All 21 Chapters Covered
SOLUTIONS
,Chapter 1
Exercises
Section 1.1
1.1 From the yield data in Table 1.1 in the text, and using the given expression, we
obtain
s2A = 2.05
sB2 = 7.64
is greater than s2B.
from where we observe that s2 A
1.2 A table of values for di is easily generated; the histogram along with sum- mary
statistics obtained using MINITAB is shown in the Figure below.
Summary for d
A nderson-Darling Normality Test
A -Squared 0.27
P-V alue 0.653
M ean 3.0467
StDev 3.3200
V ariance 11.0221
Skew ness -0.188360
Kurtosis -0.456418
N 50
M inimum -5.1712
1st Q uartile 1.0978
M edian 2.8916
3rd Q uartile 5.2501
-3 0 3 6 9 M aximum 9.1111
95% C onfidence Interv al for M ean
2.1032 3.9903
95% C onfidence Interv al for M edian
1.8908 4.2991
95% C onfidence Interv al for StDev
9 5 % Confidence Inter vals
2.7733 4.1371
Mean
Median
2.0 2.5 3.0 3.5 4.0 4.5
Figure 1.1: Histogram for d = YA −YB data with superimposed theoretical distribution
1
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,2 CHAPTER 1.
From the data, the arithmetic average, d̄, is obtained as
d̄ = 3.05 (1.1)
And now, that this average is positive, not zero, suggests the possibility that YA may
be greater than YB . However conclusive evidence requires a measure of intrinsic
variability.
1.3 Directly from the data in Table 1.1 in the text, we obtain ȳA = 75.52; ȳB =
72.47; and s2 A = 2.05; s2 B = 7.64. Also directly from the table of differences, d i,
generated for Exercise 1.2, we obtain: d̄ = 3.05; however s2 = 11.02,
d not 9.71.
Thus, even though for the means,
d̄ = ȳ A − ȳ B
for the variances, s2 2 2
d 6= sA + sB
The reason for this discrepancy is that for the variance equality to hold, YA must be
completely independent of YB so that the covariance between YA and YB is precisely
zero. While this may be true of the actual random variable, it is not always strictly the
case with data. The more general expression which is valid in all cases is as follows:
s2 2 2
d = sA + sB − 2sAB (1.2)
where sAB is the covariance between yA and yB (see Chapters 4 and 12). In this
particular case, the covariance between the yA and yB data is computed as
sAB = −0.67
Observe that the value computed for s2 (11.02) is obtained by adding −2sAB
to s2 + s2 , as in Eq (1.2). d
A B
Section 1.2
1.4 From the data in Table 1.2 in the text, s2 = 1.2.
x
1.5 In this case, with x̄ = 1.02, and variance, s2 = 1.2, even though the num- bers
are not exactly equal, within limits of random variation,
x they appear to be close
enough, suggesting the possibility that X may in fact be a Poisson random variable.
Section 1.3
1.6 The histograms obtained with bin sizes of 0.75, shown below, contain 10 bins for
YA versus 8 bins for the histogram of Fig 1.1 in the text, and 14 bins for YB versus 11
bins in Fig 1.2 in the text. These new histograms show a bit more detail but the
general features displayed for the data sets are essentially unchanged. When the bin
sizes are expanded to 2.0, things are slightly different,
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Histogram of YA (Bin size 0.75)
18
16
14
12
Frequency
10
8
6
4
2
0
72.0 73.5 75.0 76.5 78.0 79.5
YA
Histogram of YB (Bin size 0.75)
6
5
4
Frequency
3
2
1
0
67.5 69.0 70.5 72.0 73.5 75.0 76.5 78.0
YB
Figure 1.2: Histogram for YA, YB data with small bin size (0.75)
Histogram of YA (Bin size 2.0)
25
20
15
Frequency
10
5
0
72 74 76 78 80
YA
Histogram of YB(Bin Size 2.0)
14
12
10
Frequency
8
6
4
2
0
67 69 71 73 75 77 79
YB
Figure 1.3: Histogram for Y A, YB data with larger bin size (2.0)
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,4 CHAPTER 1.
as shown below. These histograms now contain fewer bins (5 for YA and 7 for
YB); and, hence in general, show less of the true character of the data sets.
1.7 The values computed from the data for ȳA and sA imply that the interval of
interest, ȳA ± 1.96sA , is 75.52 ± 2.81, or (72.71, 78.33). From the frequency distribution
of Table 1.3 in the text, 48 of the 50 points lie in this range, the
excluded points being (i) the single point in the 71.51–72.50 bin and (ii) the single
point in the 78.51–79.50 bin. Thus, this interval contains 96% of the data.
1.8 For the YB data, the interval of interest, ȳB ± 1.96sB , is 72.47 ± 5.41, or (67.06,
77.88). From Table 1.4 in the text,we see that approximately 48 of the
50 points lie in this range (excluding the 2 points in the 77.51–78.50 bin). Thus, this
interval also contains approximately 96% of the data.
1.9 From Table 1.4 in the text, we observe that the relative frequency associated with x
= 4 is 0.033; that associated with x = 5 is 0.017 and 0 thereafter. The implication is
that the relative frequency associated with x > 3 = 0.050. Hence, the value of x such
that only 5% of the data exceeds this value is x = 3.
1.10 Using µ = 75.52 and σ = 1.43, the theoretical values computed for the
function in Eq 1.3 in the text, (for y = 72, 73, . . . , 79) are shown in the table below
along with the the corresponding relative frequency values from Table 1.3 in the text.
Theoretical Relative
YA Group y f(y) Frequency
71.51-72.50 72 0.014 0.02
72.51-73.50 73 0.059 0.04
73.51-74.50 74 0.159 0.18
74.51-75.50 75 0.261 0.34
75.51-76.50 76 0.264 0.14
76.51-77.50 77 0.163 0.16
77.51-78.50 78 0.062 0.10
78.51-79.50 79 0.014 0.02
TOTAL 50 0.996 1.00
The agreement between the theoretical values and the relative frequency is rea-
sonable but not perfect.
1.11 This time time with µ = 72.47 and σ = 2.76 and for y = 67, 68, 69, . . . , 79, we obtain
the table shown below for the YB data (along with the the corre- sponding relative
frequency values from Table 1.4 in the text).
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Theoretical Relative
YB Group y f(y) Frequency
66.51-67.50 67 0.020 0.02
67.51-68.50 68 0.039 0.06
68.51-69.50 69 0.066 0.08
69.51-70.50 70 0.097 0.16
70.51-71.50 71 0.125 0.04
71.51-72.50 72 0.142 0.14
72.51-73.50 73 0.142 0.08
73.51-74.50 74 0.124 0.12
74.51-75.50 75 0.095 0.10
75.51-76.50 76 0.064 0.12
76.51-77.50 77 0.038 0.00
77.51-78.50 78 0.019 0.04
78.51-79.50 79 0.009 0.00
TOTAL 50 0.980 1.00
There is reasonable agreement between the theoretical values and the relative
frequency.
1.12 Using λ = 1.02, the theoretical values of the function f (x|λ) of Eq 1.4 in the
text at x = 0, 1, 2, . . . 6 are shown in the table below along with the corresponding
relative frequency values from Table 1.5 in the text.
Theoretical Relative
X f(x|λ = 1.02) Frequency
0 0.3606 0.367
1 0.3678 0.383
2 0.1876 0.183
3 0.0638 0.017
4 0.0163 0.033
5 0.0033 0.017
6 0.0006 0.000
TOTAL 1.0000 1.000
The agreement between the theoretical f (x) and the data relative frequency is
reasonable. (This pdf was plotted in Fig 1.6 of the text.)
Application Problems
1.13 (i) The following is one way to generate a frequency distribution for this data:
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,6 CHAPTER 1.
Relative
X Frequency Frequency
1.00-3.00 4 0.047
3.01-5.00 9 0.106
5.01-7.00 11 0.129
7.01-9.00 20 0.235
9.01-11.00 10 0.118
11.01-13.00 9 0.106
13.01-15.00 3 0.035
15.01-17.00 6 0.070
17.01-19.00 6 0.070
19.01-21.00 5 0.059
21.01-23.00 1 0.012
23.01-25.00 1 0.012
TOTAL 85 0.999
The histogram resulting from this frequency distribution is shown below where we
observe that it is skewed to the right. Superimposed on the histogram is a theoretical
gamma distribution, which fits the data quite well. The variable in question, time-to-
publication, is (a) non-negative, (b) continuous, and (c) has the potential to be a
large number (if a paper goes through several revisions before it is finally accepted,
or if the reviewers are tardy in completing their reviews in the first place). It is
therefore not surprising that the histogram will be skewed to the right as shown.
Histogram of x
Gamma
Shape 3.577
20 Scale 2.830
N 85
Frequency
15
10
5
0
0 4 8 12 16 20 24
x
Figure 1.4: Histogram for time-to-publication data
(ii) From this frequency distribution and the histogram, we see that the “most popular”
time-to-publication is in the range from 7-9 months (centered at 8 months); from the
relative frequency values, we note that 41/85 or 0.482 is the
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, 7
fraction of the papers that took longer than this to publish.
1.14 (i) A plot of the histogram for the 20-sample averages, yi, generated as
prescribed is shown in the top panel of the figure below. We note the narrower range
occupied by this data set as well as its more symmetric nature. (Super- imposed on
this histogram is a theoretical normal distribution distribution.)
(ii) A histogram of the average of averages, z i, is shown in the bottom panel of the
figure. The “averaging” significantly narrows the range of the data and also makes the
data set somewhat more symmetric.
Histogram of y
Normal
Mean 10.12
12
StDev 0.8088
N 85
10
8
Frequency
6
4
2
0
8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0
y
Histogram of z
Normal
Mean 10.38
20 StDev 0.7148
N 85
15
Frequency
10
5
0
9.0 9.6 10.2 10.8 11.4 12.0
z
Figure 1.5: Histogram for time-to-publication data
1.15 (i) Average number of safety incidents per month, x̄ = 0.500; the associated
variance, s2 = 0.511. The frequency table is shown below:
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,8 CHAPTER 1.
Relative
X Frequency Frequency
0 30 0.625
1 12 0.250
2 6 0.125
3 0 0.000
TOTAL 48 1.000
The resulting histogram is shown below.
Histogram of SafetyIncidents
30
25
20
Frequency
15
10
5
0
0 1 2
SafetyIncidents
Figure 1.6: Histogram for safety incidents data
(ii) It is reasonable to consider the relative frequency of occurrence of the safety
incidents as an acceptable measure of the “chances” of obtaining each indicated
number of occurrences: since fr (0) = 0.625, fr (1) = 0.250, fr (2) = 0.125, fr (3) =
0.000 = fr (4) = fr (5), these may then be considered as reasonable estimates of
the chances of observing the indicated occurrences.
(iii) From the postulated model:
e−0.50.5x
f(x) = x!
we obtain the following table which shows the theoretical probability of occur- rence
side-by-side with the relative frequency data; it indicates that the model actually fits
the data quite well.
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TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
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SOLUTIONS
,Chapter 1
Exercises
Section 1.1
1.1 From the yield data in Table 1.1 in the text, and using the given expression, we
obtain
s2A = 2.05
sB2 = 7.64
is greater than s2B.
from where we observe that s2 A
1.2 A table of values for di is easily generated; the histogram along with sum- mary
statistics obtained using MINITAB is shown in the Figure below.
Summary for d
A nderson-Darling Normality Test
A -Squared 0.27
P-V alue 0.653
M ean 3.0467
StDev 3.3200
V ariance 11.0221
Skew ness -0.188360
Kurtosis -0.456418
N 50
M inimum -5.1712
1st Q uartile 1.0978
M edian 2.8916
3rd Q uartile 5.2501
-3 0 3 6 9 M aximum 9.1111
95% C onfidence Interv al for M ean
2.1032 3.9903
95% C onfidence Interv al for M edian
1.8908 4.2991
95% C onfidence Interv al for StDev
9 5 % Confidence Inter vals
2.7733 4.1371
Mean
Median
2.0 2.5 3.0 3.5 4.0 4.5
Figure 1.1: Histogram for d = YA −YB data with superimposed theoretical distribution
1
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,2 CHAPTER 1.
From the data, the arithmetic average, d̄, is obtained as
d̄ = 3.05 (1.1)
And now, that this average is positive, not zero, suggests the possibility that YA may
be greater than YB . However conclusive evidence requires a measure of intrinsic
variability.
1.3 Directly from the data in Table 1.1 in the text, we obtain ȳA = 75.52; ȳB =
72.47; and s2 A = 2.05; s2 B = 7.64. Also directly from the table of differences, d i,
generated for Exercise 1.2, we obtain: d̄ = 3.05; however s2 = 11.02,
d not 9.71.
Thus, even though for the means,
d̄ = ȳ A − ȳ B
for the variances, s2 2 2
d 6= sA + sB
The reason for this discrepancy is that for the variance equality to hold, YA must be
completely independent of YB so that the covariance between YA and YB is precisely
zero. While this may be true of the actual random variable, it is not always strictly the
case with data. The more general expression which is valid in all cases is as follows:
s2 2 2
d = sA + sB − 2sAB (1.2)
where sAB is the covariance between yA and yB (see Chapters 4 and 12). In this
particular case, the covariance between the yA and yB data is computed as
sAB = −0.67
Observe that the value computed for s2 (11.02) is obtained by adding −2sAB
to s2 + s2 , as in Eq (1.2). d
A B
Section 1.2
1.4 From the data in Table 1.2 in the text, s2 = 1.2.
x
1.5 In this case, with x̄ = 1.02, and variance, s2 = 1.2, even though the num- bers
are not exactly equal, within limits of random variation,
x they appear to be close
enough, suggesting the possibility that X may in fact be a Poisson random variable.
Section 1.3
1.6 The histograms obtained with bin sizes of 0.75, shown below, contain 10 bins for
YA versus 8 bins for the histogram of Fig 1.1 in the text, and 14 bins for YB versus 11
bins in Fig 1.2 in the text. These new histograms show a bit more detail but the
general features displayed for the data sets are essentially unchanged. When the bin
sizes are expanded to 2.0, things are slightly different,
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Histogram of YA (Bin size 0.75)
18
16
14
12
Frequency
10
8
6
4
2
0
72.0 73.5 75.0 76.5 78.0 79.5
YA
Histogram of YB (Bin size 0.75)
6
5
4
Frequency
3
2
1
0
67.5 69.0 70.5 72.0 73.5 75.0 76.5 78.0
YB
Figure 1.2: Histogram for YA, YB data with small bin size (0.75)
Histogram of YA (Bin size 2.0)
25
20
15
Frequency
10
5
0
72 74 76 78 80
YA
Histogram of YB(Bin Size 2.0)
14
12
10
Frequency
8
6
4
2
0
67 69 71 73 75 77 79
YB
Figure 1.3: Histogram for Y A, YB data with larger bin size (2.0)
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,4 CHAPTER 1.
as shown below. These histograms now contain fewer bins (5 for YA and 7 for
YB); and, hence in general, show less of the true character of the data sets.
1.7 The values computed from the data for ȳA and sA imply that the interval of
interest, ȳA ± 1.96sA , is 75.52 ± 2.81, or (72.71, 78.33). From the frequency distribution
of Table 1.3 in the text, 48 of the 50 points lie in this range, the
excluded points being (i) the single point in the 71.51–72.50 bin and (ii) the single
point in the 78.51–79.50 bin. Thus, this interval contains 96% of the data.
1.8 For the YB data, the interval of interest, ȳB ± 1.96sB , is 72.47 ± 5.41, or (67.06,
77.88). From Table 1.4 in the text,we see that approximately 48 of the
50 points lie in this range (excluding the 2 points in the 77.51–78.50 bin). Thus, this
interval also contains approximately 96% of the data.
1.9 From Table 1.4 in the text, we observe that the relative frequency associated with x
= 4 is 0.033; that associated with x = 5 is 0.017 and 0 thereafter. The implication is
that the relative frequency associated with x > 3 = 0.050. Hence, the value of x such
that only 5% of the data exceeds this value is x = 3.
1.10 Using µ = 75.52 and σ = 1.43, the theoretical values computed for the
function in Eq 1.3 in the text, (for y = 72, 73, . . . , 79) are shown in the table below
along with the the corresponding relative frequency values from Table 1.3 in the text.
Theoretical Relative
YA Group y f(y) Frequency
71.51-72.50 72 0.014 0.02
72.51-73.50 73 0.059 0.04
73.51-74.50 74 0.159 0.18
74.51-75.50 75 0.261 0.34
75.51-76.50 76 0.264 0.14
76.51-77.50 77 0.163 0.16
77.51-78.50 78 0.062 0.10
78.51-79.50 79 0.014 0.02
TOTAL 50 0.996 1.00
The agreement between the theoretical values and the relative frequency is rea-
sonable but not perfect.
1.11 This time time with µ = 72.47 and σ = 2.76 and for y = 67, 68, 69, . . . , 79, we obtain
the table shown below for the YB data (along with the the corre- sponding relative
frequency values from Table 1.4 in the text).
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Theoretical Relative
YB Group y f(y) Frequency
66.51-67.50 67 0.020 0.02
67.51-68.50 68 0.039 0.06
68.51-69.50 69 0.066 0.08
69.51-70.50 70 0.097 0.16
70.51-71.50 71 0.125 0.04
71.51-72.50 72 0.142 0.14
72.51-73.50 73 0.142 0.08
73.51-74.50 74 0.124 0.12
74.51-75.50 75 0.095 0.10
75.51-76.50 76 0.064 0.12
76.51-77.50 77 0.038 0.00
77.51-78.50 78 0.019 0.04
78.51-79.50 79 0.009 0.00
TOTAL 50 0.980 1.00
There is reasonable agreement between the theoretical values and the relative
frequency.
1.12 Using λ = 1.02, the theoretical values of the function f (x|λ) of Eq 1.4 in the
text at x = 0, 1, 2, . . . 6 are shown in the table below along with the corresponding
relative frequency values from Table 1.5 in the text.
Theoretical Relative
X f(x|λ = 1.02) Frequency
0 0.3606 0.367
1 0.3678 0.383
2 0.1876 0.183
3 0.0638 0.017
4 0.0163 0.033
5 0.0033 0.017
6 0.0006 0.000
TOTAL 1.0000 1.000
The agreement between the theoretical f (x) and the data relative frequency is
reasonable. (This pdf was plotted in Fig 1.6 of the text.)
Application Problems
1.13 (i) The following is one way to generate a frequency distribution for this data:
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,6 CHAPTER 1.
Relative
X Frequency Frequency
1.00-3.00 4 0.047
3.01-5.00 9 0.106
5.01-7.00 11 0.129
7.01-9.00 20 0.235
9.01-11.00 10 0.118
11.01-13.00 9 0.106
13.01-15.00 3 0.035
15.01-17.00 6 0.070
17.01-19.00 6 0.070
19.01-21.00 5 0.059
21.01-23.00 1 0.012
23.01-25.00 1 0.012
TOTAL 85 0.999
The histogram resulting from this frequency distribution is shown below where we
observe that it is skewed to the right. Superimposed on the histogram is a theoretical
gamma distribution, which fits the data quite well. The variable in question, time-to-
publication, is (a) non-negative, (b) continuous, and (c) has the potential to be a
large number (if a paper goes through several revisions before it is finally accepted,
or if the reviewers are tardy in completing their reviews in the first place). It is
therefore not surprising that the histogram will be skewed to the right as shown.
Histogram of x
Gamma
Shape 3.577
20 Scale 2.830
N 85
Frequency
15
10
5
0
0 4 8 12 16 20 24
x
Figure 1.4: Histogram for time-to-publication data
(ii) From this frequency distribution and the histogram, we see that the “most popular”
time-to-publication is in the range from 7-9 months (centered at 8 months); from the
relative frequency values, we note that 41/85 or 0.482 is the
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, 7
fraction of the papers that took longer than this to publish.
1.14 (i) A plot of the histogram for the 20-sample averages, yi, generated as
prescribed is shown in the top panel of the figure below. We note the narrower range
occupied by this data set as well as its more symmetric nature. (Super- imposed on
this histogram is a theoretical normal distribution distribution.)
(ii) A histogram of the average of averages, z i, is shown in the bottom panel of the
figure. The “averaging” significantly narrows the range of the data and also makes the
data set somewhat more symmetric.
Histogram of y
Normal
Mean 10.12
12
StDev 0.8088
N 85
10
8
Frequency
6
4
2
0
8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0
y
Histogram of z
Normal
Mean 10.38
20 StDev 0.7148
N 85
15
Frequency
10
5
0
9.0 9.6 10.2 10.8 11.4 12.0
z
Figure 1.5: Histogram for time-to-publication data
1.15 (i) Average number of safety incidents per month, x̄ = 0.500; the associated
variance, s2 = 0.511. The frequency table is shown below:
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,8 CHAPTER 1.
Relative
X Frequency Frequency
0 30 0.625
1 12 0.250
2 6 0.125
3 0 0.000
TOTAL 48 1.000
The resulting histogram is shown below.
Histogram of SafetyIncidents
30
25
20
Frequency
15
10
5
0
0 1 2
SafetyIncidents
Figure 1.6: Histogram for safety incidents data
(ii) It is reasonable to consider the relative frequency of occurrence of the safety
incidents as an acceptable measure of the “chances” of obtaining each indicated
number of occurrences: since fr (0) = 0.625, fr (1) = 0.250, fr (2) = 0.125, fr (3) =
0.000 = fr (4) = fr (5), these may then be considered as reasonable estimates of
the chances of observing the indicated occurrences.
(iii) From the postulated model:
e−0.50.5x
f(x) = x!
we obtain the following table which shows the theoretical probability of occur- rence
side-by-side with the relative frequency data; it indicates that the model actually fits
the data quite well.
@
@SS
eeisim
smiciicsiolala
so tio nn
tio
, THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
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tio
