organization and locate the repo titled `ae-4-YOUR_GITHUB_USERNAME` to get started.

## Packages

``````library(tidyverse)
library(tidymodels)
library(ggfortify)
library(knitr)``````

## Restaurant tips

What factors are associated with the amount customers tip at a restaurant? To answer this question, we will use data collected in 2011 by a student at St. Olaf who worked at a local restaurant.1

The variables we’ll focus on for this analysis are

• `Tip`: amount of the tip
• `Party`: number of people in the party

View the data set to see the remaining variables.

``tips <- read_csv("data/tip-data.csv")``

## Exploratory analysis

1. Visualize, summarize, and describe the relationship between `Party` and `Tip`.
``# add your code here``

## Modeling

Let’s start by fitting a model using `Party` to predict the `Tip` at this restaurant.

1. Write the statistical model.

2. Fit the regression line and write the regression equation. Name the model `tips_fit` and display the results with `kable()` and a reasonable number of digits.

``# add your code here``
1. Interpret the slope.

2. Does it make sense to interpret the intercept? Explain your reasoning.

3. What is the R-squared and RMSE for this model? How do you evaluate your model’s fitting performance? Explain your reasoning.

## Inference

### Inference for the slope

1. The following code can be used to create a bootstrap distribution for the slope (and the intercept, though we’ll focus primarily on the slope in our inference). Describe what each line of code does, supplemented by any visualizations that might help with your description.
``````set.seed(1234)

boot_dist <- tips %>%
specify(Tip ~ Party) %>%
generate(reps = 100, type = "bootstrap") %>%
fit()``````
1. Use the bootstrap distribution created in Exercise 7, `boot_dist`, to construct a 90% confidence interval for the slope using bootstrapping and the percentile method and interpret it in context of the data.
``# add your code here``
1. Conduct a hypothesis test at the equivalent significance level using permutation. State the hypotheses and the significance level you’re using explicitly. Also include a visualization of the null distribution of the slope with the observed slope marked as a vertical line.
``# add your code here``
1. Now repeat Exercises 8 and 9 using approaches based on mathematical models.
``# add your code here``

### Inference for a prediction

1. Based on your model, predict the tip for a party of 4.
``# add your code here``
1. Suppose you’re asked to construct a confidence and a prediction interval. Which one would you expect to be wider and why? In your answer clearly state the difference between these intervals.

2. Now construct the intervals from Exercise 11 and comment on whether your guess is confirmed.

``# add your code here``

## Footnotes

1. Dahlquist, Samantha, and Jin Dong. 2011. “The Effects of Credit Cards on Tipping.” Project for Statistics 212-Statistics for the Sciences, St. Olaf College.↩︎