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statistical techniques in business
Questions and Answers of
Statistical Techniques in Business
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,
Let X1,X2, . . .,Xn be a random sample from a distribution with pdf f(x; θ) = θxθ−1, 0 < x < 1, zero elsewhere, where θ > 0. Show the likelihood has mlr in the statistic Πni=1 Xi. Use
Suppose X1, . . . , Xn is a random sample on X which has a N(μ, σ20) distribution, where σ20 is known. Consider the two-sided hypothesesH0 : μ = 0 versus H1 : μ ≠ 0.Show that the test based on
Assume that same situation as in the last exercise but consider the test with critical region C∗ = { ‾X > √σ20/nzα}. Show that the test based on C∗ has significance level α but that it
Let X1,X2,X3 be a random sample from the normal distribution N(0,σ2). Are the quadratic forms X21 +3X1X2+X22 +X1X3+X23 and X21−2X1X2 + 2/3X22 − 2X1X2 − X23 independent or dependent?
Compute the mean and variance of a random variable that is χ2(r, θ).
If at least one ϒij ≠ 0, show that the F, which is used to test that each interaction is equal to zero, has non-centrality parameter equal to c Σ=1Σ=11/02.
Let X' = [X1,X2] be bivariate normal with matrix of means μ' = [μ1, μ2] and positive definite covariance matrix Σ. LetShow that Q1 is χ2(r, θ) and find r and θ. When and only when does Q1 have
A random sample of size n = 6 from a bivariate normal distribution yields a value of the correlation coefficient of 0.89. Would we accept or reject, at the 5% significance level, the hypothesis that
The following are observations associated with independent random samples from three normal distributions having equal variances and respective means μ1, μ2, μ3.Compute the F-statistic that is
Compute the mean of a random variable that has a noncentral F-distribution with degrees of freedom r1 and r2 > 2 and non centrality parameter θ.
Extend the Bonferroni procedure described in the last problem to simultaneous testing. That is, suppose we have m hypotheses of interest: H0i versus H1i, i = 1, . . . , m. For testing H0i versus H1i,
Let X1,X2,X3,X4 denote a random sample of size 4 from a distribution which is N(0, σ2). Let Y = Σ41 aiXi, where a1, a2, a3, and a4 are real constants. If Y2 and Q = X1X2 − X3X4 are independent,
Show that the square of a noncentral T random variable is a noncentral F random variable.
Suppose X1, . . . , Xn are independent random variables with the common mean μ but with unequal variances σ2i = Var(Xi).(a) Determine the variance of ‾X.(b) Determine the constant K so that Q = K
Let X1,X2,X3,X4 be a random sample of size n = 4 from the normal distribution N(0,1). Show that Σ4i=1(Xi − ‾X)2 equalsand argue that these three terms are independent, each with a chi-square
Let A be the real symmetric matrix of a quadratic form Q in the observations of a random sample of size n from a distribution which is N(0, σ2). Given that Q and the mean ‾X of the sample are
Given the following observations associated with a two-way classification with a = 3 and b = 4, compute the F-statistic used to test the equality of the column means (β1 = β2 = β3 = β4 = 0) and
Let X1 and X2 be two independent random variables. Let X1 and Y = X1+X2 be χ2(r1, θ1) and χ2(r, θ), respectively. Here r1 < r and θ1 ≤ θ. Show that X2 is χ2(r − r1, θ − θ1).
Let μ1, μ2, μ3 be, respectively, the means of three normal distributions with a common but unknown variance σ2. In order to test, at the α = 5% significance level, the hypothesis H0 : μ1 = μ2
Suppose X1, . . . , Xn are correlated random variables, with common mean μ and variance σ2 but with correlations ρ (all correlations are the same).(a) Determine the variance of ‾X.(b) Determine
With the background of the two-way classification with c > 1 observations per cell, show that the maximum likelihood estimators of the parameters areShow that these are unbiased estimators of the
Two experiments gave the following results:Calculate r for the combined sample. n T y 100 10 20 200 12 22 Sx Sy 5 8 6 10 r 0.70 0.80
The driver of a diesel-powered automobile decided to test the quality of three types of diesel fuel sold in the area based on mpg. Test the null hypothesis that the three means are equal using the
Given the following observations in a two-way classification with a = 3, b = 4, and c = 2, compute the F-statistics used to test that all interactions are equal to zero (ϒij = 0), all column means
We wish to compare compressive strengths of concrete corresponding to a = 3 different drying methods (treatments). Concrete is mixed in batches that are just large enough to produce three cylinders.
Let Q1 and Q2 be two nonnegative quadratic forms in the observations of a random sample from a distribution which is N(0, σ2). Show that another quadratic form Q is independent of Q1 +Q2 if and only
Show that the covariance between ˆα and ˆβ is zero.
Fit y = a + x to the databy the method of least squares. X y 0 1 1 3 2 4
Suppose X is an n × p matrix with rank p.(a) Show that ker(X'X) = ker(X).(b) Use part (a) and the last exercise to show that if X has full column rank, then X'X is nonsingular.
Fit by the method of least squares the plane z = a + bx + cy to the five points (x, y, z) : (−1,−2, 5), (0,−2, 4), (0, 0, 4), (1, 0, 2), (2, 1, 0).
Suppose Y is an n × 1 random vector, X is an n × p matrix of known constants of rank p, and β is a p × 1 vector of regression coefficients. Let Y have a N(Xβ, σ2I) distribution. Discuss the
Let the independent normal random variables Y1, Y2, . . . , Yn have, respectively, the probability density functions N(μ, ϒ2x2i), i = 1, 2, . . ., n, where the given x1, x2, . . . , xn are not all
Let Y1, Y2, . . . , Yn be n independent normal variables with common unknown variance σ2. Let Yi have mean βxi, i = 1, 2, . . ., n, where x1, x2, . . . , xn are known but not all the same and β is
Let X be a continuous random variable with pdf f(x). Suppose f(x) is symmetric about a; i.e., f(x − a) = f(−(x − a)). Show that the random variables X − a and −(X − a) have the same pdf.
Obtain the sensitivity curves for the sample mean, the sample median and the Hodges–Lehmann estimator for the following data set. Evaluate the curves at the values −300 to 300 in increments of 10
Let X be a random variable with cdf F(x) and let T (F) be a functional. We say that T (F) is a scale functional if it satisfies the three propertiesShow that the following functionals are scale
Let ^Fn(x) denote the empirical cdf of the sample X1,X2, . . .,Xn. The distribution of ^Fn(x) puts mass 1/n at each sample item Xi. Show that its mean is ‾X. If T (F) = F−1(1/2) is the median,
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