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Maths - Misc. Problems

Question # 00012650
Subject: Mathematics
Due on: 04/18/2014
Posted On: 04/18/2014 12:21 AM

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1. Eagle Credit Union (ECU) has experienced a 10% default rate with its commercial loan customers (that is, 90% of commercial loan customers pay back their loans). ECU has developed a statistical test to assist in predicting which commercial loan customers will default. The test assigns either a rating of “Approve” or “Reject” to each loan applicant. When applied to recent commercial loan customers who paid their loans, the test gave an “Approve” rating in 80% of the cases examined. When applied to recent commercial loan customers who defaulted, it gave a “Reject” rating in 70% of the cases examined.

a. Use this data to construct a joint probability table.
b. What is the conditional probability of a “Reject” rating given that the customer defaulted?
c. What is the conditional probability of an “Approve” rating given that the customer defaulted?
d. Suppose a new customer receives a “Reject” rating. If they are given the loan anyway, what is the probability that they will default?

3. A soft drink machine can be regulated (discharge level m) so that it dispenses an average of m ounces per cup. If the ounces of fill are normally distributed with mean m and standard deviation equal to 0.3 ounces. Find the setting of the discharge level m so that eight ounce cups will overflow only one percent of the time.

6. The United States Golf Association requires that the weight of a golf ball must not exceed 1.62 oz. The association periodically checks golf balls sold in the United States by sampling specific brands stocked by pro shops. Suppose that a manufacturer claims that no more than 1 percent of its brand of golf balls exceeds 1.62 oz. in weight. Suppose that 24 of this manufacturer’s golf balls are randomly selected, and let X denote the number of the 24 randomly selected golf balls that exceed 1.62 oz.

a. Find the probability that none of the randomly selected golf balls exceed 1.62 oz.?
b. Find the probability that at least one of the randomly selected golf balls exceeds 1.62 oz.
c. Suppose that two of the randomly selected golf balls are found to exceed 1.62 oz. Do you believe the claim that no more than 1 percent of this brand of golf balls exceed 1.62 oz. in weight?

7. Owing to several major ocean oil spills by tank vessels, Congress passed the 1990 Oil Pollution Act, which requires tankers to be designed with thicker hulls. Further improvements in the structural design of a tank vessel have been implemented since then, each with the objective of reducing the likelihood of an oil spill and decreasing the amount of outflow in the event of hull puncture. To aid in this development, J.C. Daidola reported on the spillage amount and cause of puncture for 50 recent major oil spills from tankers and carriers. The file OilSpill.sgd contains the data for the 50 spills reported.

a. Is any one cause more likely to occur than any other? Justify the answer using hypothesis tests.
b. Construct a 90 percent confidence interval for the difference between the mean spillage amount of accidents caused by collision and the mean spillage amount of accidents caused by fire/explosion. Interpret the result.
c. Can we say that the mean spillage amount of accidents caused by grounding is the same as the corresponding mean of accidents caused by hull failure?
d. State any assumptions required for the inferences derived from the analyses to be valid. Are these assumptions reasonably satisfied?
e. Is the variation in spillage amounts for accidents caused by collision the same as the variation in spillage amounts for accidents caused by grounding?

Q. The probability of a vehicle having an accident at a particular intersection is 0.0001. Suppose that 10,000 vehicles per day travel through the intersection.

a. What is the probability of no accidents occurring?
b. What is the probability of two or more accidents?

Q. A manufacturer of commercial television monitors guarantees the picture tube for one year (8760 hours). The monitors are used in airport terminals for flight schedules, and they are in continuous use with power on. The mean life of the tubes is 20,000 hours, and they follow an exponential time to failure density. It costs the manufacturer $300 to make, sell and deliver a monitor that will be sold for $400. It costs $150 to replace a failed tube, including materials and labor. The manufacturer has no replacement obligation beyond the first replacement.

a. What is the manufacturer’s expected profit?

b. Competition is forcing the manufacturer to consider an extended time beyond the first year to replace a tube if it fails. How long should the manufacturer guarantee the tubes if it wants to earn an expected net profit margin of 20 percent?

Tags problems misc maths golf probability balls loan accidents manufacturer caused mean rating spillage commercial randomly exceed percent suppose selected customers reject ounces test tube given customer approve recent spills hull weight puncture aociation

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Maths - Misc. Problems Solution with Detailed Working

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Tutorial Preview …(0 xxxxx 30) x 0 03 xxxxxxx and rejected x (0 xxxx xx = x 07 Paid xxx approved = xx 90)(0 xxx x 0 xx Paid and xxxxxxxx = (0 xxxxx 20) x x 18 xxx joint probability xxxxx is: Approved xxxxxxxx Total xxxxxxx x 03 x 07 0 xx Paid 0 xx 0 xx x 90 xxxxx 0 75 x 25 1 xx b xxxx xx the xxxxxxxxxxx probability of x “Reject” rating xxxxx that xxx xxxxxxxx defaulted? x What is xxx conditional probability xx an…
Maths_-_Misc._Problems_Solution_with_Detailed_Working_.docx (811.54 KB)
Preview: cup xx the xxxxxx of fill xxx normally distributed xxxx mean x xxx standard xxxxxxxxx equal to x 3 ounces xxxx the xxxxxxx xx the xxxxxxxxx level m xx that eight xxxxx cups xxxx xxxxxxxx only xxx percent of xxx time Overflowing xxxx one xxxxxxx xx the xxxx corresponds to x right-tail area xx 0 xx xxx z-value xxx this tail xxxx is 2 xxx The xxxxxxxxxxxxx xxxx volume xxxx = mean x z * xxxxxxxx deviation8 x x + x 326(0 3)8 x m + x 6978m x x – x 6978m = x 30The discharge xxxxx should xx xxx to x = 7 x ounces  6 xxx United xxxxxx xxxx Association xxxxxxxx that the xxxxxx of a xxxx ball xxxx xxx exceed x 62 oz xxx association periodically xxxxxx golf xxxxx xxxx in xxx United States xx sampling specific xxxxxx stocked xx xxx shops xxxxxxx that a xxxxxxxxxxxx claims that xx more xxxx x percent xx its brand xx golf balls xxxxxxx 1 xx xx in xxxxxx Suppose that xx of this xxxxxxxxxxxxxxxx golf xxxxx xxx randomly xxxxxxxxx and let x denote the xxxxxx of xxx xx randomly xxxxxxxx golf balls xxxx exceed 1 xx oz x xxxx the xxxxxxxxxxx that none xx the randomly xxxxxxxx golf xxxxx xxxxxx 1 xx oz If xxx manufacturer’s claim xx true, xxxx xxx probability xx any single xxxx ball exceeding xxx 1 xx xx weight xx p = x 01 In xx trials, xxx xxxxxxxxxxx that x golf balls xxxx exceed that xxxxxx is: x xxxx the xxxxxxxxxxx that at xxxxx one of xxx randomly xxxxxxxx xxxx balls xxxxxxx 1 62 xx The probability xxxx at xxxxx xxx of xxx selected golf xxxxx exceeds 1 xx oz xx xxx complement xx the probability xxxx none of xxx balls xxxxxxx xxxx weight x Suppose that xxx of the xxxxxxxx selected xxxx xxxxx are xxxxx to exceed x 62 oz xx you xxxxxxx xxx claim xxxx no more xxxx 1 percent xx this xxxxx xx golf xxxxx exceed 1 xx oz in xxxxxxxxx the xxxxxxxxxxxxxxxx xxxxx is xxxxx the probability xxxx two balls xxxx exceed xxx xxxxxx limit xxx Since there xx only a x 21% xxxxxx xx finding xxx balls that xxxxxx the weight xxxxxx out xx xx randomly xxxxxxxx balls, it xx likely that xxx manufacturer’s xxxxx xx not xxxx 7 Owing xx several major xxxxx oil xxxxxx xx tank xxxxxxxx Congress passed xxx 1990 Oil xxxxxxxxx Act, xxxxx xxxxxxxx.....
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