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mathematics
statistics

Questions and Answers of Statistics

Complete the proof of Theorem 11.2.8 by showing that
Show that under the oneway ANOVA assumptions, for any set of constants a = (a1,..., ak), the quantity ∑ai i. is normally distributed with mean ∑aiθi and variance σ2 ∑a2i / ni.
Using an argument similar to that which led to the t test in (11.2.7), show how to construct a t test for (a) H0: ∑aiθi = δ versus Hi: ∑aiθi ≠ δ. (b) H0: ∑aiθi ≤ δ versus Hi: ∑aiθi
Verify the expressions in (12.2.7).
(a) Show that for random variables X and Y and constants a, b, c, d,Cov(ay + bX, cY + dX) = acVar Y + (be + ad)Cov(X, Y) + bdVar X.(b) Use the result in part (a) to verify that in the structural
There is an interesting connection between the Creasy-Williams confidence set of (12.2.24) and the interval CG() of (12.2.22).€ƒ(a) Show thatwhere is the MLE of Î² and
For the logistic regression function in (12.3.2), verify these relationships. (a) π(-α/β) = 1/2 (b) π((-α/β) + c) = 1 - π((-α/β) - c) for any c (c) (12.3.3) for dπ(x)/dx (d) (12.3.4) about
In probit regression, the link function is the standard normal cdf Φ(x) = P(Z ≤ x), where Z ~ n(0, 1). Thus, in this model we observe (Y1, x1), (V2, X2), ..., (Yn, xn), where Yi ~ Bernoulli(πi)
For the logistic regression model:(a) Show thatIs a sufficient statistic for (Î±, Î²). (b) Verify the formula for the logistic regression information matrix in (12.3.10).
Let Y ~ binomial(n, π), and let = Y/n denote the MLE of π. Let W = log (/(l - )) denote the sample logit, the MLE of log (π/(1 - π)). Use the Delta Method to show that l/(n(l - )) is a
In Example 12.4.1, in contrast to Example 10.2.1, when we introduced the contaminated distribution for Îµi, we did not introduce a bias. Show that if we had, it would not have mattered.
A problem with the LAD regression line is that it is not always uniquely defined. (a) Show that, for a data set with three observations, (x1, y1), (x1, y2), and (x3, y3) (note the first two xs are
(a) Verify that(b) Verify that and, with part (a), conclude that
Using a Taylor series argument as in Example 12.4.3, derive the asymptotic distribution of the median in iid sampling.
In maximizing the likelihood (12.2.13), we first minimized, for each value of Î±, Î², and Ïƒ2Î´, the functionwith respect to Î¾1,...,
For the data of Table 12.4.1, we could also use the nonparametric bootstrap to assess the standard error from the LAD and M-estimator fit.(a) Fit the line y = Î± + Î²x to get
In the EIV functional relationship model, where λ = σ2δ / σ2ε is assumed known, show that the MLE of σ2δ is given by (12.2.18).
Consider a linear structural relationship model where we assume that Î¾i, has an improper distribution, Î¾i ~ uniform(-ˆž, ˆž).(a) Show that for each
In the structural relationship model, the solution to the equations in (12.2.20) implies a restriction on , the same restriction seen in the functional relationship case.(a) Show that in (12.2.20),
What is the sample space when a coin is tossed three times?
An experiment has three outcomes. I, II and III. If outcome I is twice as likely as outcome II and outcome II is three times as likely as outcome III, what are the probability values of the three
A probability value p is often reported as an odds ratio, which is p / (1 - p). This is the ratio of the probability that the event happens to the probability that the event does not happen.(1) If
An experiment has live outcomes. I, II, III, IV and V. If P(I) = 0.13. P(II) = 0.24, P(III) = 0.07 and P(III) = 0.38, Calculate P(V)?
An experiment has three outcomes. I, II, III, IV and V. If P(I) = 0.08, P(II) = 0.20 and P(III) = 0.33, what are the possible values for the probability of outcome V? If outcomes IV and V are equally
Consider the sample space in Figure 1.22 with outcomes a, b, c, d and e. Calculate:(a) P(b)(b) P(A)(c) P(AÊ¹)
Three types of batteries are being tested, type I, type II, and type III. The outcome (I, II, III) denotes that the battery of type I fails first, the battery of type II next, and the battery of type
A factory has two assembly lines, each of which is shut down (S), at partial capacity (P), or M full capacity (F). The sample space is given in Figure 1.25, where, for example. (S, P) denotes that
A fair coin is tossed three times. What is the probability that two heads will be obtained in succession?
Consider the sample space in Figure 1.23 with outcomes a, b, c, d, e and f. If P(A) = 0.27, calculate:(a) P(b)(b) P(AÊ¹)(c) P(d)
If birthdays are equally likely to fall on any day, what is the probability that a person chosen at random has a birthday in January? What about February?
If a lair die is thrown, what is the probability of scoring a prime number (suppose that the number 1 is considered to be a prime number)?
If two fair dice are thrown, what is the probability that at least one score is a prime number? What is the complement of this event? What is its probability?
Two fair dice are thrown, one red and one blue. What is the probability that the red die has a score that is strictly greater than the score of the blue die? Why is this probability less than 0.5?
If a card is chosen at random from a pack of cards, what is the probability that the card is from one of the two black suits?
If a card is chosen at random from a pack of cards, what is the probability that it is an Ace?
A winner and a runner-up are decided in a tournament of four players, one of whom is Terica. [fall the outcomes are equally likely, what is the probability that (a) Terica is the winner? (b) Terica
A card is drawn from a pack of cards. A is the event that an Ace is obtained, B is the event that a card from one of the two red suits is obtained, and C is the event that a picture card is obtained.
A car repair can be performed either on time or late and either satisfactorily or unsatisfactorily. The probability of a repair being on time and satisfactory is 0.26. The probability of a repair
A bag contains 200 balls that are either red or blue and either dull or shiny. There are 55 shiny red balls, 91 shiny balls and 79 red balls. If a ball is chosen at random, what is the probability
In a study of patients arriving at a hospital emergency room, the gender of the patients is considered, together with whether the patients are younger or older than 30 years of age, and whether or
Consider the sample space and events in Figure 1.55. Calculate the probabilities of the events:(a) B(b) B ˆ© C(c) A ˆª C(d) A ˆ© B ˆ© C(e) A ˆª B
Let A be the event that a person is female, let B be the event that a person has black hair, and let C be the event that a person has brown eyes. Describe the kinds of people in the following
A card is chosen from a pack of cards. Are the events that a card from one of the two red suits is chosen and that a card from one of the two black suits is chosen mutually exclusive? What about the
If P(A) = 0.4 and P(A ∩ B) = 0.3, what are the possible values for P(B)?
If P(A) = 0.5, P(A ∩ B) = 0.1, and P(A ∪ B) = 0.8, what is P(B)?
A fair die is thrown. A is the event that an even score is obtained, and B is the event that a prime score is obtained. Give the probabilities: (a) A ∩ B (b) A ∪ B (c) A ∩ Bʹ
A card is drawn at random from a pack of cards. A is the event that a heart is obtained, B is the event that a club is obtained, and C is the event that a diamond is obtained. Are these three events
Consider again Figure 1.55 and calculate the probabilities:(a) P(A | B)(b) P(C | A)(c) P(B | A ˆ© B)(d) P(B | A ˆª B)(e) P(A | A ˆª B ˆª C)(f) P(A ˆ©
Consider again Figure 1.25 and the two assembly lines. Calculate the probabilities:(a) Both lines are at full capacity conditional on neither line being shut down 1.4.14(b) At least one line is at
The length, width, and height of a manufactured part are classified as being either within or outside specified tolerance limits. In a quality inspection 86% of the parts are found to be within the
A gene can be either type A or type B. and it can be either dominant or recessive. If the gene is type B. then there is a probability of 0.31 that it is dominant. There is also a probability of 0.22
A manufactured component has its quality graded on its performance, appearance, and cost. Each of these three characteristics is graded as either pass or fail. There is a probability old.40 that a
An agricultural research establishment grows vegetables and grades each one as either good or bad for its taste, good or bad for its size, and good or bad for its appearance. Overall 78% of the
There is a 4 % probability that the plane used for a commercial Might has technical problems, and this causes a delay in the flight. If there are no technical problems with the plane, then there is
In a reliability test there is a 42% probability that a computer chip survives more than 500 temperature cycles. If a computer chip does not survive more than 500 temperature cycles, then there is a
Let A be the event that a prime number is obtained from the roll of a fair die. Calculate P(5 | A), P(6 | A), and P(A | 5).
A card is drawn at random from a pack of cards. Calculate: (a) P(A(|card from red suit) (b) P(heart | card from red suit) (c) P(card from red suit | heart) (d) P(heart | card from black suit) (e)
A ball is chosen at random from a bag containing 150 balls that are either red or blue and either dull or shiny. There are 36 red shiny balls and 54 blue balls. What is the probability of the chosen
A car repair is either on time or late and either satisfactory or unsatisfactory. If a repair is made on time, then there is a probability of 0.85 that it is satisfactory. There is a probability of
Assess whether the probabilities of the events (i) increase, decrease, or remain unchanged when they are conditioned on the events (ii). (a) (i) It rains tomorrow, (ii) It is raining today. (b) (i) A
Suppose that births are equally likely to be on any day. What is the probability that somebody chosen at random has a birthday on the first day of a month? How does this probability change
Consider again Figure 1.24 and the battery lifetimes. Calculate the probabilities:(a) A type I battery lasts longest conditional on it not failing first(b) A type I battery lasts longest conditional
Two cards are chosen from a pack of cards without replacement. Calculate the probabilities: (a) Both are picture cards. (b) Both are from red suits. (c) One card is from a red suit and one card is
Repeat Problem 1.5.9, except that the drawings are made with replacement. Compare your answers with those from Problem 1.5.9. Problem 1.5.9 Suppose that 17 light bulbs in a box of 100 light bulbs are
Suppose that a bag contains 43 red balls, 54 blue balls, and 72 green balls, and that 2 balls are chosen at random without replacement. Construct a probability tree for this problem. What is the
Repeat Problem 1.5.11, except that the drawings are made with replacement. Compare your answers with those from Problem 1.5.11. Problem 1.5.11 Suppose that a bag contains 43 red balls, 54 blue balls,
A biased coin has a probability p of resulting in a head. If the coin is tossed twice, what value of p minimizes the probability that the same result is obtained on both throws?
If a fair die is rolled six times, what is the probability that each score is obtained exactly once? If a fair die is rolled seven times, what is the probability that a 6 is not obtained at all?
(a) If a fair die is rolled five times, what is the probability that the numbers obtained are all even numbers? (b) If a lair die is rolled three times, what is the probability that the three numbers
A system has four computers. Computer 1 works with a probability of 0.88; computer 2 works with a probability of 0.78; computer 3 works with a probability of 0.92; computer 4 works with a probability
Repeat Problem 1.5.1, except that the second drawing is made with replacement. Compare your answers with those from Problem 1.5.1. Problem 1.5.1 (a) Both are picture cards. (b) Both are from red
Two cards are chosen from a pack of cards without replacement. Are the following events independent? (a) (i) The first card is a picture card, (ii) The second card is a picture card. (b) (i) The
Four cards are chosen from a pack of cards without replacement. What is the probability that all four cards are hearts? What is the probability that all four cards are from red suits? What is the
Repeat Problem 1.5.4, except that the drawings are made with replacement. Compare your answers with those from Problem 1.5.4. Problem 1.5.4 Four cards are chosen from a pack of cards without
Show that if the events A and B are independent events, then so are the events (a) A and Bʹ (b) Aʹ and B (c) Aʹ and Bʹ.
Consider the network given in Figure 1.66 with three switches. Suppose that the switches operate independently of each other and that switch 1 allows a message through 1.5, with probability 0.88,
Suppose that birthdays are equally likely to be on any day of the year (ignore February 29 as a possibility). Show 1.5, that the probability that two people chosen a; random have different birthdays
Suppose that 17 light bulbs in a box of 100 light bulbs are broken and that 3 are selected at random without replacement. Construct a probability tree for this problem. What is the probability that
Suppose it is known that 1% of the population suffers from a particular disease. A blood test has a 97% chance of identifying the disease for diseased individuals, but also has a 6 % chance of
Bag A contains 3 red balls and 7 blue balls. Bag B contains 8 red balls and 4 blue balls. Bag C contains 5 red balls and 11 blue balls. A bag is chosen at random, with each bag being equally likely
A class had two sections. Section I had 55 students of whom 10 received A grades. Section II had 45 students of whom 11 received A grades. Now I of the 100 students is chosen at random, with each
An island has three species of bird. Species 1 accounts for 45 % of the birds, of which 10 % have been tagged. Species 2 accounts for 38 % of the birds, of which 15 % have been tagged. Species 3
After production, an electrical circuit is given a quality score of A, B, C, or D. Over a certain period of time. 77% of the circuits were given a quality score A, 11% were given a quality score B,
The weather on a particular day is classified as either cold, warm, or hot. There is a probability of 0.15 that it is cold and a probability of 0.25 that it is warm. In addition, on each day it may
A valve can be used at four temperature levels. If the valve is used at a cold temperature, then there is a probability of 0.003 that it will leak. If the valve is used at a medium temperature, then
A company sells live types of wheelchairs, with type A being 12% of the sales, type B being 34 % of the sales, type C being 7% of the sales, type D being 25% of the sales, and type E being 22% of the
Evaluate: (a) 7! (b) 8! (c) 4! (d) 13!
A poker hand consists of live cards chosen at random from a pack of cards. (a) How many different hands are there? (b) How many hands consist of all hearts? (c) How many hands consist of cards all
In an arrangement of n objects in a circle, an object's neighbors are important, but an object's place in the circle is not important. Thus, rotations of a given arrangement are considered to be the
In how many ways can six people sit in six seats in a line at a cinema? In how man) ways can the six people sit around a dinner table eating pizza after the movie?
Repeat Problem 1.7.12 with the condition that one of the six people, Andrea, must sit next to Scott. In how many ways can the seating arrangements be made if Andrea refuses to sit next to
A total of n balls are to be put into k boxes with the conditions that there will be n1 balls in box I, n2 balls in box 2, and so on, with nk balls being placed in box k (n1 + ... + nk = n) Explain
Explain why the following two problems are identical and solve them. (a) In how many ways can 12 balls be placed in 3 boxes, when the first box can hold 3 balls, the second box can hold 4 balls, and
A quality inspector selects a sample of 12 items at random from a collection of 60 items, of which 18 have excellent quality, 25 have good quality. 12 have poor quality, and 5 are defective. (a) What
A salesman has to visit 10 different cities. In how many different ways can the ordering of the visits be made? If he decides that 5 of the visits will be made one week, and the other 5 visits will
Evaluate: (a) P72 (b) P95 (c) P52 (d) P174
A hand of 8 cards is chosen at random from an ordinary deck of 52 playing cards without replacement. (a) What is the probability that the hand does not have any hearts? (b) What is the probability
Evaluate: (a) C62 (b) C84 (c) C52 (d) C146
A menu has live appetizers, three soups, seven main courses, six salad dressings, and eight desserts. In how many ways can a full meal be chosen? In how many ways can a meal be chosen if either an
Four players compete in a tournament and are ranked from 1 to 4. They then compete in another tournament and are again ranked from 1 to 4. Suppose that their performances in the second tournament are

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