Question

# In a small-scale regression study, we collected data on the number of children in a family...

In a small-scale regression study, we collected data on the number of children in a family Xi and the number of hours per week spent shopping Yi. 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 Yi on Xi, i.e. E(Yi) = ?0 + ?1Xi

(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(Yh) and a 95% prediction interval for Yh(new) at Xh = 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

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)

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