Portland Community College Probability & Statistical Variance Worksheet see the attached word document and please use excel, one sheet per question. Please

Portland Community College Probability & Statistical Variance Worksheet see the attached word document and please use excel, one sheet per question. Please use formulas. I have to get full mark for this exam in order to pass so please do your best 1. Medical researchers have developed a new artificial heart constructed primarily of titanium and
plastic. The heart will last and operate almost indefinitely once it is implanted in the patient’s body, but
the battery pack needs to be recharged about every 8 hours. A random sample of 60 battery packs is
selected and subjected to a life test. The average life of these batteries is 8.04 hours. Assume that
battery life is normally distributed with standard deviation σ = 0.15 hour. The claim is that mean battery
life exceeds 8 hours. Use α = 0.01.
1.1 (2 points) What is the P-value?
1.2 (2 points) Is there evidence to support the claim? Explain by constructing a one-sided confidence
bound on the mean life.
1
2. Overhead Door (OD) Corporation’s founder, C. G. Johnson, invented the upward-lifting
garage door in 1921 and the electric garage door opener in 1926. Since then OD has been a
leading supplier of commercial, industrial, and residential garage doors sold through a
nationwide network of more than 450 authorized distributors. They have built a solid
reputation as a premier door supplier, commanding 15 % share of the market.
Suppose that customers assess door quality first in terms of the ease of operation, followed by
its durability. The quality improvement team (QIT) might then assign an engineering team to
determine the factors that contribute to these two main problems.
Smooth operation of a garage door is a critical quality characteristic that affects both problems:
If a door is too heavy, it’s difficult and unsafe to balance and operate; if it’s too light, it tends to
buckle and break down frequently or may not close properly.
Suppose the design engineers determine that a standard garage door should weigh a minimum
of 77.5 kg. and a maximum of 82.5 kg., which thus specifies its design quality specification. To
determine the quality of conformance, suppose the QIT decides to collect data on the actual
weights of 90 standard garage doors sampled randomly from their monthly production of
almost 2,000 doors. See the tables on the next page.
2.1 (2 point) What is the sample mean over all 90 doors? What is the sample standard
deviation? You can use an Excel function STDEV.S() to evaluate the sample standard deviation.
2.2 (2 points) Is there evidence to support the claim that a standard garage door weigh 80 kg.?
What is the P-value? Use α = 0.01.
2.3 (2 points) What is the 99% confidence interval?
2
Table 2:
TD
1
2
3
4
5
6
7
8
9
10
9
81
82
80
74
75
81
83
86
88
82
1
73
77
83
81
76
76
82
83
79
84
5
85
78
76
81
82
83
76
82
86
79
TD
11
12
13
14
15
16
17
18
19
20
9
80
78
84
75
84
78
77
79
84
84
1
80
84
82
83
75
81
78
85
85
80
5
76
76
78
72
84
76
74
85
82
79
TD
21
22
23
24
25
26
27
28
29
30
9
84
83
79
86
85
82
76
85
84
80
1
81
78
83
80
81
83
83
82
83
80
5
81
80
83
79
78
84
79
77
82
83
3
3. An article shows data to compare several methods for predicting the shear strength for steel plate
girders. Data for two of these methods, the Karlsruhe and Lehigh procedures, when applied eight times
to a girder, are shown in the following table. We wish to determine whether there is any difference
between the two methods. Assume the population standard deviations are known to be 1 = 2 =
0.027.
Karlsruhe Method
Lehigh Method
1.2150
1.1419
1.2418
1.1595
1.2218
1.1546
1.2539
1.1821
1.2179
1.2219
1.1907
1.2204
1.2472
1.1607
1.1921
1.1775
3.1. (2 points) Do the data suggest that the two methods provide the same mean shear strength? Use α
= 0.01. Find the P-value rounded to 6 decimal places (using Microsoft Excel.)
3.2. (2 points) Find a 99% confidence interval on the mean difference between the two methods and use
it to answer the question in 3.1.
4
4. Wonder Shed Inc. is a manufacturer of storage sheds. The manufacturing process involves the
procurement of sheets of steel that will be used to form both the roof and the base of each shed.
The first step involves separating the material need for the roof from that needed for the base. Then the
roof and the base can be fabricated in parallel, or simultaneously. Roof fabrication involves first
punching and then forming the roof to shape. Base fabrication entails the punching-and-forming process
plus a subassembly operation.
Fabricated roofs and bases are then assembled into finished sheds that are subsequently inspected for
quality assurance. A list of activities needed to fabricate a roof, fabricate a base, and assemble a shed is
given in Table 4.1. A flowchart of the process is shown in Figure 4.1.
5
4.1 (2 points) We use a beta distribution to represent an activity duration. (Note that it is a BETA
DISTRIBUTION.) We assume that the activity durations are independent. We denote the optimistic, the
most likely and the pessimistic times by a, c and b. Calculate the mean time and the variance of each
activity duration and fill in the blanks of the table below.
Activity
a
c
b
1
5
10
20
2
15
20
35
3
10
20
25
4
5
7
10
5
5
10
20
6
10
20
30
7
10
12
15
8
15
30
40
Mean
Variance
6
4.2. (2 points) The process contains two paths:
Path 1 : Start → 1 → 3 → 5 → 7 → 8 → End
Path 2 : Start → 1 → 2 → 4 → 6 → 7 → 8 → End
What are the mean and the variance of the duration of each path? Fill in the following table.
Mean
Variance
Path 1
Path 2
4.3. (2 points) Based on Central Limit Theorem, we assume that the duration of each path is normally
distributed with mean and variance given in 4.2. We also assume that the durations of the two paths are
independent random variables. What is the probability that both paths are completed within 110
minutes?
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