---
output: word_document
---
  Project #2
========================================================

```{r echo=FALSE}
library(readr)
#need to use the slunova() function to provide modified ANOVA output - put a path to it here
source("../../RScripts/slunova.R")
```

```{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)
```

**Model #1 Age & Mileage**
```{r}
mod1=lm(price~age+mileage,data=cars)
```

```{r echo=F}
x=summary(mod1)
x$coeff
a=anova(mod1)
```
(1) R-sq=`r round(x$r.squared,3)` , adjR-sq=`r round(x$adj.r.squared,3)`, $\hat{\sigma}_\epsilon=$ `r round(x$sigma,3)`, 
F=`r round(x$fstatistic[1],2)` with 2 and `r x$df[2]` df,  p-value=`r 1-pf(x$fstatistic[1],2,x$df[2])`
```{r echo=F, results=F}
aage=cars$age[1]
apred=round(mod1$fitted[1],2)
aprice=cars$price[1]
amiles=cars$mileage[1]
ares=round(mod1$residual[1],2)
```
(2) For age=`r aage` and mileage=`r amiles` we have $\widehat{price}=$ `r apred` and residual=`r ares`

```{r echo=F}
source("../../RScripts/VIF.R")
v=vif(mod1)
```
(5) VIF=`r round(v[1],2)`



**Model #2: Polynomial**  
```{r}
mod2=lm(price~age+I(age^2),data=cars)
```
```{r echo=F}
m=mod2$coeff
```
(6-7) $\widehat{price}=`r m[1]`+`r m[2]`age+`r m[3]`age^2$
  
```{r fig.width=4.3, fig.height=4.3,echo=F}
plot(price~age,main="Quadratic",xlim=c(0,30),ylim=c(0,max(cars$price)+2),data=cars)
curve(m[1]+m[2]*x+m[3]*x^2,add=T)
abline(0,0)
```
```{r}
newx=data.frame(age=3)
forecast=predict.lm(mod2,newx,interval="prediction")
```
(7) The predicted price for a 3 year old car is \$`r round(forecast[1],3)` thousand and we are 95\% sure that the price will be between \$`r round(forecast[2],3)` thousand and \$`r round(forecast[3],3)` thousand.

(10) Cubic model
```{r}
modCube=lm(price~age+I(age^2)+I(age^3),data=cars)
```
```{r echo=F}
x=summary(modCube)
x$coeff
```
   Cubic: R-sq=`r round(x$r.squared,3)` , adjR-sq=`r round(x$adj.r.squared,3)`, $\hat{\sigma}_\epsilon=$ `r round(x$sigma,3)`, 
F=`r round(x$fstatistic[1],2)` with 3 and `r x$df[2]` df,  p-value=`r noquote(format(1-pf(x$fstatistic[1],2,x$df[2]),digits=2))`
```{r echo=F}
x=summary(mod2)
```
Quadratic: R-sq=`r round(x$r.squared,3)` , adjR-sq=`r round(x$adj.r.squared,3)`, $\hat{\sigma}_\epsilon=$ `r round(x$sigma,3)`, 
F=`r round(x$fstatistic[1],2)` with 2 and `r x$df[2]` df,  p-value=`r noquote(format(1-pf(x$fstatistic[1],2,x$df[2]),digits=2))`

**Model #3: Complete Second Order**

(11) $price=\beta_0+\beta_1age+\beta_2miles+\beta_3age^2+\beta_4miles^2+\beta_5age*miles+\epsilon$ 

```{r}
mod3=lm(price~age+mileage+I(age^2)+I(mileage^2)+I(age*mileage),data=cars)
```

(12) Fit the second order model  
```{r echo=F}
x=summary(mod3)
noquote(format(mod3$coeff,scientific=FALSE,digits=4))
```
```{r echo=F}
xa=slunova(mod3)
xa[1:5]
```

(13) $H_0:\beta_3=\beta_5=0$  vs. $H_a: \beta_3 \ne 0$ or $\beta_5 \ne 0$

```{r echo=F}
mod4=lm(price~age+mileage+I(mileage^2),data=cars)
xb=slunova(mod4)
xb[1:5]
```

  
```{r echo=F}
SSMFull=xa[1,2]
SSMRed=xb[1,2]
SSE=xa[2,2]
dfE=xa[2,1]
MSEFull=xa[2,3]
Delta=SSMFull-SSMRed
Fstat=(Delta/2)/MSEFull
pv=1-pf(Fstat,2,dfE)
```
$F=\frac{(`r round(SSMFull,1)`-`r round(SSMRed,1)`)/2}{`r round(SSE,1)`/`r dfE`}=\frac{(`r round(Delta,1)`)/2}{`r round(SSE,1)`/`r dfE`}=\frac{`r round(Delta/2,1)`}{`r round(MSEFull,1)`}=`r round(Fstat,2)`$
  
  p-value with 2 and `r dfE` df is `r round(pv,5)`.