Maths - Misc. Problems Solution with Detailed Working

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?