STAT GU5205 Homework 1 [100 pts]
Due 8:40am Monday, September 23rd
Problem 1 (1.22 KNN)
Sixteen batches of the plastic were made, and from each batch one test item was molded.
Each test item was randomly assigned to one of the four predetermined time levels, and
the hardness was measured after the assigned elapsed time. The results are shown below;
X is the elapsed time in hours, and Y is hardness in Brinell units. Assume the first-order
regression model (1.1) is appropriate (model (2.1) in the notes).
Data not displayed
Perform the following tasks:
i. Use R to obtain the estimated regression function.
ii. Use R to create a scatter plot with the line of best fit. Make the line of best fit red.
iii. Use R to calculate the best point estimate of σ2.
iv. Use R to calculate the sample correlation coefficient and coefficient of determination.
Problem 2
Recall the sample residual is defined by ei = yi− yˆi, where yi is the ith response value and yˆi
is its corresponding fitted value computed by least squares estimates yˆi = βˆ0 + βˆ1xi. Prove
the following properties:
i.
n∑
i=1
xiei = 0
ii.
n∑
i=1
yˆiei = 0
1
Problem 3
Recall that the ith fitted value Yˆi can be expressed as a linear combination of the response
values, i.e.,
Yˆi =
∑
j=1
hijYj,
where
hij =
1
n
+
(xi − x¯)(xj − x¯)
Sxx
,
and
Sxx =
∑
i=1
(xi − x¯)2.
Prove the following properties of the hat-values hij.
i. ∑
j=1
h2ij = hii
ii. ∑
j=1
hijxj = xi
Problem 4
Consider the regression through the origin model given by
(1) Yi = βxi + i i = 1, 2, . . . , n i
iid∼ N(0, σ2).
The estimated model at observed point (x, y) is
yˆ = βˆx,
where
(2) βˆ =
∑n
i=1 xiyi∑n
i=1 x
2
i
.
Complete the following tasks
i. Show that
βˆ =
∑n
i=1 xiYi∑n
i=1 x
2
i
is an unbiased estimator of β.
ii. Compute the standard error of estimator βˆ.
iii. Identify the probability distribution of estimator βˆ.
2