(d)
0.1
=
,
0.05
=
. For a two-sided alternative, the common sample size is
( )( )
1 2 1 2 2
/2 1 1 2 2
()
2
p p q q
z z p q p q
++
++
Reserve Problems Chapter 10 Section 6 Problem 3
A random sample of 1900 adult women indicated that 941 of them asked for medical assistance
last time they felt sick. The same sized sample of men indicated that 893 of them asked for
medical assistance.
Use the z-values rounded to three decimal places to obtain the answers.
a) Is it reasonable to conclude that women ask for medical assistance more often than men? Use
0.05
=
. Find the P-value for this test.
b) Suppose that
10.5p=
and
20.45p=
. Determine the sample size needed to detect this
difference with a probability of at least 0.95.
SOLUTION
a)
1) The parameters of interest are the proportions of people who asked for medical treatment,
b)
0.05
=
,
0.05
=
. For a one-sided alternative, the common sample size is
Reserve Problems Chapter 10 Section 6 Problem 4
A random sample of 2100 adult women indicated that 1141 of them asked for medical assistance
last time they felt sick. The same sized sample of men indicated that 1093 of them asked for
medical assistance.
Use the z-values rounded to three decimal places to obtain the answers.
a) Construct a 98% confidence interval for the difference in the two tests.
b) Reconstruct the CI using the alternative procedure described in this section. Compare those
intervals.
SOLUTION
a) Two-sided confidence interval:
b) Alternate CI:
˜˜
2, 1n n x x= + = +
Reserve Problems Chapter 10 Section 6 Problem 5
Consider the following computer output
Sample
X
N
Sample p
1
146
250
0.584000
2
196
370
0.529730
Difference = p(1) – p(2)
Estimate for difference: 0.054270
95% lower bound for difference: ?
Test for difference = 0 (vs > 0 ) : Z = ?
P-Value = ?
(a) Is this one-sided or a two-sided test?
(b) Fill in the missing values.
SOLUTION
(a)
(b)
Reserve Problems Chapter 10 Section 6 Problem 6
Two different types of polishing solutions are being evaluated for possible use in a tumble-polish
operation for manufacturing interocular lenses used in the human eye following cataract surgery.
240 lenses were tumble polished using the first polishing solution, and of this number 150 had no
polishing-induced defects. Another 240 lenses were tumble-polished using the second polishing
solution, and 123 lenses were satisfactory upon completion.
(a) What is the P-Value for this test?
(b) Is there any reason to believe that the two polishing solutions differ? Use
0.01
=
.
SOLUTION
(a)
The parameters of interest are the proportion of satisfactory lenses, p1 and p2
Reserve Problems Chapter 10 Section 6 Problem 7
Air pollution has been linked to lower birthweight in babies. In a study reported in the Journal of
the American Medical Association, researchers examined the proportion of low-weight babies
born to mothers exposed to heavy doses of soot and ash during the World Trade Center attack of
September 11, 2001. Of the 182 babies born to these mothers, 15 were classified as having low
weight. Of 2300 babies born in the same time period in New York in another hospital, 92 were
classified as having low weight.
(a) Determine the P-value.
(b) Is there evidence to suggest that the exposed mothers had a higher incidence of low-weight
babies? Use
0.05
=
.
SOLUTION
(a)
The parameters of interest are the proportion of satisfactory lenses, p1 and p2
(b)
Reserve Problems Chapter 10 Section 6 Problem 8
In the 2004 presidential election, exit polls from the critical state of Ohio provided the following
results: For respondents with college degrees, 53% voted for Bush and 46% voted for Kerry.
There were 2020 respondents.
Find the two-sided CI for the difference in the two proportions with
0.05
=
. Use the alternate
CI procedure.
SOLUTION
1 2 1 2
2020, 2020, 1071, 929,n n x x= = = =
Reserve Supplemental Exercises Chapter 10 Problem 1
Consider the following computer output:
Two-Sample T-Test CI
Sample
N
Mean
StDev
SeMean
1
21
16.01
5.38
1.1740
2
40
24.97
10.25
1.6207
Difference = μ(1)-μ(2)
Estimate for difference: -8.96
T-Test of difference = 0 ( vs < 0 ) :
T-Value = -3.73
P-Value = ?
DF = ?
(a) Is this one-sided or a two-sided test?
(b) Fill in the missing values.
SOLUTION
(a) I
This is one-sided test because the alternative hypothesis is μ1 – μ2 < 0.
Reserve Supplemental Exercises Chapter 10 Problem 2
An article in the Journal of Materials Engineering [1989, Vol. 11(4), pp. 275–282] reported the
results of an experiment to determine failure mechanisms for plasmasprayed thermal barrier
coatings. The failure stress for one particular coating (NiCrAlZr) under two different test
conditions is as follows:
Failure stress (× 106 Pa) after nine 1-hour
cycles
19.8
18.5
17.6
16.7
16.7
14.8
15.4
14.1
13.6
Failure stress (× 106 Pa) after six 1-hour
cycles
14.9
12.7
11.9
11.4
10.1
7.9
(a) Find a 90% confidence interval on the difference in mean failure stress under the two
different test conditions.
(b) Construct a 95% confidence interval on the ratio of the variances,
2
1
2
2
, of failure stress under
the two different test conditions.
SOLUTION
(a)
Reserve Supplemental Exercises Chapter 10 Problem 3
The breaking strength of yarn supplied by two manufacturers is being investigated. You know
from experience with the manufacturers’ processes that
16.2 psi
=
and
23.9 psi
=
. Suppose
that prior to collecting the data, you decide that you want the error in estimating
12
−
by
12
xx−
to be less than
1psi
.
(a) Specify the sample size for the following percentage confidence:
(1) 90%
(2) 98%
(b) Repeat part (a) with an error of less than
0.5 psi
instead of
1psi
.
(c) Comment on the effect of increasing the percentage confidence and the effect of decreasing
the error on the sample size needed.
SOLUTION
(a)
Reserve Supplemental Exercises Chapter 10 Problem 4
In a random sample of 200 Phoenix residents who drive a domestic car, 165 reported wearing
their seat belt regularly, and another sample of 250 Phoenix residents who drive a foreign car
revealed 198 who regularly wore their seat belt.
(a) Do you think there is a reason not to believe these data?
(b) Is it reasonable to use the hypothesis-testing results from the previous problem to draw an
inference about the difference in proportion of seat belt usage
(i) of the spouses of these drivers of domestic and foreign cars? Explain your answer.
(ii) of the children of these drivers of domestic and foreign cars? Explain your answer.
(iii) of all drivers of domestic and foreign cars? Explain your answer.
(iv) of all drivers of domestic and foreign trucks? Explain your answer.
SOLUTION
(a)
Reserve Supplemental Exercises Chapter 10 Problem 5
An article in Fire Technology investigated two different foam-expanding agents that can be used
in the nozzles of firefighting spray equipment. A random sample of five observations with an
aqueous film-forming foam (AFFF) had a standard deviation of 0.6. A random sample of five
observations with alcohol-type concentrates (ATC) had a standard deviation 0.8. Suppose that if
agent 1 produces a mean expansion that differs from the mean expansion of agent 2 by 1.5, you
would like to reject the null hypothesis with probability at least 0.9.
What sample size is required?
SOLUTION
0.05, 1 0.90, 1.5, 5n
= = − = =
*
Reserve Supplemental Exercises Chapter 10 Problem 6
Two different gauges can be used to measure the depth of bath material in a Hall cell used in
smelting aluminum.
Each gauge is used once in 15 cells by the same operator.
Cell
Gauge 1
Gauge 2
1
46 in.
47 in.
2
50
53
3
47
45
4
53
50
5
49
51
6
48
48
7
53
54
8
56
53
9
52 in.
51 in.
10
47
45
11
49
51
12
45
45
13
47
49
14
46
43
15
50
51
(a) Apply an appropriate statistical procedure to determine whether the data support the claim
that the two gauges produce different mean bath depth readings.
Use
0.1
=
.
(b) Suppose that if the two gauges differ in mean bath depth readings by as much as 1.34 inch,
you want the power of the test to be at least 0.8. For
0.01
=
, how many cells should have been
used?
SOLUTION
(a)
The parameter of interest is the mean difference in depth using the two gauges,
d
0:0
d
H
=
The test statistic is
0/
d
d
tsn
=
Reserve Supplemental Exercises Chapter 10 Problem 7
An article in Polymer Degradation and Stability (2006, Vol. 91) presented data from a nine-year
aging study on S537 foam. Foam samples were compressed to 50% of their original thickness
and stored at different temperatures for nine years. At the start of the experiment as well as
during each year, sample thickness was measured, and the thicknesses of the eight samples at
each storage condition were recorded. The data for two storage conditions follow.
50°C
0.047
0.060
0.061
0.064
0.080
0.090
0.118
0.165
0.183
60°C
0.062
0.105
0.118
0.137
0.153
0.197
0.210
0.250
0.375
Is there any indication that the variances of foam thickness are different at the two different
levels of temperature?
SOLUTION
The parameters of interest are the foam thickness variances,
22
12
,
.
22
0 1 2
:H
=
22
1 1 2
:H
Reserve Supplemental Exercises Chapter 10 Problem 8
An article in Quality Engineering [2012, Vol. 24(1)] described an experiment on a grinding
wheel. The following are some of the grinding force data (in N) from this experiment at two
different vibration levels.
Low
242
249
265
250
254
244
258
311
237
261
314
252
High
301
421
419
399
317
311
350
363
392
367
301
302
Is there any indication that the variances of grinding force are different at the two different levels
of temperature?
SOLUTION
The parameters of interest are the grinding force variances,
22
12
,
.
22
0 1 2
:H
=
Reserve Supplemental Exercises Chapter 10 Problem 9
In a random sample of 30 Phoenix residents who drive a domestic car, 25 reported wearing their
seat belt regularly, and another sample of 89 Phoenix residents who drive a foreign car revealed
71 who regularly wore their seat belt.
Find 95% CIs on the difference in the proportions of seat belt usage for drivers of foreign and
domestic cars using both procedures described in this chapter.
SOLUTION
95% traditional confidence interval on the difference:
1 2 1 2
30, 89, 25, 71,n n x x= = = =
