---
title: "Using RMarkdown to Produce Reports from R Scripts"
output: pdf_document
---

This document shows you how to use the \texttt{R Markdown} in  \texttt{RStudio} to create pdf documents based on \texttt{R} scripts. This $x=1$

\[
X=(x_1,x_2,\cdots,x_p)
\]

The file that created this pdf document is \texttt{1-lm.Rmd} for use in \texttt{RStudio}.

## Create an R Markdown file

Using \texttt{RStudio}:  you can open the \texttt{1-lm.Rmd} file directly in \texttt{RStudio}, and compile
it using the `Knit' button at the top of the R script window that opens automatically when you click on
the \texttt{.Rmd} file.  Note that you can change the setting of \texttt{R Markdown} in *Tools $\rightarrow$ Global Options* pulldown menu, and hitting the R Markdown symbol in the left-hand sidebar. 

## Load and plot the data

Here is a plot of the rocket propulsion data from the textbook, as studied in class:


```{r}
data.source<-"http://www.math.mcgill.ca/yyang/regression/data/2-1-RocketProp.csv"
RocketProp<-read.csv(file=data.source)
names(RocketProp)<-c('i','Strength','Age')
x<-RocketProp$Age
y<-RocketProp$Strength
plot(x,y,pch=19,cex=0.6,xlab='Age',ylab='Shear Strength')
xmean<-mean(x)
ymean<-mean(y)
abline(v=xmean,h=ymean,lty=2)
```


The mean of the $x$ values is `r xmean`, and the mean of the $y$ values is `r ymean`.

## Fit the regression

```{r}
############################################
#Fit the simple linear regression using lm

fit.RP<-lm(y~x)
summary(fit.RP)
coef(fit.RP)
```

From this, we can deduce that the estimate of the intercept is $\widehat \beta_0= `r coef(fit.RP)[1]`$, 
and the estimate of the slope is $\widehat \beta_1= `r  coef(fit.RP)[2]`$, that is, the line of best fit is
\[
y = `r coef(fit.RP)[1]` + `r coef(fit.RP)[2]` x
\]
However, this is reporting the estimates to too many decimal places: we can reduce that by using the \texttt{round}
function: for example \texttt{round(coef(fit.RP)[1],3)} gives the result to 3 decimal places.  We have
\[
y = `r round(coef(fit.RP)[1],3)` + `r round(coef(fit.RP)[2],3)` x.
\]
The line of best fit is good (see plot below).


Here is an alternative way to use the \texttt{lm} function in \texttt{R}, by using 
the \texttt{data=RocketProp} argument; this allows you to call the variables by name 
in the command
\[
\texttt{fit.RP<-lm(Strength~Age,data=RocketProp)}
\]
```{r}
fit.RP<-lm(Strength~Age,data=RocketProp)
plot(x,y,pch=19,cex=0.6,xlab='Age',ylab='Shear Strength')
coef(fit.RP)
abline(coef(fit.RP),col='red')
title('Line of best fit for Rocket Propulsion Data')
```