6) Reject the null hypothesis if
12
0 1 /2, 1, 1 0.975,12,13
0.025,13,12
11
0.308
3.23
nn
f f f f
− − −
= = = =
or
7) Conclusion: Because
0.308 0.623 3.153
, we fail to reject the null hypothesis. There is no
sufficient evidence that there is a difference in the standard deviations of water hardness.
b) 95% confidence interval:
2 1 2 1
2 2 2
1 1 1
1 /2, 1, 1 /2, 1, 1
2 2 2
2 2 2
n n n n
ss
ff
ss
− − − − −
Reserve Problems Chapter 10 Section 5 Problem 3
Dark current is the relatively small electric current that flows through photosensitive devices
(such as charge couple device (CCD) and photodiode) even when no photons are entering the
device. It consists of electrons and other charges randomly generated in the detector. The mean
value of dark current increases with temperature. Dark frames (images captured with the sensor
in the dark) and dark current generation measurements are used to reduce noise due to dark
current.
Dark current generation is measured for sample CCDs of the same type at two different
temperatures. The data follow.
0°C
15.4
16
14.3
14.2
15
14.5
14.4
14.5
15.4
15.3
14.7
14.7
14.2
14.3
14.7
15.5
5°C
31.9
31.4
32.7
32.8
31.8
30.8
32.4
32
31
31.6
30.2.0
31
33
32.6
a) Is there evidence to support the claim that the dark current generation variance increases with
temperature? Use
0.05
=
. Find the P-value of this test.
b) Construct a 95% two-sided confidence interval
2
1
2
2
.
0.05
=
SOLUTION
a) 1) The parameters of interest are the standard deviations,
12
,
.
4) The test statistic is
0.9,15,13 0.5f=
b) 95% two-sided confidence interval:
Reserve Problems Chapter 10 Section 5 Problem 4
Surface roughness of steel is investigated. The method of surface manufacturing largely
determines roughness of the surface. One of the parameters to describe the roughness is Rz, the
maximum height of the surface irregularities profile.
Rz(in μm) is measured for 21 random places on the surface of forged details and 21 random
places on the surface of casted details as follows.
forge
102
82
94
117
94
105
88
92
90
81
102
96
88
80
105
101
90
110
91
110
94
cast
105
99
97
105
96
101
90
104
92
106
104
104
90
104
107
91
92
89
93
105
96
a) Is there evidence to support the claim that the surface of forged steel has a larger surface
roughness variance than the surface of casted details? Use
0.05
=
. Find the P-value of this
test.
b) Construct a 95% two-sided confidence interval
2
1
2
2
.
SOLUTION
a) 1) Because the rough surface variance is considered more important, the parameters of interest
are the standard deviations,
12
,
4) The test statistic is
b) 95% two-sided confidence interval:
Reserve Problems Chapter 10 Section 5 Problem 5
For an F-distribution, find the following:
(a)
0.25,6,15
f=
(b)
0.1,8,12
f=
(c)
0.01,20,8
f=
(d)
0.75,6,15
f=
(e)
0.9,8,12
f=
(f)
0.99,20,8
f=
SOLUTION
Use the tables for F-distribution in book’s appendix and rule
1 , ,
,,
1
uv
vu
ff
−=
.
Reserve Problems Chapter 10 Section 5 Problem 6
Consider the hypothesis test
22
0 1 2
:H
=
against
22
1 1 2
:H
. Suppose that the sample sizes
are
120n=
and
28n=
, and that
22
12
4.4; 2.1ss==
. Use
0.01
=
.
(a) Test the hypothesis.
The test statistic is
0
f=
The critical value is
f=
Conclusion: ______ the null hypothesis at
0.01
=
.
(b) Construct the confidence interval on
22
12
/
which can be used to test the hypothesis.
SOLUTION
(a)
1) Parameters of interest are the standard deviations
12
,
.
Reserve Problems Chapter 10 Section 5 Problem 7
Consider the hypothesis test
22
0 1 2
:H
=
against
22
1 1 2
:H
. Suppose that the sample sizes
are
115n=
and
215n=
, and that
22
12
2.1; 2.0ss==
. Use
0.05
=
.
0.01
=
0.01
=
0.01
=
(a) Test the hypothesis.
The test statistic is
0
f=
Conclusion: _____ the null hypothesis at
0.05
=
.
Find a confidence interval interval on
22
12
/
which can be used to test the hypothesis.
(b) What is the power of the test in part (a) if
12
2.0
=
?
(с) Assuming equal sample sizes, what approximately sample size should be used to obtain
0.05
=
if
12
1.8
=
?
SOLUTION
(a)
1) Parameters of interest are the standard deviations
12
,
.
0.05
=
22
12
/
Reserve Problems Chapter 10 Section 5 Problem 8
The diameter of steel rods manufactured on two different machines is being investigated. Two
random samples of sizes
115n=
and
217n=
are selected. Sample means and variances are
22
1 1 2 2
8.73, 0.34, 8.68, 0.39x s x s= = = =
.
(a) Construct a 90% two-sided confidence interval on.
(b) Construct a 95% two-sided confidence interval on.
(c) Construct a 90% lower-sided confidence interval.
SOLUTION
(a)
two-sided confidence interval for
0.1
=
(b)
two-sided confidence interval for
0.05
=
2 1 2 1
2 2 2
1 1 1
1 /2, 1, 1 /2, 1, 1
2 2 2
2 2 2
n n n n
ss
ff
ss
− − − − −
Reserve Problems Chapter 10 Section 5 Problem 9
An article in Technometrics (1999, Vol41, pp.202-211) studied the capability of a gauge by
measuring the weights of two sheet of paper. Consider data in table:
paper 1
paper 2
3.481
3.258
3.448
3.254
3.485
3.256
3.475
3.249
3.472
3.241
3.477
3.254
3.472
3.247
3.464
3.257
3.472
3.239
3.470
3.250
3.470
3.258
3.470
3.239
3.477
3.245
3.473
3.240
3.474
3.254
Is there evidence that the variance of measurment differs for the sheets of paper? Use
0.1
=
.
Find the test statistic.
SOLUTION
1) The parameters of interest are the variances of the weight measurements between the two
Reserve Problems Chapter 10 Section 5 Problem 10
Olympic swimmers are seeded into the difference heats. Data of times from heats five, six and
seven are in seconds for 100m-swim in the following table.
Times of swimmers in heat 5
Times of swimmers in heat 6
Times of swimmers in heat 7
49.74
48.19
48.58
49.49
48.29
48.67
49.6
48.54
48.93
49.78
48.6
48.91
49.95
48.67
48.97
50.8
49.18
49.03
49.18
49.29
Use
0.05
=
.
(a) Is there evidence to suggest that the standard deviations of the heats differ for heat five and
heat seven? Find the P-value of that test.
(b) Repeat (a) for heats six and seven.
SOLUTION
The parameters of interest are the time variances,
22
12
,
.
22
0 1 2
:H
=
,
22
1 1 2
:H
.
Test statistic is
2
1
02
2
s
fs
=
. Reject the null hypothesis if the p-value less than
0.05
=
.
(a) Is there evidence to suggest that the standard deviations of the heats differ for heat five and
heat seven? Find the P-value of that test.
(b)
F-test two-sample for variances of heat 6 and 7.
Variable1
Heat 6
Variable2
Heat7
Mean
48.664
48.911
Variance
0.153
0.055
Observations(n)
7
7
dF(n-1)
6
6
Variable1
Heat 5
Variable2
Heat7
Mean
49.893
48.911
Variance
0.222
0.055
Observations(n)
6
7
dF(n-1)
5
6
Reserve Problems Chapter 10 Section 5 Problem 11
Following is a random sample of 15 measurements from high-flow rivers and 13 from low-flow
rivers of a total algae concentration.
High
Low
23.3
18.4
23.8
59.6
33.6
37.4
42.4
47.3
56.0
34.1
78.8
33.3
17.8
52
31.0
43.1
23.4
26.0
49.5
41.8
65.5
38.7
75.8
12.8
43.9
16.4
48.8
56.4
Is there evidence to suggest that the standard deviations of the aglae concentration in the two
types of rivers differ? What is the test statistic?
SOLUTION
1) The parameters of interest are the algae concentration variances,
22
12
,
.
Reserve Problems Chapter 10 Section 6 Problem 1
Rafting down two different rivers took place. 324 boats rafted down the first river, and accidents
(capsizing, boat damage, etc.) happened to 35 of them. 92 boats rafted down the second river,
and accidents happened to 18 of them.
Use the z-values rounded to two decimal places to obtain the answers.
a) The second river is considered to be a more complicated route to raft. Is there evidence for this
assumption? Find the P-value of the test. Use
0.10
=
.
b) Construct a 90% one-sided confidence limit for the difference in proportions that can be used
to answer the question in part (a).
SOLUTION
a)
2
p
5) The test statistic is
( )
12
0
12
ˆˆ
ˆˆ
11
1
pp
z
pp
nn
−
=
−+
b) One-sided confidence interval:
( ) ( ) ( )
1 1 2 2
1 2 1 2
12
1
ˆ ˆ ˆ ˆ
ˆˆ 1p p p p
p p p p z nn
−−
− − + +
Reserve Problems Chapter 10 Section 6 Problem 2
Two different rifles are tested at the shooting range. 190 shots were fired from the first rifle and
the target was hit 158 times. 140 shots were fired from the second rifle, with 107 hits.
Use the z-values rounded to two decimal places to obtain the answers.
(a) Is it reasonable to conclude that both rifles have the same accuracy? Use
0.05
=
. Find the
P-value for this test.
(b) Construct a 95% confidence interval for the difference in the two rifles accuracy.
(c) Suppose that
10.85p=
and
20.75p=
. With the sample size given here, what is the power
of the test for this two-sided alternate?
(d) Determine the sample size needed to detect the difference from (c) with a probability of at
least 0.9.
SOLUTION
(a) Two different rifles
5) The test statistic is
12
0
ˆˆ
pp
z
−
=
(c) Power of test is
1
−
.
( ) ( )
1 2 1 2
ˆ
/2 1 2 /2 1 2
1 2 2
ˆ
1
ˆˆ
1 1 1
ˆˆ
1
p p p p
z pq p p z pq p p
n n n n
−−
+ − − − + − −
= −