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mathematics
statistics

Questions and Answers of Statistics

A woman claims that she is a fortune-teller. Specifically, she claims that she can predict the direction of the change (up or down) in the Dow Jones Industrial Average for the next 10 days (such as
The annual number of industrial accidents occurring in a particular manufacturing plant is known to follow a Poisson distribution with mean 12.a. What is the probability of observing exactly 12
Suppose that the number of customers arriving each hour at the only checkout counter in a local pharmacy is approximately Poisson distributed with an expected arrival rate of 20 customers per hour.
Suppose the number of baskets scored by the Indiana University basketball team in one minute follows a Poisson distribution with λ = 1.5. In a 10-minute span of time, what is the probability that
Suppose that the times between arrivals at a bank during the peak period of the day are exponentially distributed with a mean of 45 seconds. If you just observed an arrival, what is the probability
Consider a Poisson random variable X with parameter λ = 2.a. Find the probability that X is within one standard deviation of its mean.b. Find the probability that X is within two standard deviations
Based on historical data, the probability that a major league pitcher pitches a no-hitter in a game is about 1/1300.a. Use the binomial distribution to determine the probability that in 650 games 0,
A production manager is interested in determining the proportion of defective items in a typical shipment of one of the computer components that her company manufactures. The proportion of defective
The manager of a local fast-food restaurant is interested in improving the service provided to customers who use the restaurant’s drive-up window. As a first step in this process, the manager asks
The operations manager of a tollbooth located at a major exit of a state turnpike is trying to estimate the average number of vehicles that arrive at the tollbooth during a one-minute period during
A finance professor has just given a midterm examination in her corporate finance course and is interested in learning how her class of 250 students performed on this exam. The data are in the file
Suppose the annual return on XYZ stock follows a normal distribution with mean 0.12 and standard deviation 0.30.a. What is the probability that XYZ’s value will decrease during a year?b. What is
Assume the annual mean return on ABC stock is around 15% and the annual standard deviation is around 25%. Assume the annual and daily returns on ABC stock are normally distributed. a. What is the
Suppose Comdell Computer receives its hard drives from Diskco. On average, 4% of all hard disk drives received by Comdell are defective.a. Comdell has adopted the following policy. It samples 50 hard
A family is considering a move from a Midwestern city to a city in California. The distribution of housing costs where the family currently lives is normal, with mean $105,000 and standard deviation
The number of traffic fatalities in a typical month in a given state has a normal distribution with mean 125 and standard deviation 31.a. If a person in the highway department claims that there will
In the game of baseball, every time a player bats, he is either successful (gets on base) or he fails (doesn’t get on base). His on-base percentage, usually expressed as a decimal, is the
In the financial world, there are many types of complex instruments called derivatives that derive their value from the value of an underlying asset. Consider the following simple derivative. A
When you sum 30 or more independent random variables, the sum of the random variables will usually be approximately normally distributed, even if each individual random variable is not normally
The daily demand for six-packs of Coke at Mr. D’s supermarket follows a normal distribution with mean 120 and standard deviation 30. Every Monday the Coke delivery driver delivers Coke to Mr.
Many companies use sampling to determine whether a batch should be accepted. An (n, c) sampling plan consists of inspecting n randomly chosen items from a batch and accepting the batch if c or fewer
Suppose that if a presidential election were held today, 53% of all voters would vote for Obama over McCain. (You can substitute the names of the current presidential candidates.) This problem shows
A soft-drink factory fills bottles of soda by setting a timer on a filling machine. It has generally been observed that the distribution of the number of ounces the machine puts into a bottle is
The time it takes you to swim 100 yards in a race is normally distributed with mean 62 seconds and standard deviation 2 seconds. In your next five races, what is the probability that you will swim
A company assembles a large part by joining two smaller parts together. Assume that the smaller parts are normally distributed with a mean length of 1 inch and a standard deviation of 0.01 inch. a.
A high roller has come to the casino to play 300 games of craps. For each game of craps played there is a 0.493 probability that the high roller will win $1 and a 0.507 probability that the high
A high roller comes to the casino intending to play 500 hands of blackjack for $1 a hand. On each hand, the high roller will win $1 with probability 0.48 and lose $1 with probability 0.52. After the
A soft-drink company produces 100,000 12-ounce bottles of soda per year. By adjusting a dial, the company can set the mean number of ounces placed in a bottle. Regardless of the mean, the standard
The weekly demand for TVs at Lowland Appliance is normally distributed with mean 400 and standard deviation 100. Each time an order for TVs is placed, it arrives exactly four weeks later. That is, TV
An elevator rail is assumed to meet specifications if its diameter is between 0.98 and 1.01 inches. Each year a company produces 100,000 elevator rails. For a cost of $10/σ2 per year the company can
A 50-question true–false examination is given. Each correct answer is worth 10 points. Consider an unprepared student who randomly guesses on each question.a. If no points are deducted for
The percentage of examinees who took the GMAT (Graduate Management Admission) exam from June 1992 to March 1995 and scored below each total score is given in the file S05_72.xlsx. For example, 96% of
What caused the crash of TWA Flight 800 in 1996? Physics professors Hailey and Helfand of Columbia University believe there is a reasonable possibility that a meteor hit Flight 800. They reason as
In the decade 1982 through 1991, 10 employees working at the Amoco Company chemical research center were stricken with brain tumors. The average employment at the center was 2000 employees.
Claims arrive at random times to an insurance company. The daily amount of claims is normally distributed with mean $1570 and standard deviation $450. Total claims on different days each have this
Your company is running an audit on the Sleaze Company. Because Sleaze has a bad habit of overcharging its customers, the focus of your audit is on checking whether the billing amounts on its
Continuing the previous problem, suppose that when Sleaze overcharges a customer, the distribution of the amount overcharged (expressed as a percentage of the correct billing amount) is normally
Your manufacturing process makes parts such that each part meets specifications with probability 0.98. You need a batch of 250 parts that meet specifications. How many parts must you produce to be at
Let X be normally distributed with a given mean and standard deviation. Sometimes you want to find two values a and b such that P(a < X < b) is equal to some specific probability such as 0.90 or
As any credit-granting agency knows, there are always some customers who default on credit charges. Typically, customers are grouped into relatively homogeneous categories, so that customers within
The Excel functions discussed in this chapter are useful for solving a lot of probability problems, but there are other problems that, even though they are similar to normal or binomial problems,
You have a device that uses a single battery, and you operate this device continuously, never turning it off. Whenever a battery fails, you replace it with a brand new one immediately. Suppose the
In the previous problem, we ran the experiment for a certain number of days and then asked about the number of failures. In this problem, we take a different point of view. Suppose you operate the
In the game of soccer, players are sometimes awarded a penalty kick. The player who kicks places the ball 12 yards from the 24-foot-wide goal and attempts to kick it past the goalie into the net.
The Euro Watch Company assembles expensive wristwatches and then sells them to retailers throughout Europe. The watches are assembled at a plant with two assembly lines. These lines are intended to
Many states supplement their tax revenues with state-sponsored lotteries. Most of them do so with a game called lotto. Although there are various versions of this game, they are all basically as
For the example in Simple Decision Problem.xlsx, are there any probabilities that make the EMV criterion equivalent to the maximin criterion? Are there any probabilities that make the EMV criterion
Using a data table in Excel, perform a sensitivity analysis on the example in Simple Decision Problem.xlsx. Specifically, keep the probabilities in row 10 (for D2) as they are, but vary the
In the SciTools example, make two changes: change all references to $115,000 to $110,000, and change all references to $125,000 to $130,000. Rework the EMV calculations and the decision tree. What is
In the SciTools example, which decision would a maximin decision maker choose? Which decision would a maximax decision maker choose? Would you defend either of these criteria for this particular
In the SciTools example, use a two-way data table to see how (or whether) the optimal decision changes as the bid cost and the company’s production cost change simultaneously. Let the bid cost vary
In the SciTools example, the probabilities for the low bid of competitors, given that there is at least one competing bid, are currently 0.2, 0.4, 0.3, and 0.1. Let the second of these be p, and let
For the example in Simple Decision Problem.xlsx, we found that decision D3 is the EMV-maximizing decision for the given probabilities. See whether you can find probabilities that make decision D1 the
A decision d is said to be dominated by another decision D if, for every outcome, the payoff from D is better than (or no worse than) the payoff from d.a. Explain why you would never choose a
Besides the maximin, maximax, and EMV criteria, there are other possible criteria for making decisions. One possibility involves regret. The idea behind regret is that if you make any decision and
Referring to the previous problem, another possible criterion is called expected regret. Here you calculate the regret for each cell, take a weighted average of these regrets in each row, weighted by
In the SciTools example, you might argue that there is a continuum of possible low competitor bids (given that there is at least one competing bid), not just four possibilities. In fact, assume the
In a tree built with Precision Tree, there are two blue values at each end node, the top one of which is a probability. Why are so many of these probabilities 0 in the finished tree in Figure 6.9?
In the SciTools example, there are two equivalent decision tree structures, shown in Figures 6.5 and 6.6. Use Precision Tree to create the first of these, and verify that it yields the same EMVs and
For the completed decision tree in Figure 6.9, the monetary values in black are those you enter. The monetary values in color are calculated automatically by Precision Tree. For this particular
For the SciTools example, once you build the tree as in Figure 6.9 and then run a one-way sensitivity analysis with the dialog box filled in as in Figure 6.22, you obtain three strategy charts.
The tornado chart in Figure 6.24 and the spider chart in Figure 6.25 show basically the same information in slightly different forms. Explain in words exactly what information they provide.
Explain in words what information a two-way sensitivity chart, such as the one in Figure 6.27, provides. Demonstrate how you could provide this same information without Precision Tree’s sensitivity
For each of the following, use a one-way data table to see how the posterior probability of being a drug user, given a positive test, varies as the indicated input varies. Write a brief explanation
In the drug testing, assume there are three possible test results: positive, negative, and inconclusive. For a drug user, the probabilities of these outcomes are 0.65, 0.06, and 0.29. For a nonuser,
Referring to the previous problem, find the same probabilities through the counting argument explained in this section. Start with 100,000 athletes and divide them into the various categories.a. What
Suppose you are a heterosexual white male and are going to be tested to see if you are HIV positive. Assume that if you are HIV positive, your test will always come back positive. Assume that if you
The terms prior and posterior are relative. Assume that the drug test has been performed, and the outcome is positive, which leads to the posterior probabilities in row 20 of Figure 6.28. Now assume
In the OJ Simpson trial it was accepted that OJ had battered his wife. OJ’s lawyer tried to negate the impact of this information by stating that in a one year period, only 1 out of 2500 battered
In deciding whether to perform mandatory drug testing, we claimed that it is difficult to justify such testing under reasonable conditions. Check this yourself in the following questions.a. Drug
In the drug testing decision, find and interpret EVSI and EVPI. Here, “sample” information refers to the information from the imperfect drug test, whereas “perfect” information refers to
Explain in general why EVSI is the same, regardless of the actual cost of the information. For example, in the Acme problem EVSI is the same regardless of whether the actual cost of the test market
Following up on the previous problem, the expected net gain from information is defined as the expected amount gained by having access to the information, at its given cost, as opposed to not having
Prior probabilities are often educated guesses at best, so it is worth performing a sensitivity analysis on their values. However, you must make sure that they are varied so that all probabilities
In the Acme problem, perform a sensitivity analysis on the quantity sold from a great national market (the value in cell B11). Let this value vary over a range of values greater than the current
Using trial and error on the prior probabilities in the Acme problem, find values of them that make EVSI equal to 0. These are values where Acme will make the same decision, regardless of the
We related EVPI to the value of an envelope that contains the true ultimate outcome. This concept can be extended to “less than perfect” information. For example, in the Acme problem suppose that
The concept behind EVPI is that you purchase perfect information (the envelope), then open the envelope to see which outcome occurs, and then make an easy decision. You do not, however, get to choose
For the risky venture example, create a line chart that includes three series—that is, three lines (or curves). Each line should show the expected utility of a particular decision for a sequence of
In the risky venture example, the more risky alternative, in spite of its dominating EMV, is not preferred by a decision maker with a risk tolerance of $1.92 million. Now suppose everything stays the
In the risky venture example, suppose there is no riskless alternative; the only two possible decisions are the less risky venture and the more risky venture. Explore which of these is the preferred
Do the absolute magnitudes of the monetary outcomes matter in the risky venture example? Consider the following two possibilities. In each case, multiply all monetary values in the example by a
The Sweet Tooth Candy Company knows it will need 10 tons of sugar six months from now to implement its production plans. Jean Dobson, Sweet Tooth’s purchasing manager, has essentially two options
Carlisle Tire and Rubber, Inc., is considering expanding production to meet potential increases in the demand for one of its tire products. Carlisle’s alternatives are to construct a new plant,
A local energy provider offers a landowner $180,000 for the exploration rights to natural gas on a certain site and the option for future development. This option, if exercised, is worth an
Techware Incorporated is considering the introduction of two new software products to the market. In particular, the company has four options regarding these two proposed products: introduce neither
Consider an investor with $10,000 available to invest. He has the following options regarding the allocation of his available funds: (1) He can invest in a risk-free savings account with a guaranteed
A buyer for a large department store chain must place orders with an athletic shoe manufacturer six months prior to the time the shoes will be sold in the department stores. In particular, the buyer
Each day the manager of a local bookstore must decide how many copies of the community newspaper to order for sale in her shop. She must pay the newspaper’s publisher $0.40 for each copy, and she
Two construction companies are bidding against one another for the right to construct a new community center building in Bloomington, Indiana. The first construction company, Fine Line Homes,
Suppose that you have sued your employer for damages suffered when you recently slipped and fell on an icy surface that should have been treated by your company’s physical plant department.
One of your colleagues has $2000 available to invest. Assume that all of this money must be placed in one of three investments: a particular money market fund, a stock, or gold. Each dollar your
Consider a population of 2000 individuals, 800 of whom are women. Assume that 300 of the women in this population earn at least $60,000 per year, and 200 of the men earn at least $60,000 per year. a.
Yearly automobile inspections are required for residents of the state of Pennsylvania. Suppose that 18% of all inspected cars in Pennsylvania have problems that need to be corrected. Unfortunately,
Consider again the landowner’s decision problem described in Problem 39. Suppose now that, at a cost of $90,000, the landowner can request that a soundings test be performed on the site where
The chief executive officer of a firm in a highly competitive industry believes that one of her key employees is providing confidential information to the competition. She is 90% certain that this
A customer has approached a bank for a $100,000 one year loan at a 12% interest rate. If the bank does not approve this loan application, the $100,000 will be invested in bonds that earn a 6% annual
A company is considering whether to market a new product. Assume, for simplicity, that if this product is marketed, there are only two possible outcomes: success or failure. The company assesses that
The U.S. government is attempting to determine whether immigrants should be tested for a contagious disease. Assume that the decision will be made on a financial basis. Furthermore, assume that each
The senior executives of an oil company are trying to decide whether to drill for oil in a particular field in the Gulf of Mexico. It costs the company $600,000 to drill in the selected field.

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