---
title:  "STAT340: Discussion 7: Linear regression"
documentclass: article
classoption: letterpaper
output:
  html_document:
    highlight: tango
    fig_caption: false
---

<style>
table{width:50%!important;margin-left:auto!important;margin-right:auto!important;}
ol[style*="decimal"]>li{margin-top:40px!important;}
</style>

```{r setup, include=FALSE}
# check packages installed
knitr::opts_chunk$set(echo=T,tidy=F,strip.white=F,fig.align="center",comment=" #",message=F,warning=F)
options(width=120)
```

<br/>

[link to source](ds07.Rmd)

## XKCD comic

<center><a href="https://xkcd.com/1725/"><img id="comic" src="https://imgs.xkcd.com/comics/linear_regression.png" title="The 95% confidence interval suggests Rexthor's dog could also be a cat, or possibly a teapot."></a></center>

---

## Exercise

Today we're going to use the built-in `mtcars` dataset to practice simple linear regression. Note this is a built-in dataset provided as part of the `datasets` package in `R`.


### Background

Run `?(mtcars)` in the console (do NOT add it to this `Rmd` file) and briefly read the help page. Specifically, take note of the following:

1. What is the source of this data?
2. What is this dataset measuring? (i.e. what is the response variable?)
3. What predictors are available and what do they mean?

Feel free to also run `head(mtcars, 10)` or `View(mtcars)` to inspect the data frame briefly before moving on.


### Fitting

Uncomment the line below and finish it. Specifically, use `lm` to run a regression of `mpg` on one other predictor (an easy way to do this is to use `mpg ~ var` where `var` is the predictor you're using). Make sure to also include `data = mtcars` as an argument or it won't know where to get the variable names from.

```{r}
# lm.mtcars = lm(...)
```

View a summary of the regression by uncommenting and running the line below

```{r}
# summary(lm.mtcars)
```


Briefly inspect the residuals plot by running `plot(lm.mtcars,which=1:2)` . What do you observe, and what does it mean?

> _REPLACE TEXT WITH RESPONSE_


### Interpretation

Uncomment the line below to get the estimated coefficients along with their standard errors.

```{r}
# summary(lm.mtcars)$coefficients[,1:2]
```

Give an interpretation of the estimate and standard error for your predictor. Be careful in your wording of the interpretation.

> _REPLACE TEXT WITH RESPONSE_

What does the intercept here mean? (Except for special situations, we generally don't care much about the intercept, but you should still understand what it means.)

> _REPLACE TEXT WITH RESPONSE_

What is the R² for this model? (Hint: look at the output of `summary`) Give an interpretation of this value.

> _REPLACE TEXT WITH RESPONSE_

Briefly read about the [adjusted R² here](https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/adjusted-r2/). What is the adjusted R² of this model and how does this differ from the normal R² value? (Hint: again, look at the output of `summary`).

> _REPLACE TEXT WITH RESPONSE_

Generate $95\%$ confidence intervals for the coefficients using the `confint` function. Give an interpretation of these confidence intervals.

```{r}
# confint(...)
```

> _REPLACE TEXT WITH RESPONSE_


# Try with others!

Repeat the steps above for at least 1 other predictor. Which of these two predictors seems to offer a better "predictive" ability for `mpg`? How do you know?