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Questions and Answers of Statistics

Consider a two-sided confidence interval for the mean μ when σ is known;Where a1 + a2 = a. If a1 = a2 = a/2, we have the usual 100(1 €“ a) % confidence interval for μ. In the
It is possible to construct a nonparametric tolerance interval that is based on the extreme values in a random sample of size n from any continuous population. If p is the minimum proportion of the
Suppose that X1, X2., Xn is a random sample from a continuous probability distribution with median μ(a) Show thatHint: The complement of the event [min(Xi) (b) Write down a 100 (1 €“
Students in the industrial statistics lab at ASU calculate a lot of confidence intervals on μ suppose all these CIs are independent of each other. Consider the next one thousand 95% confidence
Suppose we have a random sample of size 2n from a population denoted by X, and and E(X = μ and V(X) = σ2. Let be two estimators of μ.Which is the better estimator of μ? Explain your
Let X1, X2, .. , X7 denote a random sample from a population having mean and variance. Consider the following estimators of μ:(a) Is either estimator unbiased?(b) Which estimator is best? In what
Suppose that Ф1 and Ф2 are unbiased estimators of the parameter θ. We know that V (Ф1) = 10 and V(Ф2) = 4. Which estimator is best and in what sense is it best?
Calculate the relative efficiency of the two estimators in Exercise 7-2.
Calculate the relative efficiency of the two estimators in Exercise 7-2.
Suppose that Ф1 and Ф2 are estimators of the parameter θ. We know that E(Ф1) = θ, E(Ф2) = θ/2m V (Ф1) = 10, V(Ф2) = 4. Which estimator is best? In
Suppose that Ф1, Ф2 and Ф3 are estimators of θ. We know that E(Ф1) = 10 and E(Ф2)– θ, E(Ф3)  θ, V(Ф2) = 12, V(Ф2) = 10, and
Let three random samples of sizes n1 = 20, n2 = 10, and n3 = 8 be taken from a population with mean μ and variance σ2. Let S2/1, S2/2, and S2/3 be the sample variances. Show that S2 =
a) Show that Σ = 1 (Xi – X)2/n. is a biased estimator of σ2. (b) Find the amount of bias in the estimator. (c) What happens to the bias as the sample size n increases? (
Let X1, X2, . , Xn be a random sample of size n from a population with mean μ and variance σ2. (a) Show that X2 is a biased estimator for μ2. (b) Find the amount of bias in this
Data on pull-off force (pounds) for connectors used in an automobile engine application are as follows: 79.3, 75.1, 78.2, 74.1, 73.9, 75.0, 77.6, 77.3, 73.8, 74.6, 75.5, 74.0, 74.7, 75.9, 72.9, 73.8,
Data on oxide thickness of semiconductors are as follows: 425, 431, 416, 419, 421, 436, 418, 410, 431, 433, 423, 426, 410, 435, 436, 428, 411, 426, 409, 437, 422, 428, 413, 416.(a) Calculate a point
X1, X2, .., Xa is a random sample from a normal distribution with mean and variance σ2. Let Xmin and Xmax be the smallest and largest observations in the sample. (a) Is (Xmin + X man)/2) an
Suppose that X is the number of observed “successes” in a sample of n observations where p is the probability of success on each observation. (a) Show that P = X/n is an unbiased estimator of
X1 and S2/1 are the sample mean and sample variance from a population with mean μ and variance similarly, and are the sample mean and sample variance from a second independent population with
Continuation of Exercise 7-15. Suppose that both populations have the same variance; that is σ2/1 = σ2/2 = σ. Show that is an unbiased estimator of σ2.
Two different plasma etchers in a semiconductor factory have the same mean etch rate μ. However, machine 1 is newer than machine 2 and consequently has smaller variability in etch rate. We know
Of n1 randomly selected engineering students at ASU, owned an HP calculator, and of randomly selected engineering students at Virginia Tech owned an HP calculator. Let p1 and p2 be the probability
Consider the Poisson distribution.Find the maximum likelihood estimator of λ, based on a random sample of size n.
Consider the shifted exponential distribution.When θ = 0, this density reduces to the usual exponential distribution. When θ > 0, there is only positive probability to the right of
Let X be a geometric random variable with parameter p. Find the maximum likelihood estimator of p, based on a random sample of size n.
Let X be a random variable with the following probability distribution:
Consider the Weibull distributionf(x) = {β/δ(x/δ) β€“1 e €“ (x/δ), 0 (a) Find the likelihood function based on a random sample of size n. Find the
Consider the probability distribution in Exercise 7-22. Find the moment estimator of θ.
Let X1, X2, , Xn be uniformly distributed on the interval 0 to a. Show that the moment estimator of a is a = aX. Is this an unbiased estimator? Discuss the reasonableness of this estimator.
Let X1, X2, , Xn be uniformly distributed on the interval 0 to a. Recall that the maximum likelihood estimator of a is a = max (Xi).(a) Argue intuitively why cannot be an unbiased estimator for a.(b)
For the continuous distribution of the interval 0 to a, we have two unbiased estimators for a: the moment estimator a1 = 2X and a2 = [(n + 1/n] max (Xi), where max (Xi) is the largest observation in
Consider the probability density functionFind the maximum likelihood estimator for θ.
The Rayleigh distribution has probability density function(a) It can be shown that E(X2) = 20. Use this information to construct an unbiased estimator for θ.(b) Find the maximum likelihood
Consider the probability density function(a) Find the value of the constant c.(b) What is the moment estimator for θ?(c) Show that is an unbiased estimator for θ.(d) Find the maximum
Suppose that the random variable X has the continuous uniform distribution.f(x) = { 1, when 0Suppose that a random sample of n = 12 observations is selected from this distribution. What is the
Continuation of Exercise 7-31. Suppose that for the situation of Exercise 7-12, the sample size was larger (n = 40) but the maximum likelihood estimates were numerically equal to the values obtained
PVC pipe is manufactured with a mean diameter of 1.01 inch and a standard deviation of 0.003 inch. Find the probability that a random sample of n = 9 sections of pipe will have a sample mean diameter
Suppose that samples of size n = 25 are selected at random from a normal population with mean 100 and standard deviation 10. What is the probability that the sample mean falls in the interval from
A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. Find the probability that a random sample of n = 6
Consider the synthetic fiber in the previous exercise. How is the standard deviation of the sample mean changed when the sample size is increased from n = 6 to n = 49?
The compressive strength of concrete is normally distributed with μ = 2500 psi and σ = 50 psi. Find the probability that a random sample of n = 5 specimens will have a sample mean diameter
A normal population has mean 100 and variance 25. How large must the random sample be if we want the standard error of the sample average to be 1.5?
Suppose that the random variable X has the continuous uniform distributionSuppose that a random sample of n = 12 observations is selected from this distribution. What is the probability distribution
Suppose that X has a discrete uniform distributionA random sample of n = 36 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.5, assuming
The amount of time that a customer spends waiting at an airport check-in counter is a random variable with mean 8.2 minutes and standard deviation 1.5 minutes. Suppose that a random sample of n = 49
A random sample of size n1 = 16 is selected from a normal population with a mean of 75 and a standard deviation of 8. A second random sample of size n2 = 9 is taken from another normal population
A consumer electronics company is comparing the brightness of two different types of picture tubes for use in its television sets. Tube type A has mean brightness of 100 and standard deviation of 16,
The elasticity of a polymer is affected by the concentration of a reactant. When low concentration is used, the true mean elasticity is 55, and when high concentration is used the mean elasticity is
Suppose that a random variable is normally distributed with mean μ and variance σ2, and we draw a random sample of five observations from this distribution. What is the joint probability
Transistors have a life that is exponentially distributed with parameter λ. A random sample of n transistors is taken. What is the joint probability density function of the sample?
Suppose that X is uniformly distributed on the interval from 0 to 1. Consider a random sample of size 4 from X. What is the joint probability density function of the sample?
A procurement specialist has purchased 25 resistors from vendor 1 and 30 resistors from vendor 2. Let X1,1, X1,2, , X1,25 represent the vendor 1 observed resistances, which are assumed to be normally
A random sample of 36 observations has been drawn from a normal distribution with mean 50 and standard deviation 12. Find the probability that the sample mean is in the interval47 < X < 53.
Is the assumption of normality important in Exercise 7-51? Why?
A random sample of n = 9 structural elements is tested for compressive strength. We know that the true mean compressive strength μ = 5500 psi and the standard deviation is σ = 100 psi. Find
A normal population has a known mean 50 and known variance σ2 = 2. A random sample of n = 16 is selected from this population, and the sample mean is How unusual is this result?
Let X be a random variable with mean μ and variance σ2. Given two independent random samples of sizes n1 and n2, with sample means X1 and X2, show thatis an unbiased estimator for . If X1
A random variable x has probability density functionFind the maximum likelihood estimator for θ.
Let f(x) = θxθ–1, 0 < θ < ∞ and 0 < x < 1.Show that Θ = - n/(In xi) is the maximum likelihood estimator for θ.
Let f(x) = θxθ–1, 0 < θ < ∞ and 0 < x < 1.Show that Θ = - n/(In xi) is the maximum likelihood for θ. Show that Θ = -(1/n) Σ= 1 in(Xi) is the maximum
A lot consists of N transistors, and of these M (MDetermine the joint probability function for X1 and X2. What are the marginal probability functions for X1 and X2? Are X1 and X2 independent random
When the sample standard deviation is based on a random sample of size n from a normal population, it can be shown that S is a biased estimator for . Specifically,(a) Use this result to obtain an
A collection of n randomly selected parts is measured twice by an operator using a gauge. Let Xi and Yi denote the measured values for the ith part. Assume that these two random variables are
Consistent Estimator Another way to measure the closeness of an estimator to the parameter Θ is in terms of consistency. If Θn is an estimator of Θ based on a random sample of n
Order Statistics. Let X1, X2, , Xn be a random sample of size n from X, a random variable having distribution function F(x). Rank the elements in order of increasing numerical magnitude, resulting in
Continuation of Exercise 7-65. Let X1, X2, , Xn be a random sample of a Bernoulli random variable with parameter p. Show thatUse the results of Exercise 7-65.
Continuation of Exercise 7-65. Let X1, X2, , Xn be a random sample of a normal random variable with mean μ and variance σ2. Using the results of Exercise 7-65, derive the probability
Continuation of Exercise 7-65. Let X1, X2, , . , Xn be a random sample of an exponential random variable of parameter. Derive the cumulative distribution functions and probability density functions
Let X be a random variable with mean μ and variance σ2, and let X1, X2,., Xn be a random sample of a continuous random variable with cumulative distribution function F(x). Find E[F(X(n))]
Let X be a random variable with mean μ and variance σ2, and let X1, X2, , Xn be a random sample of size n from X. Show that the statistic V = k Σ (Xi+1 – Xi)2is an unbiased estimator
When the population has a normal distribution, the estimator is sometimes used to estimate the population standard deviation.This estimator is more robust to outliers than the usual sample standard
Censored Data a common problem in industry is life testing of components and systems. In this problem, we will assume that lifetime has an exponential distribution with parameter λ, so is an
Modify the program Coin Tosses to toss coin n times and print out after every 100 tosses the proportion of heads minus 1/2. Do these numbers appear to approach 0 as n increases? Modify the program
In the early 1600s, Galileo was asked to explain the fact that, although the number of triples of integers from 1 to 6 with sum 9 is the same as the number of such triples with sum 10, when three
The Labouchere system for roulette is played as follows. Write down a list of numbers, usually 1, 2, 3, 4. Bet the sum of the first and last, 1 + 4 = 5, on red. If you win, delete the first and last
Modify the program HTSimulation so that it keeps track of the maximum of Peter’s winnings in each game of 40 tosses. Have your program print out the proportion of times that your total winnings
The psychologist Tversky and his colleagues11 say that about four out of five people will answer (a) to the following question: A certain town is served by two hospitals. In the larger hospital about
Let Ω = {a, b, c} be a sample space. Let m (a) = 1/2, m (b) = 1/3, and m (c) = 1/6. Find the probabilities for all eight subsets of Ω
John and Mary are taking a mathematics course. The course has only three grades: A, B, and C. The probability that John gets a B is .3. The probability that Mary gets a B is .4. The probability that
Assume that the probability of a “success” on a single experiment with n outcomes is 1/n. Let m be the number of experiments necessary to make it a favourable bet that at least one success will
If A, B, and C are any three events, show that
Let Ω be the sample space Ω = {0, 1, 2 . . .}, And define a distribution function by m (j) = (1 − r)j r , For some fixed r, 0 < r < 1, and for j = 0, 1, 2,. Show that this is a
Tversky and Kahneman23 asked a group of subjects to carry out the following task. They are told that:Linda is 31, single, outspoken, and very bright. She majored in philosophy in college. As a
A life table is a table that lists for a given number of births the estimated number of people who will live to a given age. In Appendix C we give a life table based upon 100,000 births for ages from
(From Sholander24) In a standard clover-leaf interchange, there are four ramps for making right-hand turns, and inside these four ramps, there are four more ramps for making left-hand turns. Your car
(From vos Savant26) A reader of Marilyn vos Savant’s column wrote in with the following question:My dad heard this story on the radio. At Duke University, two students had received A’s in
Barbara Smith is interviewing candidates to be her secretary. As she interviews the candidates, she can determine the relative rank of the candidates but not the true rank. Thus, if there are six
Show that
If a set has 2n elements, show that it has more subsets with n elements than with any other number of elements.
An elevator takes on six passengers and stops at ten floors. We can assign two different equiprobable measures for the ways that the passengers are discharged: (a) We consider the passengers to be
John claims that he has extrasensory powers and can tell which of two symbols is on a card turned face down (see Example 3.11). To test his ability he is asked to do this for a sequence of trials.
Using the technique of Exercise 16, show that the number of ways that one can put n different objects into three boxes with a in the first, b in the second, and c in the third is n! / (a! b! c!).
A gin hand consists of 10 cards from a deck of 52 cards. Find the probability that a gin hand has(a) All 10 cards of the same suit.(b) Exactly 4 cards in one suit and 3 in two other suits.(c) A 4, 3,
Using the method for the hint in Exercise 22, show that r indistinguishable objects can be put in n boxes in Different ways
A drug is assumed to be effective with an unknown probability p. To estimate p the drug is given to n patients. It is found to be effective for m patients. The method of maximum likelihood for
Prove the following binomial identityConsider an urn with n red balls and n blue balls inside. Show that each side of the equation equals the number of ways to choose n balls from the urn.
Lucas27 proved the following general result relating to Exercise 37. If p is any prime number, then (n/j) (mod p) can be found as follows: Expand n and j in base p as n = s0 + s1p + s2p2 + · · · +
2n balls are chosen at random from a total of 2n red balls and 2n blue balls. Find a combinatorial expression for the probability that the chosen balls are equally divided in colour. Use Stirling’s
A die is rolled twice. What is the probability that the sum of the faces is greater than 7, given that?(a) The first outcome was a 4?(b) The first outcome was greater than 3?(c) The first outcome was
A coin is tossed twice. Consider the following events. A: Heads on the first toss. B: Heads on the second toss. C: The two tosses come out the same. (a) Show that A, B, C are pairwise independent

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