Reserve Problems Chapter 9 Section 5 Problem 7
In a random sample of 500 handwritten zip code digits, 464 were read correctly by an optical
character recognition (OCR) system operated by the U.S. Postal Service (USPS). USPS would
like to know whether the rate is at least 90% correct. Do the data provide evidence that the rate is
at least 90% at
0.05
=
? State the appropriate hypotheses. Test the appropriate hypotheses
using
0.05
=
. Find
0
z
. Is it possible to reject
0
H
hypothesis at the 0.05 level of significance?
SOLUTION
The parameter of interest is the rate of zip codes that can be correctly read, p.
0: 0.90Hp=
and
1: 0.90Hp
.
0.05
=
0.05
=
0.05
=
Reserve Problems Chapter 9 Section 5 Problem 8
In a random sample of 500 handwritten zip code digits, 464 were read correctly by an optical
character recognition (OCR) system operated by the U.S. Postal Service (USPS). USPS would
like to know whether the rate is at least 90% correct. Construct a 90% confidence interval for the
proportion of handwritten zip codes that were read correctly. Does this confidence interval
support the claim that at least 90% of the zip codes can be correctly read?
SOLUTION
The one-sided 90% confidence interval
Reserve Problems Chapter 9 Section 5 Problem 9
An article in the British Medical Journal [“Comparison of Treatment of Renal Calculi by
Operative Surgery, Percutaneous Nephrolithotomy, and Extra-Corporeal Shock Wave
Lithotrips” (1986, Vol. 292, pp. 879–882)] repeated that percutaneous nephrolithotomy (PN) had
a success rate in removing kidney stones of 289 of 350 patients. The traditional method was 78%
effective. Can we claim that at least 78% of procedures are effective? Construct a 95% lower
confidence interval for the proportion of patients with kidney stones successfully removed. Does this confidence
interval support the claim that at least 78% of procedures are successful?
SOLUTION
Construct a 95% lower confidence interval for the proportion of patients with kidney stones
successfully removed.
Reserve Problems Chapter 9 Section 8 Problem 1
300 people of different age were asked if they knew PewDiePie. We wish to know whether the
fame of PewDiePie among people is independent of their age group, and we want to use
0.05
=
.
Age
Young
Middle-age
Elder
Totals
They know
95
37
22
154
They do not know
5
63
78
146
Totals
100
100
100
300
Determine the value of test statistic.
Because
2
0
is _____ than the critical value for
0.05
=
, we _____
0
H
. The evidence ______
sufficient to claim that the fame of PewDiePie is not independent of the viewers age group at
0.05
=
.
Determine the P-value for this test.
SOLUTION
The expected frequencies are shown in the table below:
Age
Young
Middle-age
Elder
Totals
They know
51.333
51.333
51.333
154
H
0.05
=
0
H
They do not know
48.667
48.667
48.667
146
Totals
100
100
100
300
Reserve Problems Chapter 9 Section 8 Problem 2
An independent research was made to understand if the job choice depends on the gender of an
employee. Use
0.1
=
. Statistics of a random sample are shown in the table.
Job
Social workers
Teachers
Engineers
IT
Total
Male
54
108
210
196
568
Female
67
158
167
154
546
Total
121
266
377
350
1114
Determine the value of test statistic.
Because
2
0
is _____ than the critical value for
0.05
=
, we ____
0
H
. The evidence _____
sufficient to claim that the job choice is not independent of the employee’s gender at
0.1
=
.
Determine the P-value for this test.
SOLUTION
The expected frequencies are shown in the table below:
Job
Social workers
Teachers
Engineers
IT
Total
Male
178.456
568
Female
171.544
546
Total
121
266
377
1114
Reserve Problems Chapter 9 Section 9 Problem 1
An experiment was made to measure the mean number of axe strikes that is necessary to cut a
tree trunk. 16 identical tree trunks were cut by a randomly chosen person.
Observation
Number of strikes
1
52
2
54
3
56
4
55
5
57
6
47
7
53
8
45
9
46
10
48
11
55
12
57
13
51
14
54
15
56
16
52
Test the hypothesis that the median number of strikes is 50, using
0.05
=
. Assume that the
alternative is two-sided. Determine the value of test statistic. Calculate the P-value for this test.
We _____
0
H
. There ______ evidence to conclude that the median of the number of strikes
differs from 50.
SOLUTION
1) The parameter of interest is the median of the number of strikes.
Reserve Problems Chapter 9 Section 9 Problem 2
An experiment was made to measure the mean number of axe strikes that is necessary to cut a
tree trunk. 16 identical tree trunks were cut by a randomly chosen person.
Observation
Number of strikes
1
52
2
54
3
49
4
55
5
44
6
47
7
53
8
45
9
46
10
48
11
55
12
43
13
51
14
54
15
56
16
52
Use the Wilcoxon signed-rank test to investigate the claim that the median number of strikes is
50. Use
0.05
=
. Assume that the alternative is two-sided.
Determine the sum of the positive ranks. Determine the sum of the absolute values of negative
ranks. Determine the value of test statistic.
We ______ the null hypothesis that the mean number of strikes is 50 at the 0.05 level of
significance.
SOLUTION
Observation
Difference
Signed Rank
13
1
1.5
3
-1
-1.5
2
4
9
9
-4
-9
14
4
9
4
5
12
8
-5
-12
11
5
12
5
-6
15
6
12
-16
10
-2
-4
1
2
4
16
2
4
7
3
6.5
6
-3
-6.5
2)
˜
0: 50H
=
Reserve Problems Chapter 9 Section 10 Problem 1
The gym is looking for a new equipment supplier. The candidate manufacturer provides for the
equivalence testing a set of ten 12 kg dumbbells, the sample mean and standard deviation of the
weight of dumbbell are
12.06x=
and
0.06 kgs=
. The gym is going to sign a contract if the
mean weight of a dumbbell is within 0.05 of the standards of 12 kg
( )
0.05
=
.
Assume that the weight of the dumbbells is normally distributed. Is the candidate manufacturer a
good choice?
SOLUTION
1) The parameter of interest is the mean weight of a dumbbell from a candidate supplier
.
Test of hypothesis (I):
4) The test statistic is:
Test of hypothesis (II):
4) The test statistic is:
0.05
=
Reserve Problems Chapter 9 Section 10 Problem 2
A fast-food company is looking for a place for a new point. The company finds a place
appropriate if its daily attendance is at least 10 thousand people. One of the shopping malls has
the following daily attendance data for the previous month (30 days):
9500x=
and
1040s=
people. Assume that the number of daily visitors of this place is normally distributed.
Is it an appropriate place at
0.01
=
?
SOLUTION
1) The parameter of interest is the mean daily attendance of the mall
.
3)
1
H
:
10000
Reserve Problems Chapter 9 Section 10 Problem 1
A ping pong robot performed 15 throws with the mean speed of 23.7 m/s and its standard
deviation of 1.9 m/s. The standard speed for an exercising player is 20 m/s, the player is able to
take the ball if its speed is no more than 25 m/s.
At what level of significance, we can claim that this robot is suitable for this player? Assume that
the speed of balls is normally distributed.
SOLUTION
1) The parameter of interest is the mean speed of a ball
.
1
H
H
H
Reserve Problems Chapter 9 Section 10 Problem 2
The mean length of produced wires must equal 30 meters. Ten independent samples were
selected, and the hypotheses about the mean length were tested. The P-values from these tests are
as follows:
.030
0.062
0.0.075
0.22
0.004
0.322
0.103
0.067
079
0.038
Is there sufficient evidence to conclude that the mean length of the wire is not equal to 30
meters?
SOLUTION
0
H
:
30
=
Reserve Supplemental Exercises Chapter 9 Problem 1
Reserve Supplemental Exercises Chapter 9 Problem 1
An article in Transfusion Science [“Early White Blood Cell Recovery Is a Predictor of Low
Number of Apheresis and Good CD34+ Cell Yield” (2000, Vol. 23, pp. 91-100)] studied the
0
H
white blood cell recovery of patients with haematological malignancies after a new
chemotherapy treatment. Data (in days) on white blood cell recovery (WBC) for 19 patients
consistent with summary data reported in the paper follow: 18, 16, 13, 16, 15, 12, 9, 14, 12, 8,
16, 12, 10, 8, 14, 9, 5, 18, and 12.
(a) Is the sufficient evidence to support a claim that the mean WBC recovery exceeds 12 days?
(b) Find s 95% two-sided CI on the mean WBC recovery.
SOLUTION
(a)
The null hypothesis is
12
=
versus
12
.
Reserve Supplemental Exercises Chapter 9 Problem 2
Suppose that we wish to test the hypothesis
0: 76H
=
versus the alternative
1: 76H
where
16
=
. Suppose that the true mean is77 and that in the practical context of the problem,
this is not a departure from
076
=
that has practical significance.
(a) For a test with
0.01
=
, compute
for the sample sizes assuming that
76
=
:
25n=
,
=
100n=
,
=
400n=
,
=
2500n=
,
=
(b) Suppose that the sample average is
76x=
. Find the P-value for the test statistic for the
different sample sizes specified in part (a).
25n=
100n=
400n=
2500n=
SOLUTION
(a)
25n=
,
0.01
76 77 1
2.33 0.98
16 / 25 16 / 25
Ф z Ф
−−
= + = + =
.
(b)
25n=
,
( )
00
77 76 ; :1 0.38
16 / 25
z P value Фz
−
= − − =
.
Reserve Supplemental Exercises Chapter 9 Problem 3
An inspector of flow metering devices used to administer fluid intravenously will perform a
hypothesis test to determine whether the mean flow rate is different from the flow rate setting of
207 milliliters per hour. Based on prior information, the standard deviation of the flow rate is
assumed to be known and equal to 10 milliliters per hour. For each of the following sample sizes,
and a fixed
0.05
=
, find the probability of a type II error if the true mean is 209 milliliters per
hour.
(a)
20:n=
=
(b)
50:n=
=
(c)
100:n=
=
(d) Does the probability of a type II error increase or decrease as the sample size increases?
Explain your answer.
SOLUTION
0.025
10, 209 207 2, 0.025, 1.96
2z
= = − = = =
20:n=