---
title: "Lab 9"
author: "Your Name Here"
date: ""
output: html_document
---

##### Remember to change the `author: ` field on this Rmd file to your own name.

### Learning objectives

> In today's Lab you will gain practice with the following concepts from today's class:

>- Interpreting linear regression coefficients of numeric covariates
- Interpreting linear regression coefficients of categorical variables
- Applying the "2 standard error rule" to construct approximate 95% confidence intervals for regression coefficients
- Using the `confint` command to construct confidence intervals for regression coefficients
- Using `pairs` plots to diagnose collinearity
- Using the `update` command to update a linear regression model object
- Diagnosing violations of linear model assumptions using `plot`

We'll begin by loading some packages.
```{r}
library(tidyverse)
library(knitr)

Cars93 <- as_tibble(MASS::Cars93)
# If you want to experiment with the ggpairs command,
# you'll want to run the following code:
# install.packages("GGally")
# library(GGally)
```

### Linear regression with Cars93 data

**(a)** Use the `lm()` function to regress Price on: EngineSize, Origin, MPG.highway, MPG.city and Horsepower.

```{r}
# Edit me
```

**(b)** Use the `kable()` command to produce a nicely formatted coefficients table.  Ensure that values are rounded to an appropriate number of decimal places.

```{r}
# Edit me
```

> Replace this text with your answer.


**(c)** Interpret the coefficient of `Originnon-USA`.  Is it statistically significant?

```{r}
# Edit me
```

> Replace this text with your answer.

**(d)** Interpret the coefficient of `MPG.highway`.  Is it statistically significant?

```{r}
# Edit me
```

> Replace this text with your answer.


**(d)** Use the "2 standard error rule" to construct an approximate 95% confidence interval for the coefficient of `MPG.highway`.  Compare this to the 95% CI obtained by using the `confint` command.

```{r}
# Edit me
```

> Replace this text with your answer.

**(e)**  Run the `pairs` command on the following set of variables: EngineSize, MPG.highway, MPG.city and Horsepower.  Display correlations in the   Do you observe any collinearities?  

```{r}
panel.cor <- function(x, y, digits = 2, prefix = "", cex.cor, ...)
{
    usr <- par("usr"); on.exit(par(usr))
    par(usr = c(0, 1, 0, 1))
    r <- abs(cor(x, y))
    txt <- format(c(r, 0.123456789), digits = digits)[1]
    txt <- paste0(prefix, txt)
    if(missing(cex.cor)) cex.cor <- 0.4/strwidth(txt)
    text(0.5, 0.5, txt, cex = pmax(1, cex.cor * r))
}


# Edit me
```

> Replace this text with your answer.

**(f)** Use the `update` command to update your regression model to exclude `EngineSize` and `MPG.city`.  Display the resulting coefficients table nicely using the `kable()` command.

```{r}
# Edit me
```


**(g)**  Does the coefficient of `MPG.highway` change much from the original model?  Calculate a 95% confidence interval and compare your answer to part (d).  Does the CI change much from before?  Explain.

```{r}
# Edit me
```

> Replace this text with your answer.

**(h)** Run the `plot` command on the linear model you constructed in part (f).  Do you notice any issues?

```{r, fig.height = 12, fig.width = 12}
# Edit me
```

> Replace this text with your answer.
