Consider a company that produces guitar strings. Its records show that the expected failure time of a guitar string is 15 h. To meet the current demand, the company is going to produce 75,000 guitar strings. Assume an exponential probability distribution. (a) Determine the probability that a random guitar string lasts longer than t = 10, 12, 14, 16, 18, 20, 25, 30 h. (b) Graph the survival function versus time. (c) Determine the proportion of the 75,000 guitar strings that fail during the 17th hour. (d) Determine the proportion of guitar strings that fail during the 13th hour given that they have survived at least 12 h.
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Consider a company that produces guitar strings. Its records show that the expected failure time of a guitar string is 15 h. To meet the current demand, the company is going to produce 75,000 guitar strings. Assume an exponential probability distribution.
(a) Determine the probability that a random guitar string lasts longer than t = 10, 12, 14, 16, 18, 20, 25, 30 h.
(b) Graph the survival
(c) Determine the proportion of the 75,000 guitar strings that fail during the 17th hour.
(d) Determine the proportion of guitar strings that fail during the 13th hour given that they have survived at least 12 h.
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2.2 Consider a company that produces light bulbs. Its records show that the expected failure time of a light bulb is 1500 h. To meet the current demand, the company is going to produce 25,000 light bulbs. Assume an exponential probability distribution.
(a) Determine the probability that a random light bulb lasts longer than t = 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1800, 2000, 2500, and 3000 h.
(b) Graph the survival function versus time.
(c) Determine the proportion of the 25,000 light bulbs that fail during the 13th hundered hours.
(d) Determine the proportion of light bulbs that fail during the 16th hundered hour given that they have survived 15 hundered hours.
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2.3 Consider a company that produces a machine part. Its records show that the expected failure time of the part is 8 years. To meet the current demand, the company is going to produce 700 parts. Assume an exponential probability distribution.
(a) Determine the probability that a random machine part lasts longer than t = 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20 years.
(b) Graph the survival function versus time.
(c) Determine the proportion of the 700 parts that fail during the 7th year.
(d) Determine the proportion of parts that fail during the 9th year given that they have survived 8 years.
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Consider five design alternatives for guitar strings. The cost and expected life of each design are presented below. The target life is 16h.
Design 1 2 3 4 5
Expected life (h) 12 15 18 19 20
Cost per item, $ 0.16 0.1 0.12 0.14 0.15
Generate five tricriteria alternatives (expected life, total cost, and probability of failure before target life), and identify efficient and inefficient alternatives. (Assume an exponential distribution function.)
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Consider five design alternatives for guitar strings. The cost and expected life of each design are presented below. The target life is 16h.
Design 1 2 3 4 5
Expected life (h) 12 15 18 19 20
Cost per item, $ 0.16 0.1 0.12 0.14 0.15
Generate five tricriteria alternatives (expected life, total cost, and probability of failure before target life), and identify efficient and inefficient alternatives. (Assume an exponential distribution function.)
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