The XYZ Corporation is considering plant expansion to enable it to begin production of a new military helicopter. The company’s CEO must decide whether to order a medium- or a large scale expansion.
Demand for the new military helicopter is uncertain because of continuing Congressional funding and the interest of our allies. Demand could be low, moderate, or high depending on domestic and foreign sales. The probability estimates for demand are 0.20, 0.50, and 0.30, respectively.
The profit from a medium-sized expansion is forecast to be 4, 18, and 10 billion dollars from low, moderate, or high demand, respectively. The profit from a large-sized expansion is forecast to be 0, 12, and 28 billion dollars from low, moderate, or high demand, respectively.
1.a.What is the proper decision if the objective is to maximize the expected profit?
b.How much profit is expected from this decision?
2.a.What is the proper decision if the objective is to minimize the expected risk? (Hint: risk is measured by standard deviation.)
b.How much profit is expected from this decision?
B.A manufacturer receiving a very large shipment of parts decides to accept delivery if, in a random sample of twenty parts, not more than one of the sampled parts is defective. (That is, the shipment will not be accepted if there is more than one defective part in the sample of 20.) The supplier of the parts claims that 4% of its parts are defective.
The manufacturer selects a sample of 20 parts from the shipment for quality testing.
NOTE: In this problem, report probabilities as decimals rounded to four (4) decimal places and expected values as decimals rounded to two decimal places.
3.What is the chance that none of the sampled parts is defective?
4.What is the chance that exactly one of the sampled parts is defective?
5.What is the chance that 19 of the sampled parts are not defective?
6.What is the chance that the shipment will be accepted by the manufacturer?
7.In a sample of 20 parts, how many defective parts should be expected?
8.What is the chance that fewer parts than expected are defective?
9.A quality control engineer for the parts supplier needs at least four defective parts for testing to determine the causes of failure. To expect to find at least four defects, at least how many parts should the quality control engineer expect to have to sample?
ABC Paper Corporation manufactures a special grade of laminated (plastic coated) paper. For the current process being used, the quality control staff has shown that, on average, one defect in the lamination can be expected in every 1.25 square feet of laminated paper.
10.A square foot of the product is randomly selected for inspection after the coating is applied. What is the chance that there are no defects in the coating in this area?
11.A square foot of the product is randomly selected for inspection after the coating process. What is the chance that there is at least one defect in this area?
12.What is the chance that in the next five square feet of laminated paper, exactly as many defects as expected will be found?
13.What is the chance that in the next five square feet of laminated paper, half as many defects will be found as expected?
By corporate policy, the coating process must be stopped if the coating process if it is more likely than not that there will be more than one defect in the coating in any two square feet of laminated paper. (In other words, the coating process is stopped if the chance of finding more than one defect in the coating in any two square feet of the laminated paper exceeds 0.50).
14.a.What is the probability of finding more than one defect in any two square feet of laminated paper?
b.Does this particular paint shop meet the stated probability criterion so that it will be shut down? Pick either yes or no.
D.The average price of a 42-inch television sets listed on the website of large national retailer is $790. Assume that the price of these televisions is normally distributed with a standard deviation of $160.
NOTE: In this problem, report your calculated probabilities as decimals rounded to four (4) decimal places. Money values should be reported as decimals rounded to two (2) decimal places.
15.What is the probability that a randomly selected television from the site sells for
a.less than $700?
b.between $400 and $500?
c.between $900 and $1,000?
16.The price intervals about the mean in parts 15.b and 15.c are equal ($100). Why are the probabilities so different?
17.Suppose you are shopping for a new 42-inch television. Televisions in the top five percent of the distribution are too expensive to be considered. What is the most a television can cost to stay below the top five percent threshold?
18.Suppose you are shopping for a new 42-inch television. You have budgeted no more than $750 for the purchase. There are 15 42-inch televisions listed on the website. How many of these 15 are within your budget?
The manager of a paint supply store wants to know the actual amount of paint contained in 1-gallon cans purchased from a nationally known manufacturer. A random sample of 60 cans is selected, and on average, the amount of paint per can is 0.991 gallon with a standard deviation of 0.012 gallon.
NOTE: In this problem, report your calculated results for the confidence limits to four (4) decimal places.
19.a.What is the point estimate for the mean amount of paint per can?
b.On the basis of this point estimate, what do you think the manager suspects?
20.For estimation, the manager sets the confidence level at 95 percent.
a.What is the margin of error at 95 percent confidence?
b.Calculate and state a 95 percent confidence interval for the actual mean amount of paint per can.
21.a. At 95 percent confidence, what is the chance that the true mean is actually larger than the upper bound of the 95 percent confidence interval stated in part 20.b?
b.At 95 percent confidence, what is the chance that the cans are filled with 1 gallon, on average?
c.At 95 percent confidence, is the manager justified in his opinion stated in part 19.b?
22.The manufacturer responds by telling the manager to redo the estimate at the 90 percent confidence level.
a.Calculate the 90 percent confidence interval.
b.At 90 percent confidence, what is the chance that on average a can is actually filled with 1 gallon of paint?
23.a.On the basis of your results, does the manager still have a legitimate right to doubt the claim? Choose either yes or no.
b.Why? (Be specific!)
24.What assumption or assumptions had to be made regarding the population to calculate the interval estimate in both parts 20.b and 22.a?
The IRS reported that 62% of individual tax returns were filed electronically in 2010. In a sample of 225 tax returns from 2013, 163 were filed electronically.
NOTE: In this problem, report all calculated results as decimals rounded to four (4) decimal places.
25.What is the point estimate for the true but unknown proportion of electronic filings in 2013?
26.Calculate and state a 95 percent confidence interval for the true proportion of taxpayers who filed electronically in 2013.
27.What is the margin of error for this interval?
28.What is the chance that the true proportion of electronic filings in 2013 is not contained within the interval calculated in part 26?
29.What is the chance that true proportion of electronic filings in 2013 is actually higher than the upper limit of the confidence interval calculated in part 26.
30.What conditions are necessary for the estimation performed in part 26? Show that these conditions are satisfied here.
31.Is there any evidence that the proportion of electronic filings has changed significantly from 2010 to 2013? Briefly state your evidence and reasoning.