Questions and Answers of Statistical Techniques in Business

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,
Let X be a continuous random variable with cdf F(x). Suppose Y = X+Δ, where Δ > 0. Show that Y is stochastically larger than X.
Suppose X is a random variable with mean 0 and variance σ2. Recall that the function Fx,ϵ(t) is the cdf of the random variable U = I1−ϵX + [1 − I1−ϵ]W, where X, 1−ϵ, and W are
Suppose that the hypothesis H0 concerns the independence of two random variables X and Y . That is, we wish to test H0 : F(x, y) = F1(x)F2(y), where F, F1, and F2 are the respective joint and
Let the scores a(i) be generated by aϕ(i) = ϕ[i/(n+ 1)], for i = 1, . . . , n, where ∫10 ϕ(u) du = 0 and ∫10 ϕ2(u) du = 1. Using Riemann sums, with subintervals of equal length, of the
Suppose the random variable e has cdf F(t). Let ϕ(u) =√12[u − (1/2)],0 < u < 1, denote the Wilcoxon score function.(a) Show that the random variable ϕ[F(ei)] has mean 0 and variance 1.(b)
The following amounts are bets on horses A,B,C,D, and E to win.Suppose the track wants to take 20% off the top, namely, $200,000. Determine the payoff for winning with a $2 bet on each of the five
Let Y have a binomial distribution in which n = 20 and p = θ. The prior probabilities on θ are P(θ = 0.3) = 2/3 and P(θ = 0.5) = 1/3. If y = 9, what are the posterior probabilities for θ = 0.3
Consider the Bayes modelBy performing the following steps, obtain the empirical Bayes estimate of θ.(a) Obtain the likelihood function(b) Obtain the mle ^β of β for the likelihood m(x|β).(c) Show
Show that P(C) = 1.
Show that P(Cc) = 1 − P(C).
Let X1,X2, . . . , Xn denote a random sample from a Poisson distribution with mean θ, 0 < θ < ∞. Let Y = Σn1 Xi. Use the loss function L[θ, δ(y)] = [θ−δ(y)]2. Let θ be an observed
Show that if C1 ⊂ C2 and C2 ⊂ C1 (that is, C1 ≡ C2), then P(C1) = P(C2).
Show that if C1, C2, and C3 are mutually exclusive, then P(C1∪C2∪C3) = P(C1) + P(C2) + P(C3).
Show that P(C1 ∪ C2) = P(C1) + P(C2) − P(C1 ∩ C2).
Consider the following mixed discrete-continuous pdf for a random vector (X, Y), (discussed in Casella and George, 1992):for α > 0 and β > 0.(a) Show that this function is indeed a joint,
If computation facilities are available, write a program for the Gibbs sampler of Exercise 11.4.7. Run your program for α = 10, β = 4, m = 3000, and n = 6000. Obtain estimates (and confidence
Let Y4 be the largest order statistic of a sample of size n = 4 from a distribution with uniform pdf f(x; θ) = 1/θ, 0 < x < θ, zero elsewhere. If the prior pdf of the parameter g(θ) =
Calculate the lim and ¯lim of each of the following sequences:(a) For n = 1, 2, . . ., an = (−1)n (2 − 4/2n).(b) For n = 1, 2, . . ., an = ncos(πn/2).(c) For n = 1, 2, . . ., an = 1/n + cos
Let {an} and {dn} be sequences of real numbers. Show that lim (an + dn) lim an + lim dn. n→∞ n→∞ n→∞
Let {an} be a sequence of real numbers. Suppose {ank} is a subsequence of {an}. If {ank} → a0 as k→∞, show that limn→∞ an ≤ a0 ≤ ¯limn→∞ an.
For the test at level 0.05 of the hypotheses given by (4.6.1) with μ0 = 30,000 and n = 20, obtain the power function, (use σ = 5000). Evaluate the power function for the following values: μ =
Define the sets A1 = {x : −∞ < x ≤ 0}, Ai = {x : i − 2 < x ≤ i − 1}, i = 2, . . . , 7, and A8 = {x : 6 < x < ∞}. A certain hypothesis assigns probabilities pi0 to these sets
A die was cast n = 120 independent times and the following data resulted:If we use a chi-square test, for what values of b would the hypothesis that the die is unbiased be rejected at the 0.025
A number is to be selected from the interval {x : 0 < x < 2} by a random process. Let Ai = {x : (i − 1)/2 < x ≤ i/2}, i = 1, 2, 3, and let A4 = {x :3/2 < x < 2}. For i = 1, 2, 3,
Consider the sample of data:(a) Obtain the five-number summary of these data.(b) Determine if there are any outliers.(c) Boxplot the data. Comment on the plot. 13 5 202 15 99 4 67 83 36 11 301 23 213
Let Y2 and Yn−1 denote the second and the (n − 1)st order statistics of a random sample of size n from a distribution of the continuous type having a distribution function F(x). Compute
Suppose X is a random variable with the pdf fX(x) = b−1f((x − a)/b), where b > 0. Suppose we can generate observations from f(z). Explain how we can generate observations from fX(x).
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from a distribution of the continuous type having distribution function F(x).(a) What is the distribution of
Two different teaching procedures were used on two different groups of students. Each group contained 100 students of about the same ability. At the end of the term, an evaluating team assigned a
Consider the problem from genetics of crossing two types of peas. The Mendelian theory states that the probabilities of the classifications (a) Round and yellow, (b) Wrinkled and
Let Y1 < Y2 < · · · < Y10 be the order statistics of a random sample from a continuous-type distribution with distribution function F(x). What is the joint distribution of V1 = F(Y4) −
Assume that the weight of cereal in a “10-ounce box” is N(μ, σ2). To test H0 : μ = 10.1 against H1 : μ > 10.1, we take a random sample of size n = 16 and observe that ¯x = 10.4 and s =
Determine a method to generate random observations for the following pdf:If access is available, write an R function which returns a random sample of observations from this pdf. f(x) = { 42³ 0 0
Let f(x) = 1/6, x = 1, 2, 3, 4, 5, 6, zero elsewhere, be the pmf of a distribution of the discrete type. Show that the pmf of the smallest observation of a random sample of size 5 from this
Let the result of a random experiment be classified as one of the mutually exclusive and exhaustive ways A1,A2,A3 and also as one of the mutually exclusive and exhaustive ways B1,B2,B3,B4. Two
Let the result of a random experiment be classified as one of the mutually exclusive and exhaustive ways A1,A2,A3 and also as one of the mutually exhaustive ways B1,B2,B3,B4. Say that 180 independent
Each of 51 golfers hit three golf balls of brand X and three golf balls of brand Y in a random order. Let Xi and Yi equal the averages of the distances traveled by the brand X and brand Y golf balls
It is proposed to fit the Poisson distribution to the following data:(a) Compute the corresponding chi-square goodness-of-fit statistic.(b) How many degrees of freedom are associated with this
A certain genetic model suggests that the probabilities of a particular trinomial distribution are, respectively, p1 = p2, p2 = 2p(1−p), and p3 = (1−p)2, where 0 < p < 1. If X1,X2,X3
Let us say the life of a tire in miles, say X, is normally distributed with mean θ and standard deviation 5000. Past experience indicates that θ = 30,000. The manufacturer claims that the tires
Suppose we are interested in a particular Weibull distribution with pdfDetermine a method to generate random observations from this Weibull distribution. If access is available, write an R function
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that the kth order statistic Yk
Let z∗ be drawn at random from the discrete distribution which has mass n−1 at each point zi = xi − ¯x + μ0, where (x1, x2, . . . , xn) is the realization of a random sample. Determine
Let Y be b(300, p). If the observed value of Y is y = 75, find an approximate 90% confidence interval for p.
Suppose a random sample of size 2 is obtained from a distribution that has pdf f(x) = 2(1 − x), 0 < x < 1, zero elsewhere. Compute the probability that one sample observation is at least
Let Y1 < Y2 be the order statistics of a random sample of size 2 from a distribution of the continuous type which has pdf f(x) such that f(x) > 0, provided that x ≥ 0, and f(x) = 0 elsewhere.
It is known that a random variable X has a Poisson distribution with parameter μ. A sample of 200 observations from this distribution has a mean equal to 3.4. Construct an approximate 90% confidence
Let X1,X2, . . .,Xn be a random sample from N(μ, σ2), where both parameters μ and σ2 are unknown. A confidence interval for σ2 can be found as follows. We know that (n − 1)S2/σ2 is a random
Two numbers are selected at random from the interval (0, 1). If these values are uniformly and independently distributed, by cutting the interval at these numbers, compute the probability that the
Let X1,X2, . . . , Xn be a random sample from a gamma distribution with known parameter α = 3 and unknown β > 0. Discuss the construction of a confidence interval for β.
When 100 tacks were thrown on a table, 60 of them landed point up. Obtain a 95% confidence interval for the probability that a tack of this type lands point up. Assume independence.
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from the exponential distribution with pdf f(x) = e−x, 0 < x < ∞, zero elsewhere.(a) Show that Z1 =
Let two independent random variables, Y1 and Y2, with binomial distributions that have parameters n1 = n2 = 100, p1, and p2, respectively, be observed to be equal to y1 = 50 and y2 = 40. Determine an
In the Program Evaluation and Review Technique (PERT), we are interested in the total time to complete a project that is comprised of a large number of subprojects. For illustration, let X1, X2, X3
Let Y1 < Y2 < Y3 < Y4 < Y5 denote the order statistics of a random sample of size 5 from a distribution of the continuous type. Compute:(a) P(Y1 < ξ0.5 < Y5).(b) P(Y1 < ξ0.25
Compute P(Y3 < ξ0.5 < Y7) if Y1 < · · · < Y9 are the order statistics of a random sample of size 9 from a distribution of the continuous type.
Let {Xn} be a sequence of p-dimensional random vectors. Show thatfor all vectors a ∈ Rp. D XnNp(μ, Σ) if and only if a'XnN₁(a'µ, a'Σa),
Let y1 < y2 < y3 be the observed values of the order statistics of a random sample of size n = 3 from a continuous type distribution. Without knowing these values, a statistician is given these
Let Y1 < Y2 < · · · < Yn denote the order statistics of a random sample of size n from a distribution that has pdf f(x) = 3x2 θ3, 0 < x < θ, zero elsewhere.(a) Show that P(c <
Let ‾X denote the mean of a random sample of size 100 from a distribution that is χ2(50). Compute an approximate value of P(49 < ‾X < 51).
Let Xn and Yn be p-dimensional random vectors. Show that ifwhere X is a p-dimensional random vector, then Yn D→ X. P Xn - Yn 0 and XnDX,
Let {an} be a sequence of real numbers. Hence, we can also say that {an} is a sequence of constant (degenerate) random variables. Let a be a real number. Show that an → a is equivalent to an P→ a.
Let ‾X denote the mean of a random sample of size 128 from a gamma distribution with α = 2 and β = 4. Approximate P(7 < ‾X < 9).

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