__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.__

## 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: 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 ) ; ```

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

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

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

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