---
title: "Math 50 Fall 2017, Homework #2"
---
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__NOTE: For your homework download and use the template__ (https://math.dartmouth.edu/~m50f17/HW2.Rmd)

__Read the green comments in the rmd file to see where your answers should go.__


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## Question-1 (Sample)
Given a fixed confidence interval percentage (say 95%) at what value of x does CI on the mean response achieve its minimum width?    

Write an R-chunk using the propellant data which computes the following. 

(a) Fit a simple linear regression model relating shear strength to age.
(b) Plot scatter diagram.
(c) Plot two curves (in blue color) that traces upper and lower limits of 95% confidence interval on $E(y|x_0)$
(d) Plot two curves (in red color) that traces upper and lower limits of 95% prediction interval for $y$ 
(e) Print the 95% quantile of the corresponding t distribution 


### Answer: 
<!-- If question requires a written answer it goes here. R code goes to the chunk below.) -->

The width of the interval is 
$$ 2 t_{\alpha/2,n-2} \sqrt{MS_{Res} ((1/n) + (x_0 - \bar{x})^2 / S_{xx}   } $$ 
and all terms inside the square root are positive. Therefore it is minimized when $x_0=\bar{x}$. 



```{r}

# Computation part of the answer : 

prop<-read.table("https://math.dartmouth.edu/~m50f17/propellant.csv", header=T, sep=",")

shearS<-prop$ShearS 
age<-prop$Age

plot(age, shearS, xlab = "Propellant Age (weeks)", ylab = "Shear S. (psi)", main = "Rocket Propellant")
fitted <- lm(shearS ~ age)

ageList <- seq(0,25,0.5)
cList <- predict(fitted, list(age = ageList), int = "c", level = 0.95)
pList <- predict(fitted, list(age = ageList), int = "p", level = 0.95)

matlines(ageList, pList, lty='solid' ,  col = "red")
matlines(ageList, cList, lty = 'solid', col = "blue") 

# since n=20 we look at the t_18 distribution
wantedQuantile <- qt( 0.95, 18) ; 
cat("95% quantile is of t_18 is : ", wantedQuantile ) ; 




```



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## Question-2
Plot the same graph as in Question-1 without using R function predict, but instead directly calculating the interval limits we discussed in class. 
In particular, what are the limits of 95% confidence interval on $E(y|x_0)$?  

### Answer: 
<!-- If question requires a written answer it goes here. R code goes to the chunk below.) -->

```{r}
# Computation part of the answer : 


```


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## Question-3
Load the propellant data and fit a simple linear regression model relating shear strength to age.

(a) Test the hypothesis $\beta_1 = -30$ using confidence level 97.5%.
(b) Calculate the limits of 97.5% confidence interval for $\beta_0$ and $\beta_1$  
(c) Is there any relation between the answers you find in part (a) and (b) ?
(d) Calculate $R^2$


### Answer: 
<!-- If question requires a written answer it goes here. R code goes to the chunk below.) -->

```{r}
# Computation part of the answer : 


```


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## Question-4
Load the propellant data. This time let us consider a relation between square of shear strength and propellant age. 

(a) Fit a simple linear regression model relating __square__ of shear strength to age. Plot scatter diagram and fitted line. 
(b) Using analysis-of-variance test for significance of regression (using the formulas we discussed in class)
(c) Use t-test and check significance of regression (using the formulas we discussed in class)
(d) Does the regression analysis predict a linear relationship between square of shear strength and propellant age ?

### Answer: 
<!-- If question requires a written answer it goes here. R code goes to the chunk below.) -->

```{r}
# Computation part of the answer : 


```



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## Question-5
Once again using propellant data fit a simple linear regression model between shear strength and propellant age. 
Consider the steps that we used to obtain t-test for hypothesis $\beta_1 = G_1$, and following similar steps in order to develop a test for $\beta_1 > G_1$ instead. Then 

(a)  Test the ~~~hypothesis~~~ statement $$\beta_1 > -50$$ with confidence level 99.9%. 
(b)  Find the smallest value $G_1$ such that the above ~~~hypothesis~~~ statement  is rejected.
(c)  Similarly what is the smallest value $G_0$ such that the statement "$\beta_0 > G_0$ with probability 0.999" is rejected. 
 


### Answer: 
<!-- If question requires a written answer it goes here. R code goes to the chunk below.) -->

```{r}
# Computation part of the answer : 


```

