---
title: "Project 1 - Key"
output:
  word_document:
    reference_docx: WordforProject1.docx
  html_document: default
---

```{r echo=FALSE,warning=FALSE}
library(readr)
```
```{r echo=FALSE}
#put a path to wherever student datafiles are stored
path="../../Data/Project1/"
#edit "Mystudent.csv" in the chunk below for a specific student's data
```

```{r message=FALSE}
cars=read_csv(paste(path,"Mystudent.csv",sep=""),show_col_types = FALSE)
```
#### #2. Summary statistics
```{r echo=F, results=F}
  age=cars$age
  price=cars$price
  n=length(age)

  ssx=sum((age-mean(age))^2)
  ssy=sum((price-mean(price))^2)
  ssxy=sum((age-mean(age))*(price-mean(price)))

  SSModel=(ssxy^2)/ssx
  SSE=ssy-SSModel
```
$\overline{age}=`r round(mean(age),2)`$,  $s_{age}=`r round(sd(age),2)`$,    $\qquad \overline{price}=`r round(mean(price),2)`$, $s_{price}=`r round(sd(price),2)`$ 

$SSX=`r round(ssx,1) `$,   $SSY=`r round(ssy,1) `$,  $SSXY=`r round(ssxy,1)`$, $SSModel=`r round(SSModel,1) `$, $SSE= `r round(SSE,1) `$

#### #3. Least squares line

```{r echo=F, results=F}
  b1=ssxy/ssx
  b0=mean(price)-b1*mean(age)
  Se=sqrt(SSE/(n-2))
```
$\hat{\beta}_1=\frac{`r round(ssxy,1)`}{`r round(ssx,1)`}=`r round(b1,3)`$ and $\hat{\beta}_0=`r mean(price,1)`+`r round(b1,3)`*`r round(mean(age),1)`=`r round(b0,2)`$ so $\widehat{price}=`r round(b0,2)`+`r round(b1,3)`*age$

#### #4. Std. dev of error

$\hat{\sigma}_\epsilon=\sqrt{\frac{`r round(SSE,1)`}{`r n`-2}}=\sqrt{`r round(Se^2,2)`}=`r round(Se,2)`$

##### #5  Scatterplot with regression line & residual plots

```{r fig.width=7, fig.height=2,echo=F}
par(mfrow=c(1,3))
par(mar=c(4,4,2,2))
model=lm(price~age)
resid=model$residuals  
  plot(price~age,main=" Price vs. Age")
  abline(model)
  abline(h=0,lty=2)
  
  plot(resid~model$fitted,main="Residuals vs. Fits")
  abline(0,0)
  
 # hist(resid)
  
  qqnorm(resid)
  qqline(resid)
```

#### #6 Largest residual 
```{r echo=F,resutls=F}
pos=which.max(abs(resid))
```
Case #`r pos` has residual=`r round(resid[pos],2)` with studentized residual=`r round(rstudent(model)[pos],2)`,
     Hi =`r round(hatvalues(model)[pos],3)` and Cook's D=`r round(cooks.distance(model)[pos],3)`<br>
Criteria for leverage are 4/n=`r round(4/n,3)` (mild) and 6/n=`r round(6/n,3)` (severe)

#### #7 CI for slope

```{r echo=F,results=F}
tstar=qt(0.95,n-2)
  SEslope=Se/sqrt(ssx)
  lslope=b1-tstar*SEslope
  uslope=b1+tstar*SEslope
```
$`r round(b1,3)` \pm `r round(tstar,3)`*`r round(SEslope,3)`= (`r round(lslope,3)` , `r round(uslope,3)`)$

#### #8 Three tests for a relationship
```{r echo=F, results=F}
F=SSModel/(SSE/(n-2))
pvalue=1-pf(F,1,n-2)
r=cor(price,age)
```

$t=\frac{`r round(b1,3)` }{`r round(SEslope,3)` }=\frac{`r round(r,3)`\sqrt{`r n`-2}}{\sqrt{1-(`r round(r,3)`^2 )}}=`r round(b1/SEslope,3)`$ and $F=\frac{`r round(SSModel,1)`}{`r round(SSE/(n-2),1)`}=`r round(F,2)`$ with p-value=`r pvalue`


#### #9. R-squared

$r^2 =(`r round(r,3)`)^2 =\frac{`r round(SSModel,1)`}{`r round(ssy,1)`}=`r round(r^2,3)`$


#### #10 Prediction for age=`r agestar` we have $\widehat{price}= `r round(b0+b1*agestar,2)`$

```{r echo=F, results=T}

yhat=b0+b1*agestar
  d=1/n+(agestar-mean(age))^2/ssx
  lmean=yhat-tstar*Se*sqrt(d)
  umean=yhat+tstar*Se*sqrt(d)
  lpred=yhat-tstar*Se*sqrt(1+d)
  upred=yhat+tstar*Se*sqrt(1+d)
```
$$CI=`r round(yhat,2)` \pm `r round(tstar,3)`*`r round(Se,3)`\sqrt{\frac{1}{`r n`}+\frac{(`r agestar`-`r round(mean(age),2)`)^2}{`r ssx`}}=(`r round(lmean,2)` , `r round(umean,2)`)$$
$$ PI= (`r round(lpred,2)` , `r round(upred,2)`)$$

#### #11  Free car is at age=`r round(-b0/b1,1)`