Statistics 103: Lab 9
LAB PROBLEMS
1. Let
X
and
Y
be two random variables such that
Y
=
α
+
βX
+
Z
where
Z
is independent of
X
and
Z
∼ N
(0
, σ
2
). A
random sample of size
n
= 15 from (
X, Y
) resulted
in the following statistics:
X
= 10
.
0897
(1)
Y
= 70
.
2221
(2)
15
X
i
=1
(
X
i

X
)
2
= 16
.
7527
(3)
15
X
i
=1
(
X
i

X
)
Y
i
= 30
.
5332
(4)
Find
ˆ
β
, ˆ
α
. If ˆ
σ
2
= 3
.
7272 what is your estimated
standard deviation of
ˆ
β
and ˆ
α
?
answer:
ˆ
β
= 1
.
8226
,
ˆ
α
= 51
.
8327
;
0
.
4717
,
4
.
7852
2. Referring to the data in the problem 1 which of
the following intervals best summarizes the likely
values of the true
β
:
(1
.
8
,
2
.
3)
1
.
8226
±
4
1
.
8226
±
0
.
4717
1
.
8226
±
1
.
9306
answer:
1
.
8226
±
0
.
4717
3. Referring to the data in the problem 1 which of
the following intervals best summarizes the likely
values of the true
α
:
(50
,
60)
(46
,
56)
(25
,
75)
51
.
8327
±
1
.
9306
answer:
(46
,
56)
4. Which of the following conclusions follow from the
data presented in problem 1:
There is strong evidence that
β >
0
There is weak evidence that
β >
0
The data is consistent with
β
= 0
answer: There is strong evidence that
β >
0
5. Which of the following conclusions follow from the
data presented in problem 1:
There is strong evidence that
β <
2
There is weak evidence that
β <
2
The data is consistent with
β
= 2
answer: The data is consistent with
β
= 2
6. Which of the following conclusions follow from the
data presented in problem 1:
There is strong evidence that
α <
61
There is weak evidence that
α <
61
The data is consistent with
α
= 61
answer: There is weak evidence that
α <
61
7. Consider randomly picking a married couple from
Minnesota. Let
W
denoted the height of the wife
(in inches) and let
H
denoted the height of the
husband (in inches). Suppose
H
and
W
follow the
regression model:
H
= 51 + 0
.
27
W
+
Z
where
Z
is independent of
W
and
Z
∼ N
(0
,
6
.
83).
(a) If
W
= 66 what do you predict the husbands
height to be?
(b) Of the husbands who are married to women
5 feet tall, what percentage are over 5 feet 8
inches tall?
answer: 68.82; 37.83%
Treatment A
Treatment B
Small Kidney Stones
81
/
87 = 93%
234
/
270 = 87%
Large Kidney Stones
192
/
263 = 73%
55
/
80 = 69%
Both
273
/
350 = 78%
289
/
350 = 83%
TABLE I. Success rates for two treatments of two types of
kidney stones.
8.
This
problem
illustrates
“Sympson’s
Paradox”
which can occur when studying the relationship be
tween two variables:
Two medical procedures are
being considered for the treatment of kidney stones.
Kidney stones can be classified as either small or
large.
Table I displays some experimental results
of the two treatments on small and large kidney
stones. Based on the overall results (bottom row)
it would seem that Treatment B is better.
How
ever, on closer inspection, Treatment A does better
for both small kidney stones and for large kidney
stones. How could this be??
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
answer: Treatment A is more effective than Treat
ment B in all cases so is clearly better.
The rea
son Treatment B does better overall is that Treat
ment A is used mostly for the hard cases (large
kidney stones)...therefore Treatment B has a bet
ter “overall score” since it only works on the easy
cases (small kidney stones).
This is the end of the preview.
Sign up
to
access the rest of the document.