Stats 3y03 assignment 2 PDF

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first 3 assignments plus 2 midterms for practice...

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STATS 3Y03 ASSIGNMENT 2 Problem #1: Consider the system of components connected as in the figure below. Components 1 and 2 are connected in parallel, so that subsystem works if and only if either 1 or 2 works. Since 3 and 4 are connected in series, that subsystem works if and only if both 3 and 4 work. Suppose that P(1 works) = 0.78, P(2 works) = 0.75, P(3 works) = 0.89, and P(4 works) = 0.89. Find the probability that the system works.

Correct Answer: 0.9886 Problem #1: 0.9885655

Your Mark: 1/1 Problem #1

Attempt #1

Your Answer:

0.9885655

Your Mark:

Attempt #2

Attempt #3

1/1 ✔

Problem #2: Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let A1 be the event that the receiver functions properly throughout the warranty period. Let A2 be the event that the speakers function properly throughout the warranty period. Let A3 be the event that the CD player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with P(A1) = 0.93, P(A2) = 0.86, and P(A3) = 0.72. (a) What is the probability that at least one component needs service during the warranty period? (b) What is the probability that exactly one of the components needs service during the warranty period?

Correct Answer: 0.4241 Problem #2(a): 0.424144

Your Mark: 1/1 Correct Answer: 0.3610 Problem #2(b): 0.361032

Your Mark: 1/1 Problem #2

Attempt #1

Attempt #2

Your Answer:

2(a) 0.424144 2(b) 0.751128

2(a) 2(b) 0.361032

Attempt #3 2(a) 2(b)

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1/5

Your Mark:

2(a) 1/1 ✔ 2(b) 0/1✘

2(a) 2(b) 1/1 ✔

2(a) 2(b)

Problem #3: A box consists of 12 components, 6 of which are defective. (a) Components are selected and tested one at a time, without replacement, until a non-defective component is found. Let X be the number of tests required. Find P(X = 3). (b) Components are selected and tested, one at a time without replacement, until two consecutive non defective components are obtained. Let X be the number of tests required. Find P(X = 5). Correct Answer: 0.1364 Problem #3(a): 0.13636363

Your Mark: 1/1 Correct Answer: 0.1136 Problem #3(b): 0.11363636

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2/5

Your Mark: 1/1 Problem #3

Attempt #1

Attempt #2

Attempt #3

Your Answer:

3(a) 0.75757575 3(b) 0.85227272

3(a) 0.09090909 3(b) 0.08522727

3(a) 0.13636363 3(b) 0.11363636

3(a) 0/1✘

3(a) 0/1✘

3(a) 1/1 ✔

3(b) 0/1✘

3(b) 0/1✘

3(b) 1/1 ✔

Your Mark:

Problem #4: An assembly consists of two mechanical components. Suppose that the probabilities that the first and second components meet specifications are 0.93 and 0.89, respectively. Let X be the number of components in the assembly that meet specifications. (a) Find the mean of X. (b) Find the variance of X. Correct Answer: 1.8200 Problem #4(a): 1.82

Your Mark: 1/1 Correct Answer: 0.1630 Problem #4(b): 0.163

Your Mark: 1/1 Problem #4

Attempt #1

Your Answer:

4(a) 1.82 4(b) 0.1662

4(a) 4(b) 0.163

4(a) 4(b)

4(a) 1/1 ✔

4(a) 4(b) 1/1 ✔

4(a) 4(b)

Your Mark:

4(b) 0/1✘

Attempt #2

Attempt #3

Problem #5: A manufacturing process has 49 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically 2% of the components are identified as defective, and the components can be assumed to be independent. (a) If the manufacturer stocks 52 components, what is the probability that the 49 orders can be filled without reordering components? (b) Let X be the number of good (i.e., non-defective) components among the 52 in stock. Find the mean of X. (c) Find the variance of X  [from part (b)]. Problem #5(a):

0.97976508 Correct Answer: 0.9798

Your Mark: 1/1 Correct Answer: 50.9600 Problem #5(b): 50.96

Your Mark: 1/1 Correct Answer: 1.0192 Problem #5(c): 1.0192

Your Mark: 1/1 Problem #5

Attempt #1

Your Answer:

5(a) 0.97976508 5(b) 50.96 5(c) 1.0192

5(a) 5(b) 5(c)

5(a) 5(b) 5(c)

5(a) 1/1 ✔ 5(b) 1/1 ✔

5(a) 5(b) 5(c)

5(a) 5(b) 5(c)

Your Mark:

Attempt #2

Attempt #3

5(c) 1/1 ✔

Problem #6: The probability that a randomly selected box of a certain type of cereal has a particular prize is 0.10. Suppose that you purchase box after box until you have obtained 3 of these prizes. (a) What is the probability that you purchase exactly 8 boxes? (b) What is the probability that you purchase at least 12 boxes? (c) How many boxes would you expect to purchase, on average? Correct Answer: 0.0124 Problem #6(a): 0.01240029

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3/5

Your Mark: 1/1 Correct Answer: 0.9104 Problem #6(b): 0.91043849

Your Mark: 1/1 Correct Answer: 30.0000 Problem #6(c): 30

Your Mark: 1/1 Problem #6 Your Answer:

Your Mark:

Attempt #1

Attempt #2

Attempt #3

6(a) 0.02066715 6(b) 0.8716961 6(c) 30

6(a) 0.01240029 6(a) 6(b) 6(b) 0.91043849 6(c) 6(c)

6(a) 0/1✘ 6(b) 0/1✘ 6(c) 1/1 ✔

6(a) 1/1 ✔ 6(b) 1/1 ✔ 6(c)

6(a) 6(b) 6(c)

Problem #7: A geologist has collected 21 specimens of basaltic rock and and 16 specimens of granite. The geologist instructs a laboratory assistant to randomly select 8 of the specimens for analysis. (a) What is the probability that at least 4 of the selected specimens are granite? (b) What is the expected number of granite specimens in the sample? (c) If this same process is repeated every day, how many days (on average) will it take before getting a sample consisting entirely of granite? Correct Answer: 0.4827 Problem #7(a): 0.48271724

Your Mark: 1/1 Correct Answer: 3.4595 Problem #7(b): 3.459459459

Your Mark: 1/1

Problem #7(c): 3000

round your answer to the nearest integer Correct Answer: 3000 Your Mark: 1/1

Problem #7 Your Answer:

Your Mark:

Attempt #1

Attempt #2

Attempt #3

7(a) 0.48271724 7(b) 3.45945946 7(c) 3000

7(a) 7(b) 7(c)

7(a) 7(b) 7(c)

7(a) 1/1 ✔ 7(b) 1/1 ✔ 7(c) 1/1 ✔

7(a) 7(b) 7(c)

7(a) 7(b) 7(c)

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4/5

Problem #8: Data from the Central Hudson Laboratory determined that the mean number of insect fragments in 225-gram chocolate bars was 14.4 (http://www.centralhudsonlab.com/chocolates.shtml). In a 42-gram bar the mean number of insect fragments would then be 2.69. Assume that the number of insect fragments follows a Poisson distribution. (a) If you eat a 42-gram chocolate bar, find the probability that you will have eaten at least 3 insect fragments. (b) If you eat a 42-gram chocolate bar every week for 15 weeks, find the probability that you will have eaten no insect fragments in exactly 6 of those weeks. Correct Answer: 0.5039 Problem #8(a): 0.5039227

Your Mark: 1/1 Correct Answer: 0.0003 Problem #8(b): 0.00032115

Your Mark: 1/1 Problem #8

Attempt #1

Your Answer:

8(a) 0.5039227 8(b) 0.00032115

8(a) 8(b)

8(a) 8(b)

8(a) 1/1 ✔

8(a) 8(b)

8(a) 8(b)

Your Mark:

8(b) 1/1 ✔

Attempt #2

Attempt #3

Problem #9: Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. Suppose that X has the following Rayleigh pdf.

(x/θ2) e−x2/(2θ2) x  >  0 f (x)  =

{ 0

otherwise

(a) If θ = 100, find the probability that the vibratory stress is between 83 and 388. (b) If θ = 100, then 84% of the time the vibratory stress is greater than what value? Correct Answer: 0.7081 Problem #9(a): 0.70807173

Your Mark: 1/1

Problem #9(b): 59.05139916

round your answer to 2 decimals Correct Answer: 59.05 Your Mark: 1/1

Problem #9

Attempt #1

Your Answer:

9(a) 0.70807173 9(b) 59.05139916

9(a) 9(b)

9(a) 9(b)

9(a) 1/1 ✔

9(a) 9(b)

9(a) 9(b)

Your Mark:

9(b) 1/1 ✔

Attempt #2

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Attempt #3

5/5...