# 2 Fundamentals of Statistical Analysis question, R required, 20 hours deadline

## Questions:

Question 1 (40 points):

Regression and MLE We are interested in estimating the median home value in New England. For this, we employ a regression from the origin

as presented below:

Where

is median home value in New England town

, and

is a binary variable that equals to 1 if the house is in town

and equals to 0 otherwise.

Let

be independent where

- (15 points) Find the MLE of
β

$\beta $,

${\hat{\beta}}_{MLE}$β̂ MLE .

- (15 points) Find the MLE of
σ2

${\sigma}^{2}$,

${\hat{\sigma}}_{MLE}^{2}$σ̂ 2MLE .

- (10 points) Show that sums of squares of error, SSE, can be written as:

Question 2 (40 points): Confidence Interval

Let

still be the median home value in New England town

. Let the generated

below to be the entire population data on median value of NEw England homes, where

and

.

```
set.seed(12)
Y=rnorm(1000, mean=329108, sd=50000)
```

*For steps 1 and 2 to let’s present we do not know μ*

*$\mu $*

*.*

- (5 points) Take 100 samples of size 30 (without replacement) from the population of
Y

$Y$’s

- (10 points) Calculate a 95% confidence interval for
μ

$\mu $for all of the 100 samples.

- (10 points) How many of these samples include the true mean
μ=

$\mu =$?

- (15 points) Repeat steps b and c for 90% confidence intervals.

Question 3 (20 points) Regression Estimation

- (7 points) Using the
*synthetic data*provided below on median home values (Y

$Y$) and towns in New England

$\left(X\right)$(X) , estimate the regression from question 1, i.e.,

Are the coefficients statistically significant? Do not forget to use `factor(X)`

as opposed to `X`

in your regression!!

```
housing=read.table("https://unh.box.com/shared/static/twmyqbvx0toxhvdv0n23c55e5cc3ipe4.csv", header = TRUE, sep=",", dec=".")
head(housing)
```

```
## Y X
## 1 426419.3 7
## 2 416306.1 8
## 3 344116.1 9
## 4 453613.3 7
## 5 303323.9 5
## 6 314420.3 6
```

- (13 points) Check the residuals of the model. Are the assumptions satisfied? Why? Why not?

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