STAT 2122-002 Homework 13
Instructor: Shengwen Guo
Due date: December 6, 2023
Problem 1.
For healthy individuals the level of prothrombin in the blood is approximately normally dis-
tributed with mean 20 mg/100 mL and standard deviation 4 mg/100 mL. Low levels indicate low clotting
ability. In studying the effect of gallstones on prothrombin, the level of each patient in a sample is measured
to see if there is a deficiency. Let
µ
be the true average level of prothrombin for gallstone patients.
(a) What are the appropriate null and alternative hypotheses?
(b) Let
¯
X
denote the sample average level of prothrombin in a sample of
n
= 20 randomly selected gallstone
patients. Consider the test procedure with test statistic
¯
X
and rejection region ¯
x
≤
17
.
92. What is the
probability distribution of the test statistic when
H
0
is true? What is the probability of a type I error
for the test procedure?
(c) What is the probability distribution of the test statistic when
µ
= 16
.
7? Using the test procedure of part
(b), what is the probability that gallstone patients will be judged not deficient in prothrombin, when in
fact
µ
= 16
.
7 (a type II error)?
(d) How would you change the test procedure of part (b) to obtain a test with significance level
.
05? What
impact would this change have on the error probability of part (c)?
(e) Consider the standardized test statistic
Z
=
¯
X
−
20
σ/
√
n
=
¯
X
−
20
.
8944
. What are the values of
Z
corresponding
to the rejection region of part (b)?
Problem 2.
The melting point of each of 16 samples of a brand of hydrogenated vegetable oil was deter-
mined, resulting in ¯
x
= 94
.
32. Assume that the distribution of melting point is normal with
σ
= 1
.
20.
(a) Test
H
0
:
µ
= 95 versus
H
a
:
µ
̸
= 95 using a two-tailed level
.
01 test.
(b) If a level
.
01 test is used, what is
β
(94), the probability of a type II error when
µ
= 94?
(c) What value of
n
is necessary to ensure that
β
(94) =
.
1 when
α
=
.
01?
Problem 3.
The amount of shaft wear (.0001 in.) after a fixed mileage was determined for each of
n
= 8
internal combustion engines having copper lead as a bearing material, resulting in ¯
x
= 3
.
72 and
s
= 1
.
25.
(a) Assuming that the distribution of shaft wear is normal with mean
µ
, use the t test at level .05 to test
H
0
:
µ
= 3
.
50 versus
H
a
:
µ >
3
.
50.
(b) Using
σ
= 1
.
25, what is the type II error probability
β
(
µ
′
) of the test for the alternative
µ
′
= 4
.
00?
1
