Questions and Answers of Statistical Techniques in Business

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
A university conducted a study to assess consistency of grading in a multi-section basic statistics course. To that end, the study considered the grade distribution of the course for three
For obese patients, there is an increased risk of complications in the healing of a surgical incision. A practicing surgeon learned of a new type of suturing that might improve surgical wound
A small fourth-grade class is randomly split into two groups. Each group is taught fractions using a different method. After three weeks, both groups are given the same 100-point test. The scores of
In American football a turnover is defined as a fumble or an intercepted pass. The table below gives the number of turnovers committed by the home team in four hundred forty games. Test that these
The feathers of the frizzle chicken come in three variations, or phenotypes—extreme, mild, and smooth. The genes F, the dominant for frizzle, and f, the recessive, interact in what is called
The following is the residual plot that results from fitting the equation y = 6.0 + 2.0x to a set of n = 10 points. What, if anything, would be wrong with predicting that y will equal 30.0 when x =
Would you have any reservations about fitting the following data with a straight line? Explain. x y 3 7 5 1 10 12 6 || 8 9 2 4 20 37 29 10 59 69 39 58 47 48 18 29
Students elect to take Calculus I in different years of their college career. Does the table below suggest that this affects the distribution of grades? Use α = 0.01. Year Taken I Calculus I Grade A
Burglary and larceny both involve the illegal taking of something of value. The difference, simply put, is that burglary involves unlawful entry to a structure, while larceny does not. While the two
A common saying in golf is “You drive for show, but you putt for dough.” To see if there is any truth in this assertion, data for ninety-six top money-winning golfers were examined. For each,
The fuel economy (in miles per gallon) of an automobile can depend on a number of factors, but the table below suggests that the weight of vehicle is a very good predictor.Find the 95% confidence
Many people believe that a salary bonus is a reward for good performance. The corporate world may have a different understanding. A random sample of thirty chief executive officers of large
The Super Bowl showed steady and significant growth in popularity from its beginning in 1967. This growth was reflected in ticket prices. The table gives the ticket prices in four-year intervals from
If the joint pdf of the random variables X and Y is fX,Y (x, y) = ke−(2/3)[(1/4)x2−(1/2)xy+y2] find E(X), E(Y), Var(X), Var(Y), ρ(X,Y), and k.
Among mammals, the relationship between the age at which an animal develops locomotion and the age at which it first begins to play has been widely studied. The table below lists “onset” times
Prove that a least squares straight line must necessarily pass through the point (x̅, y̅).
Biological organisms, such as yeast, often exhibit exponential growth. However, in some cases, that rapid rate of growth cannot be sustained. Such factors as lack of nutrition to support a large
The width of a Tukey confidence interval isIf k increases, but n/k and MSE stay the same, will the Tukey intervals get shorter or longer? Justify your answer intuitively. 2√MSEQa.k.n-k k
Scientists can measure tracheobronchial clearance by having a subject inhale a radioactive aerosol and then later metering the radiation level in the lungs. In one experiment (22) seven pairs of
Suppose that the population being sampled is symmetric and we wish to test H0: ˜μ = ˜μ0. Both the sign test and the signed rank test would be valid. Which procedure, if either, would you expect
Let X1,X2, . . .,Xn denote a random sample from a distribution that is N(θ, σ2), where −∞ < θ < ∞ and σ2 is a given positive number. Let Y = ‾X denote the mean of the random sample.
Let Y be b(72, 1/3). Approximate P(22 ≤ Y ≤ 28).
Show that the mean ‾X of a random sample of size n from a distribution having pdf f(x; θ) = (1/θ)e−(x/θ), 0 < x < ∞, 0 < θ < ∞, zero elsewhere, is an unbiased estimator of θ
Let X1,X2, . . . , Xn be iid N(0, θ), 0 < θ < ∞. Show that Σn1 X2i is a sufficient statistic for θ.
If az2 + bz + c = 0 for more than two values of z, then a = b = c = 0. Use this result to show that the family {b(2, θ) : 0 < θ < 1} is complete.
Let X1,X2, . . .,Xn be a random sample from each of the following distributions involving the parameter θ. In each case find the mle of θ and show that it is a sufficient statistic for θ and hence
Show that each of the following families is not complete by finding at least one nonzero function u(x) such that E[u(X)] = 0, for all θ > 0.(a)(b) N(0, θ), where 0 < θ < ∞. = {
Let Y1 < Y2 < Y3 < Y4 denote the order statistics of a random sample of size n = 4 from a distribution having pdf f(x; θ) = 1/θ, 0 < x < θ, zero elsewhere, where 0 < θ < ∞.
Let X1,X2, . . . , Xn denote a random sample from a normal distribution with mean zero and variance θ, 0 < θ < ∞. Show that Σn1 X2i /n is an unbiased estimator of θ and has variance
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from a distribution with pdfwhere −∞ < θ1 < ∞ and 0 < θ2 < ∞. Find the joint minimal
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from the uniform distribution over the closed interval [−θ, θ] having pdf f(x; θ) =
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from a N(θ, σ2), −∞ < θ < ∞, distribution. Show that the distribution of Z = Yn − ‾X does not depend
Let the pdf f(x; θ1, θ2) be of the formexp[p1(θ1, θ2)K1(x) + p2(θ1, θ2)K2(x) + H(x) + q1(θ1, θ2)], a < x < b,zero elsewhere. Suppose that K'1(x) = cK'2(x). Show that f(x; θ1, θ2) can
Let X1,X2, . . . , Xn be a random sample of size n from a geometric distribution that has pmf f(x; θ) = (1 − θ)xθ, x = 0, 1, 2, . . . , 0 < θ < 1, zero elsewhere. Show that Σn1 Xi is a
Let f(x, y) = (2/θ2)e−(x+y)/θ, 0 < x < y < ∞, zero elsewhere, be the joint pdf of the random variables X and Y .(a) Show that the mean and the variance of Y are, respectively, 3θ/2
Consider the family of probability density functions {h(z; θ) : θ ∈ Ω}, where h(z; θ) = 1/θ, 0 < z < θ, zero elsewhere.(a) Show that the family is complete provided that Ω = {θ : 0
Let ‾X denote the mean of the random sample X1,X2, . . .,Xn from a gamma type distribution with parameters α > 0 and β = θ > 0. Compute E[X1|‾x].
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from a distribution that has pdf f(x; θ) = (1/θ)e−x/θ, 0 < x < ∞, 0 < θ < ∞, zero
Show that the sum of the observations of a random sample of size n from a gamma distribution that has pdf f(x; θ) = (1/θ)e−x/θ, 0 < x < ∞, 0 < θ < ∞, zero elsewhere, is a
Let X1,X2, . . . , X5 be iid with pdf f(x) = e−x, 0 < x < ∞, zero elsewhere. Show that (X1 + X2)/(X1 + X2 + · · · + X5) and its denominator are independent.
Let X1,X2, . . . , Xn denote a random sample from a Poisson distribution with parameter θ, 0 < θ < ∞. Let Y =Σn1 Xi and let L[θ, δ(y)] = [θ − δ(y)]2. If we restrict our
Let X1,X2, . . . , Xn be a random sample of size n from a beta distribution with parameters α = θ and β = 5. Show that the product X1X2 · · ·Xn is a sufficient statistic for θ.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from the normal distribution N(θ1, θ2), −∞ < θ1 < ∞, 0 < θ2 < ∞. Show that the joint complete
Let X1,X2, . . . , Xn denote a random sample from a distribution that is N(μ, θ), 0 < θ < ∞, where μ is unknown. Let Y = Σn1 (Xi − ‾X)2/n and let L[θ, δ(y)] = [θ−δ(y)]2.
Let X1,X2, . . . , Xn be a random sample from N(θ1, θ2).(a) If the constant b is defined by the equation P(X ≤ b) = 0.90, find the mle and the MVUE of b.(b) If c is a given constant, find the mle
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from a distribution with the pdfθ1 < x < ∞, zero elsewhere, where −∞ < θ1 < ∞, 0 < θ2 <
Show that the product of the sample observations is a sufficient statistic for θ > 0 if the random sample is taken from a gamma distribution with parameters α = θ and β = 6.
Let X have the pdf fX(x; θ) = 1/(2θ), for −θ < x < θ, zero elsewhere, where θ > 0.(a) Is the statistic Y = |X| a sufficient statistic for θ? Why?(b) Let fY (y; θ) be the pdf of Y .
What is the sufficient statistic for θ if the sample arises from a beta distribution in which α = β = θ > 0?
If X1,X2, . . . , Xn is a random sample from a distribution that has a pdf which is a regular case of the exponential class, show that the pdf of Y1 =Σn1 K(Xi) is of the form fY1 (y1; θ) = R(y1)
Let Y denote the median and let ‾X denote the mean of a random sample of size n = 2k + 1 from a distribution that is N(μ, σ2). Compute E(Y |‾X = ‾x).
Let X1,X2, . . .,Xn be a random sample from a distribution with pdf f(x; θ) = θ2xe−θx, 0 < x < ∞, where θ > 0.(a) Argue that Y = Σn1 Xi is a complete sufficient statistic for θ.(b)
Show that Y = |X| is a complete sufficient statistic for θ > 0, where X has the pdf fX(x; θ) = 1/(2θ), for −θ < x < θ, zero elsewhere. Show that Y = |X| and Z = sgn(X) are independent.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample from a N(θ, σ2) distribution, where σ2 is fixed but arbitrary. Then ‾Y = ‾X is a complete sufficient statistic
Let X1, . . .,Xn be a random sample from a distribution of the continuous type with cdf F(x). Let θ = P(X1 ≤ a) = F(a), where a is known. Show that the proportion n−1#{Xi ≤ a} is the MVUE of
Let X be N(0, θ) and, in the notation of this section, let θ' = 4, θ'' = 9, αa = 0.05, and βa = 0.10. Show that the sequential probability ratio test can be based upon the statistic Σn1 X2i .
Let X and Y have a joint bivariate normal distribution. An observation (x, y) arises from the joint distribution with parameters equal to eitherorShow that the classification rule involves a
Let the random variable X have the pdf f(x; θ) = (1/θ)e−x/θ, 0 < x < ∞, zero elsewhere. Consider the simple hypothesis H0 : θ = θ' = 2 and the alternative hypothesis H1 : θ = θ'' =
If X1,X2, . . . , Xn is a random sample from a distribution having pdf of the form f(x; θ) = θxθ−1, 0 < x < 1, zero elsewhere, show that a best critical region for testing H0 : θ = 1
Let X1,X2, . . . , Xn be a random sample from the normal distribution N(θ,1). Show that the likelihood ratio principle for testing H0 : θ = θ' where θ' is specified, against H1 : θ ≠ θ' leads
Let W' = (W1,W2) be an observation from one of two bivariate normal distributions, I and II, each with μ1 = μ2 = 0 but with the respective variance covariance matricesHow would you classify W into
Let X1,X2, . . . , X10 denote a random sample of size 10 from a Poisson distribution with mean θ. Show that the critical region C defined by Σ101 xi ≥ 3 is a best critical region for testing H0 :
A random sample X1,X2, . . .,Xn arises from a distribution given byorDetermine the likelihood ratio (Λ) test associated with the test of H0 against H1. Ho : f(x;0) = 1/2, 0 < x < 0, zero elsewhere,

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