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statistical techniques in business
Questions and Answers of
Statistical Techniques in Business
Suppose that n fair dice are rolled. What are the chances that all n faces will be the same?
Let P be the set of right triangles with a 5'' hypotenuse and whose height and length are a and b, respectively. Characterize the outcomes in P.
Suppose a baseball player steps to the plate with the intention of trying to “coax” a base on balls by never swinging at a pitch. The umpire, of course, will necessarily call each pitch either a
Does a monkey have a better chance of rearrangingACCLLU U S to spell CALCU LU SorAABEGLR to spell ALGEBRA?
In international Morse code, each letter in the alphabet is symbolized by a series of dots and dashes: the letter a, for example, is encoded as “· –”. What is the minimum number of dots and/or
Given that P(A ∩ BC) = 0.3, P((A ∪ B)C) = 0.2, and P(A ∩ B) = 0.1, find P(A|B).
Monica’s vacation plans require that she fly from Nashville to Chicago to Seattle to Anchorage. According to her travel agent, there are three available flights from Nashville to Chicago, five from
Given that P(A) + P(B) = 0.9, P(A|B) = 0.5, and P(B|A) = 0.4, find P(A).
The board of a large corporation has six members willing to be nominated for office. How many different “president/vice president/treasurer” slates could be submitted to the stockholders?
Find A ∩ B ∩ C if A = {x: 0 ≤ x ≤ 4}, B = {x: 2 ≤ x ≤ 6}, and C = {x: x = 0, 1, 2, . . .}.
The nine members of the music faculty baseball team, the Mahler Maulers, are all incompetent, and each can play any position equally poorly. In how many different ways can the Maulers take the field?
Nine cards are dealt from a 52-card deck. Write a formula for the probability that three of the five even numerical denominations are represented twice, one of the three face cards appears twice, and
Let A be the set of five-card hands dealt from a fifty-two-card poker deck, where the denominations of the five cards are all consecutive—for example, (7 of hearts, 8 of spades, 9 of spades, 10 of
A coke hand in bridge is one where none of the thirteen cards is an ace or is higher than a 9. What is the probability of being dealt such a hand?
A pinochle deck has forty-eight cards, two of each of six denominations (9, J, Q, K, 10, A) and the usual four suits. Among the many hands that count for meld is a roundhouse, which occurs when a
A coin is tossed four times and the resulting sequence of heads and/or tails is recorded. Define the events A,B, and C as follows:A: exactly two heads appearB: heads and tails alternateC: first two
If the definition of n! is to hold for all nonnegative integers n, show that it follows that 0! must equal 1.
The crew of Apollo 17 consisted of a pilot, a copilot, and a geologist. Suppose that NASA had actually trained nine aviators and four geologists as candidates for the flight. How many different crews
Stanley’s statistics seminar is graded on a Pass/Fail basis. At the end of the semester each student is given the option of taking either a two-question exam (Final A) or a three-question exam
What is the smallest number of switches wired in parallel that will give a probability of at least 0.98 that a circuit will be completed? Assume that each switch operates independently and will
In an upstate congressional race, the incumbent Republican (R) is running against a field of three Democrats (D1, D2, and D3) seeking the nomination. Political pundits estimate that the probabilities
Which state name can generate more permutations, TENNESSEE or FLORIDA?
Let A and B be any two events defined on a sample space S. Which of the following sets are necessarily subsets of which other sets?A B A∪B A∩B AC ∩ BA ∩ BC (AC ∪ BC)C
An interior decorator is trying to arrange a shelf containing eight books, three with red covers, three with blue covers, and two with brown covers.(a) Assuming the titles and the sizes of the books
Suppose a pitcher faces a batter who never swings. For how many different ball/strike sequences will the batter be called out on the fifth pitch?
Prove thatdirectly without appealing to any combinatorial arguments. n+1 (" + ¹) = ( 2 ) + (k²1) k
What is the coefficient of w2x3yz3 in the expansion of (w + x + y + z)9?
An automobile insurance company has compiled the information summarized below on its policy-holders. Suppose someone calls to file a claim. To which age group does he or she most likely belong? Age
Make an anagram out of the familiar expression STATISTICS IS FUN. In how many ways can the letters in the anagram be permuted?
How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are collinear?
The Alpha Beta Zeta sorority is trying to fill a pledge class of nine new members during fall rush. Among the twenty-five available candidates, fifteen have been judged marginally acceptable and ten
The final exam in History 101 consists of five essay questions that the professor chooses from a pool of seven that are given to the students a week in advance. For how many possible sets of
(a) Let 0 < a < b. Which number is larger?(b) Let a > 0. Which number is larger? b 2π -²/2 dz or -212 ² -2²12² dz -b. √2π
In how many ways can the letters in M I S S I S S I P P Ibe arranged so that no two I’s are adjacent?
If E(W) = μ and Var(W) = σ2, show that E = (W = ") = 0 = 0 and Var W-"). = 1
Hawaii’s fourteen-thousand-foot Mauna Loa is one of the world’s most active volcanoes. Between 1832 and 1950, it erupted thirty-seven times, which translates to a rate of 0.027 times per month.
In 2009 a donor who insisted on anonymity gave seven-figure donations to twelve universities. A media report of this generous but somewhat mysterious act identified that all of the universities
A certain hospitalization policy pays a cash benefit for up to five days in the hospital. It pays $250 per day for the first three days and $150 per day for the next two. The number of days of
For a high-risk driver, the time in days between the beginning of a year and an accident has an exponential pdf. Suppose an insurance company believes the probability that such a driver will be
Frankie and Johnny play the following game. Frankie selects a number at random from the interval [a, b]. Johnny, not knowing Frankie’s number, is to pick a second number from that same interval and
Suppose the charge for repairing an automobile averages $200 with a standard deviation of $16. If a 10% tax is added to the charge and then a $15 flat fee for environmental impact, what is the
Among the many letters sent to a popular advice to - the -lovelorn columnist, was one involving a paternity issue that raised an interesting statistical question. The distraught writer—call her
Use the method of maximum likelihood to estimate θ in the pdfEvaluate θe for the following random sample of size 4: y1 = 6.2, y2 = 7.0, y3 = 2.5, and y4 = 4.2. fy (y: 0) = 0 2√5 e √, y 20
The town of Willoughby has just been selected as the site for a new automobile assembly complex. City officials estimate that the anticipated influx of job-seeking families will add some four hundred
If the standard deviation (σ) associated with the pdf that produced the following sample is 3.6, would it be correct to claim thatis a 95% confidence interval for μ? Explain.
A random sample of size 8—x1 = 1, x2 = 0, x3 = 1, x4 = 1, x5 = 0, x6 = 1, x7 = 1, and x8 = 0—is taken from the probability function pX (k; θ) = θk(1 − θ)1−k, k = 0, 1; 0 < θ < 1Find
If ˆθ is sufficient for θ, show that any one-to-one function of ˆθ is also sufficient for θ.
Suppose that y1, y2, . . . , yn is a random sample of size n from a normal distribution where σ is known. Depending on how the tail-area probabilities are split up, an infinite number of random
Show that ˆσ2 =Σni=1 Y2i is sufficient for σ2 if Y1,Y2, . . . ,Yn is a random sample from a normal pdf with μ = 0.
What is the squared-error loss Bayes estimate for the parameter θ in a binomial pdf, where θ has a uniform distribution—that is, a noninformative prior? (Recall that a uniform prior is a beta pdf
For the negative binomial pdf pX (k; p, r) =find the maximum likelihood estimator for p if r is known. (k+r-11 (*+/-¹) (1 - p) ^ p², k
Let Y1,Y2, . . . ,Yn be a random sample of size n from the pdf fY (y; θ) = 1/(r − 1)!θr yr−1e−y/θ, y > 0(a) Show that ˆθ = 1/r y̅ is an unbiased estimator for θ.(b) Show that ˆθ =
Which of the following two intervals has the greater probability of containing the binomial parameter p? X n -0.67₁ (X/n)(1-X/n) n or X n +0.67₁ X n 8 (X/n)(1-X/n) n
During one of the first “beer wars” in the early 1980s, a taste test between Schlitz and Budweiser was the focus of a nationally broadcast TV commercial. One hundred people agreed to drink from
Use the method of moments to derive estimates for the parameters r and λ in the gamma pdf, fy (y; r, ) = T(r) les, y = 0
A criminologist is searching through FBI files to document the prevalence of a rare double-whorl fingerprint. Among six consecutive sets of 100,000 prints scanned by a computer, the numbers of
Given the massive amount of information available from the internet, public libraries might have little support. Yet, in a Pew Research Center poll of 6224 Americans age sixteen and over, 3921 said
A machine stamps out a certain type of automobile part. When working properly the part has an average weight of μ = 1.6 pounds and standard deviation σ = 0.22 pounds. To test the proper working of
How long does it take to fly from Atlanta to New York’s LaGuardia airport? There are many components of the time elapsed, but one of the more stable measurements is the actual in-air time. For a
Kepler’s third law states that “the squares of the periods of the planets are proportional to the cubes of their mean distance from the Sun.” Listed below are the periods of revolution (x), the
Given the pdf fY (y; θ) = 2y θ2 , 0 ≤ y ≤ θ. Take a sample of size 3 from this pdf. Use the statistic Ymax to test H0: θ = 5 versus H1: θ > 5. Recall that for n = 3 the pdf of Ymax is
Mandatory helmet laws for motorcycle riders remain a controversial issue. Some states have a “limited” ordinance that applies only to younger riders; others have a “comprehensive” statute
What “confidence” is associated with each of the following random intervals? Assume that the Yi’s are normally distributed.(a)(b)(c)(d) [Y- Y - 2.0930 S 20 °(5)] 20 Y + 2.0930
Evaluate the following probabilities:(a) P(χ217 ≥ 8.672)(b) P(χ26 < 10.645)(c) P(9.591 ≤ χ220 ≤ 34.170)(d) P(χ22 < 9.210)
Male cockroaches can be very antagonistic toward other male cockroaches. Encounters may be fleeting or quite spirited, the latter often resulting in missing antennae and broken wings. A study was
Find the value y that satisfies each of the following equations:(a) P(χ29 ≥ y) = 0.99(b) P(χ215 ≤ y) = 0.05(c) P(9.542 ≤ χ222 ≤ y) = 0.09(d) P(y ≤ χ231 ≤ 48.232) = 0.95
For what value of n is each of the following statements true?(a) P(χ2n ≥ 5.009) = 0.975(b) P(27.204 ≤ χ2n ≤ 30.144) = 0.05(c) P(χ2n ≤ 19.281) = 0.05(d) P(10.085 ≤ χ2n ≤ 24.769) = 0.80
Luxury suites, many costing more than $100,000 to rent, have become big-budget status symbols in new sports arenas. Below are the numbers of suites (x) and their projected revenues (y) for nine of
Aedes aegypti is the scientific name of the mosquito that transmits yellow fever. Although no longer a major health problem in the Western world, yellow fever was perhaps the most devastating
Depth perception is a life-or-death ability for lambs inhabiting rugged mountain terrain. How quickly a lamb develops that faculty may depend on the amount of time it spends with its ewe. Thirteen
In athletic contests, a wide-spread conviction exists that the home team has an advantage. However, one explanation for this is that a team schedules some much weaker opponents to play at home. To
To see whether teachers’ expectations for students can become self-fulfilling prophecies, fifteen first graders were given a standard IQ test. The children's’ teachers, though, were told it was a
Among young drivers, roughly a third of all fatal automobile accidents are speed-related; by age 60 that proportion drops to about one-tenth. Listed below are a recent year’s percentages of
Gorillas are not the solitary creatures that they are often made out to be: They live in groups whose average size is about 16, which usually includes three adult males, six adult females, and seven
Roughly 360,000 bankruptcies were filed in U.S. Federal Court during 1981; by 1990 the annual number was more than twice that figure. The following are the numbers of business failures reported year
The diversity of bird species in a given area is related to plant diversity, as measured by variation in foliage heights as well as the variety of flora. Below are indices measured on those two
Male toads often have trouble distinguishing between other male toads and female toads, a state of affairs that can lead to awkward moments during mating season. When male toad A inadvertently makes
Nine political writers were asked to assess the United States’ culpability in murders committed by revolutionary groups financed by the CIA. Scores were assigned using a scale of 0 to 100. Three of
To see whether low-priced homes are easier to sell than moderately priced homes, a national realty company collected the following information on the lengths of times homes were on the market before
Investment firms financing the construction of new shopping centers pay close attention to the amount of retail floor space already available. Listed below are population and floor space figures for
The following is a breakdown of what 120 college freshmen intend to do next summer. Male Female Work School Play 22 14 14 31 19 20
A pharmaceutical company is testing two new drugs designed to improve the blood-clotting ability of hemophiliacs. Six subjects volunteering for the study are randomly divided into two groups of size
An efficiency study was done on the delivery of first-class mail originating from the four cities listed in the following table. Recorded for each city was the average length of time (in days) that
Two methods (A and B) are available for removing dangerous heavy metals from public water supplies. Eight water samples collected from various parts of the United States were used to compare the two
To illustrate the complexity and arbitrariness of IRS regulations, a tax-reform lobbying group has sent the same five clients to each of two professional tax preparers. The following are the
An investigation was conducted of one hundred seven fatal poisonings of children. Each death was caused by one of three drugs. In each instance it was determined how the child received the fatal
Out of one hundred twenty senior citizens polled, sixty-five favored a complete overhaul of the health care system while fifty-five preferred more modest changes. When the same choice was put to
As part of an affirmative-action litigation, records were produced showing the average salaries earned by white, black, and Hispanic workers in a large manufacturing plant. Three different
In Eastern Europe a study was done on fifty people bitten by rabid animals. Twenty victims were given the standard Pasteur treatment, while the other thirty were given the Pasteur treatment in
A consumer advocacy group, investigating the prices of steel-belted radial tires produced by three major manufacturers, collects the following data. Year Company A Company B 1995 2000 2005 Company B
As part of a review of its rate structure, an automobile insurance company has compiled the following data on claims filed by five male policyholders and five female policyholders.
Among a number of beliefs concerning the socalled lunar effect is that more children are born during the full moon. The table below gives the average number of births per day during the full moon
Raynaud’s syndrome is characterized by the sudden impairment of blood circulation in the fingers, a condition that results in discoloration and heat loss. The magnitude of the problem is evidenced
Despite its complexity, the English language does not afford many options for how subordinates might address supervisors in the workplace. Calling bosses by their first names strikes a tone that many
To see if any geographical pricing differences exist, the cost of a basic-cable TV package was determined for a random sample of six cities, three in the southeast and three in the northwest. Monthly
While there are a number of diet and physical causes of childhood obesity, there could also be an economic element. Use the table below to test if childhood obesity is related to family income. Use
A public relations firm hired by a would-be presidential candidate has conducted a poll to see whether their client faces a gender gap. Out of 800 men interviewed, 325 strongly supported the
A company claims to have produced a blended gasoline that can improve a car’s fuel consumption. They decide to compare their product with the leading gas currently on the market. Three different
Some studies have suggested that pilots have an unusual distribution of the sex of their children. A study of children of military instructor pilots at NAS Whiting Field in Milton, Florida compared
In a survey conducted by State University’s Learning Center, a sample of three freshmen said they studied 6, 4, and 10 hours, respectively, over the weekend. The same question was posed to three
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