Question

In a small-scale regression study, we collected data on the number of children in a family X_i and the number of hours per week spent shopping Y_i. The following data were obtained:

i	1	2	3	4	5	6
Xi	2	6	3	1	1	9
Yi	13	17	12	12	9	22

Assume we performed a simple linear regression of Y_i on X_i, i.e. E(Y_i) = ?₀ + ?₁X_i

(a) By hand compute X?X, X?Y, (X?X)^-1, b, Y^{^}(means Y-hat), and e. What are the dimensions of each matrix?

(b) Create a simple linear regression in R(R-program) and write down the fitted equation and the value of MSE. Interpret the coefficient b₁.

(c) Use the MSE value to compute (Var(b))^{^}(means Var(b) with a hat over it). From this matrix find estimates for Var(b₀), Var(b₁), and Cov(b₀, b1).

(d) Construct a 95% confidence interval for ?₁. Use the matrix formula from Chapter 6. Interpret this interval.

(e) Construct a 95% confidence interval for E(Y_h) and a 95% prediction interval for Y_h(new) at X_h = 4.

(f) In R reproduce parts (a), (c), (d), and (e). When recreating the results from parts (d) and (e), compute the confidence intervals and prediction intervals two ways:

•Using matrix notation

•Using either the confint or predict functions

Also compute the hat matrix H. What is the dimension of H? Also use H to compute Y and compare to the result in part (a).

Template for R parts of problem below.

# Type in children (x) and hours (y) as vectors. Set n to be the length

children <-

hours <-

n <-

##############################

########## Part (b) ##########

##############################

# Create a simple linear regression Y ~ X

fit <-

# Circle the value used to find MSE

summary(fit)

MSE <- # Type a number or use sigma() function

##############################

########## Part (f) ##########

##############################

# Create X and Y matrices.

ones <-

X <-

X # Displays matrix X

Y <-

Y # Displays matrix Y

# Display the dimensions of each of the X and Y matrices. See Chapter 5 Notes

# Calculate X'X, X'Y, (X'X)^(-1), and b.

XX <-

XY <-

XXinv <-

b <-

Yhat <-

e <-

# Display matrices (don't change)

XX

XY

XXinv

b

Yhat

e

# Calculate the variance-covariance matrix Var(b) two different ways

# 1: using MSE and XXinv

# 2: using the vcov function

Var.b1 <- # Method 1

Var.b1 # Displays matrix

Var.b2 <- # Method 2

Var.b2 # Displays matrix

# Construct a 95% confidence interval for beta1 two different ways

# 1: using matrix notation

# 2: using the confint function

beta1.lower <- # Method 1 lower

beta1.upper <- # Method 1 upper

c(beta1.lower, beta1.upper) # Displays interval (don't change)

beta1.int <- # Method 2

beta1.int # Displays interval

# Construct a 95% confidence interval for E(Y) when children=4 two different ways

# 1: using matrix notation

# 2: using the predict function

Xh <- # Enter vector Xh

Yh <- # Compute vector Yh using matrices

EY.lower <- # Method 1 lower

EY.upper <- # Method 1 upper

c(EY.lower, EY.upper) # Displays interval (don't change)

EY.int <- # Method 2

EY.int # Displays interval

# Construct a 95% prediction interval for Y when children=4 two different ways

# 1: using matrix notation

# 2: using the predict function

Y.lower <- # Method 1 lower

Y.upper <- # Method 1 upper

c(Y.lower, Y.upper) # Displays interval (don't change)

Y.int <- # Method 2

EY.int # Displays interval

# Compute the Hat matrix, H and Yhat. Find the dimension of H

H <-

H # Displays matrix

Yhat <-

Yhat

Answer 1

x'x is (2x2), x'y is (2x1) and (x'x)^-1 is 2x2 matrix

y	b	yhat	e	sse	sigma2	varb
13	9.0714286	11.850649	1.1493506	7.7077922	2.5692641	1.1011132	-0.183519
17	1.3896104	17.409091	-0.409091			-0.183519	0.0500506
12		13.24026	-1.24026
12		10.461039	1.538961
9		10.461039	-1.461039
22		21.577922	0.4220779

regression equation y=b0+b1*x=9.0714+1.3896x

SE(b1)=sqrt(0.05)= 0.2236

95% confidence interval for b1=1.3896t(0.05/2,errror df=4)*SE(b1)=1.38962.7764*0.2236=1.38960.6208=

(0.7688, 2.0104)