**NOTE: For your homework download and use the template** (https://math.dartmouth.edu/~m50f17/HW3.Rmd)

**Read the green comments in the rmd file to see where your answers should go.**

Is maximum-likelihood estimator \(\tilde{\sigma}^2\) of \(\sigma^2\) an unbiased estimator ? Verify your answer. Comment on the change of the value of \[E(\tilde{\sigma}^2) - {\sigma}^2\] as \(n\) goes to infinity.

Consider the shear strength vs age relation using the propellant data.

Recalculate the coefficients of the fitted linear regression model using the vector equations we obtained.

Suppose that the expectation of the initial shear strength is known to be 2400. Write the corresponding model (should involve only one parameter \(\beta_1\)). Calculate 95% CI on \(\beta_1\).

`# Computation part of the answer : `

A scientist is trying to model the relation between phytoplankton population in the city public water supply and concentration of two substances. The sample data is at : https://math.dartmouth.edu/~m50f17/phytoplankton.csv

where headers are

- pop : population of phytoplankton (\(y\))
- subs1 : concentration of substance-1 (\(x_1\))
- subs2 : concentration of substance-2 (\(x_2\))

Lets consider a guessed model

\[ y = 200 + 10x_1 -15x_2 \] Below is the corresponding code to plot the scatter diagram and the above plane.

```
# Note: Run the following in R console if you get errors in plotting or library loading :
# install.packages("scatterplot3d")
# install.packages("plot3D")
library("plot3D")
library("scatterplot3d")
# Loading data
pData <- read.table("https://math.dartmouth.edu/~m50f17/phytoplankton.csv", header=T, sep=",")
pop <- pData$pop
subs1 <- pData$subs1
subs2 <- pData$subs2
# Create a mesh
meshP <- mesh( seq(min(subs1),max(subs1),0.03) , seq(min(subs2),max(subs2),0.03) )
x1Mesh <- meshP$x
x2Mesh <- meshP$y
myModel <- 200 + 10*x1Mesh - 15 *x2Mesh
# Below is the code to plot the scatter diagram with red markers and your model
# You need to set two variables before calling :
# myModel : your model
# PLOTTING # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #
sc1 <- scatterplot3d(subs2,subs1,pop, pch=17 , type = 'p', angle = 15 , highlight.3d = T )
sc1$points3d (x2Mesh,x1Mesh, myModel, cex=.02, col="blue")
```

```
# You can also change the view angle to visually test the model
# PLOTTING # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #
sc1 <- scatterplot3d(subs2,subs1,pop, pch=17 , type = 'p', angle = 125 , highlight.3d = T )
sc1$points3d (x2Mesh,x1Mesh, myModel, cex=.02, col="blue")
```