Probability And Statistical Inference 2nd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

Unlock the comprehensive understanding of "Probability And Statistical Inference 2nd Edition" by Robert Bartoszynski and Magdalena Niewiadomska-Bugaj with our extensive collection of solved problems and step-by-step answers. Access the complete solutions manual and instructor manual for detailed explanations. This essential resource includes answers key, chapter solutions, and a test bank for enhanced learning. Whether you are looking for solutions PDF or seeking questions and answers for better preparation, our online platform offers a textbook companion with free download options for seamless study support.

16.6.2 Measure the association of GPA and attitude toward statistics courses based on the data from Problem 16.3.5 using +, Tb, and d coefficients.
16.6.1 A study reported in Science magazine investigated the relationship between sex, handedness (right- or left-handed), and relative foot size (left foot bigger than right foot, left foot within one-half shoe size of right foot, or right foot bigger). A random sample of 150 adults gave the
16.5.4 Fifty randomly selected college students that take courses with on-line quizzes were classified according to their answer on two questions: “DO you cheat taking on-line quizzes?” and “Do other students that you know well cheat taking on-line quizzes?” Explain what can be said about
16.5.3 A study of change in employment status was conducted among residents of a certain county. A random sample of 100 adult residents was surveyed on their employment status, and their responses (Yes/No) were recorded. The same people were contacted after 12 months and asked the same question.
16.5.2 Two out of ten randomly selected men, and four out of ten randomly selected women were found to be allergic to some drug. (i) Does these data indicate that there is a difference between men and women in their propensity to develop an allergic reaction to the drug in question? (ii) Does these
16.5.1 Use the data from Problem 16.3.5 to test if high school GPA’s are positively related to college GPA’s.
16.4.3 At the beginning of the semester a random sample of 104 students was selected out of students in all introductory statistics classes. Students were then classified according to their GPA (I, “below 3.0”; 11, “between 3.0 and 3.5”; and 111,“above 3.5”) and their attitude towards
16.4.2 The data on the incidence of a certain disease, classified by age and gender, are as follows:< 20 20-39 4-59 60 +Men 10 15 30 40 Women 20 30 60 300 Find the values of statistics Q2 and G2. Find the decomposition of G2, and verify that the “source” for lack of independence is the very
16.4.1 Find a decomposition of G2 into independent components by decomposing:(i) A 2 x k table. (ii) A 3 x 4 table.
16.3.5 A random sample of 29 university students was selected and each student was then classified according to their high school GPA and college GPA. Both classifications had the same two categories: “I, below 3.0,” “11, at least 3.0.” Formulate a hypothesis to be tested. Perform the test
16.3.4 Professionals from various disciplines participated in a study on job-related stress. A random sample of size 100 was selected from each group of professionals(physicians, engineers, and lawyers) and each person was asked to evaluate the level ofjob-related stress as low, moderate, or high.
16.3.3 Mrs. Smith, who teaches an elementary statistics course, classified each student in the class according to whether the grade on the first exam was below or above the median for this exam, and then did the same for the second exam. The results obtained are:Second Exam First Exam Below Above
16.3.2 For each 3 x 4 contingency table below find such k that the hypothesis about independence of two classification variables is rejected at the 0.05 significance level.(i) k k k k (ii) 5 5 5 5 k k O k 5 5 k 5 k k k k 5 5 5 5 .
16.3.1 Show that in the case of a 2 x 2 contingency table, the statistic Q2 given by(16.9) is proportional to (N11N22 - N21 N I ~ a)n~d f,in d the proportionality constant.
16.2.5 Given the data on numbers of hits of various areas of London by V2 rockets(see Example 9.14), test the hypothesis that the numbers of hits have a Poisson distribution.
16.2.4 Suppose that counts of female offsprings in a certain animal species with four offsprings are 3,8,28,40,21. Test the hypothesis that the corresponding distribution is binomial.
16.2.3 Assume that the genders of children in a family are independent. In a human population, the probability that a child is a male is very close to 0.5. Numbers of boys (X) in a random sample of 100 families with four children are:X I 0 1 2 3 4 Count I 7 21 40 27 5 Test the hypothesis that the
16.2.2 A certain type of toy is sold with three batteries included. The number of defective batteries (X) in a random sample of 200 toys are as follows:x ( o 1 2 3 Count I 51 92 40 17 Test the hypothesis that the number of defective batteries in a toy has a binomial distribution.
16.2.1 Ladislaus von Bortkiewicz, a Russian economist and statistician, is known for the data he collected on the number of Prussian cavalryman being killed by the kick of a horse. He observed 10 army corps for 20 years, obtaining 200 observations.The total of 122 deaths was distributed as
15.6.2 STAT 102 can onlytaken be taken by students who passed STAT 101. Among 15 students in STAT 102, five took STAT 101 from instructor X , four took it from instructor Y , and the rest took the course from instructor 2. Ordered according to their performance, the students of the three
15.6.1 Prove relations (15.18).
15.5.4 Assume that n is odd and that all rn elements of the first sample are below the median of the second sample. Find the range of the test statistics and the rejection region for the appropriate alternative hypothesis using: (i) The Wilcoxon test. (ii)The Mann-Whitney test. (iii) The runs test.
15.5.3 Use a runs test for the data of Problem 15.5.2.
15.5.2 Two samples of sizes m = 20 and n = 10, respectively, are selected from two populations. Let T I , . . . ,~ 1 0 d,e noting the numbers of observations from the first sample that do not exceed the lcth (k = 1, . . . , 10) element in the ordered second sample, be 1, 1,2,4,4,6,8,9,11d, . (i)
15.5.1 Prove formula (15.14) showing first that for i # j mn mn Var(lA,) = and COV(IA,I,A , ) = -(m + n)2 (m + n)2(m + n - 1) ’
15.4.6 A machine is set up to produce items, each with a diameter above 1 inch. The diameters of 15 consecutive items produced are 1.1 1, 1.15, 0.98, 1.11, 1.08, 1.06, 0.97,0.97, 1.05, 1.02,0.98,0.99,0.96, 1.03, 1.01. Use aruns test, taking 1 inch as a threshold to test the hypothesis that the
15.4.5 Solve Problems 15.3.6 and 15.3.7 using a runs test. Compare the results obtained by different methods (use the same a = 0.05). Explain the differences, if they exist, for m = 10 and k = 5.
15.4.4 Some texts define the Wilcoxon signed rank statistic as j , = C jj,Rtiwhere jj, = 1 if X, > 00 and 0 otherwise. Determine the mean and variance of S,, and show that tests based on S, and on 8, are equivalent in the following sense: under a null hypothesis, S, = .$, - $;, where .!?, and j *
15.4.3 Twelve pairs of subjects, matched within each pair with respect to age, gender, health status, and initial weight, were put on two types of diets. The data on pounds lost after five weeks are as follows:Pair 1 1 2 3 4 5 6 7 8 9 10 11 12 DietA I 15 33 21 17 14 25 25 31 18 5 46 1 1 DietB I 18
15.4.2 Out of 15 data points one is between 0 and 1, two are between -2 and -1, three are between 2 and 3, four are between -4 and -3, and five are between 4 and 5. Use the Wilcoxon signed rank statistic to test the hypothesis that the median is:(i) 0. (ii) 1.
15.4.1 Prove the asymptotic normality of the Wilcoxon signed rank statistic S, using the Liapunov theorem.
15.3.7 Assume that each of two samples contains 2m + k elements with the following ordering:YX...YXYY...YXY...XYXX...X (15.10) ---- 2m k 2m k For given m find k such that the Kolmogorov-Smirnov test will reject the hypothesis that both samples are drawn from the same population ( a = 0.05).
15.3.6 Suppose that one sample contains 2 m data points while the other contains 2 m + k data points. The first 2m and the last 2.m data points in the joint sample alternate between samples. Thus the ordered data has the form(i) For given m find k such that the Kolmogorov-Smirnov test will reject
15.3.5 Suppose that out of 30 fires in Example 15.2 those on January 5 and 18, February 3 and 21, March 10, April 6, May 25, June 19, December 3 were caused by arson, and in the remaining cases arson was excluded. Use the Kolmogorov-Smirnov statistic to test the hypothesis that the occurrences of
15.3.4 What is the minimal possible value of the statistic Dm,n if k values of Xi precede the third in the magnitude value y3? If m = 100, k = 30, and n = 200, is there enough evidence to reject (at the level cy = 0.05) the null hypothesis that the distributions of X’s and Y’s are the same?
15.3.3 Find the joint density of (X, Y) resulting from the “time-saving scheme” of Example 15.3. Find also the expected number of random variables necessary to sample in order to obtain one pair (X,Y )u nder both schemes.
15.3.2 Suppose that the data are as in Example 15.2, except that there were only 25 fires, none of them in November. Test that the fires occur according to the Poisson process.
15.3.1 Let the observed values XI,. . . ,x, be such that 1 / 3 5 xi 5 2 / 3 for all i.What can be said about n, if the null hypothesis that the underlying distribution is U[O, 11 cannot be rejected by the Kolmogorov-Smirnov test at cy = 0.05 level?
15.2.2 Let X be a random variable such that P{X = a } = 1 / 2 , P { X = b} =1/3 and P{X = c} = 1/6. Let XI,X2 , . . . , X200 be a random sample from the distributionof X . Suppose that among the first 100 observation of X i , 55 were equal to a and 38 were equal tob. Among the next 100
15.2.1 Let F ( t ) = 0 for t < 0, F ( t ) = p for 0 5 t < 1, and F ( t ) = 1 otherwise.Use the central limit theorem to evaluate directly the distribution of sup, 1 F,(t) -F ( t ) l , and show that F,(t) tends to F ( t ) almost surely and uniformly in t .
14.13.3 Show that if E(Y, j )= c ( i , j )= p + q +Pj + ~ i jw, h ere i = 1, .. , , Z,j =1, . . . , J , then one can always find constants p , ai, &, and -yij such that = = = 0 j=1 i=1 j=1
14.13.2 Use data given in Problem 14.12.2 to test for the existence of interaction between variety and geographical notation.
14.13.1 The scores measuring the level of emotional maturity of young adult males classified by age and extent of use of marijuana are given below.Test for effects of age, extent of use, and their interaction. Use a = 0.05 (a) levels of A levels of A (b) levels of A (c) levels of A (d) Figure
14.12.3 The model for three factors is E(Yij,+=) p + cyi + Pj + T k , where the tests for the hypotheses analogous to those in the case of two factors.
14.12.2 The nutritional value of a certain vegetable is measured on 18 specimens grown in two varieties in three geographical regions. The data are as follows:Geographical Region Variety A B C 6.3 9.2 6.8 1 11.5 5 . 1 7 . 2 9.2 8.1 5.5 11.0 5 . 4 7 . 1 2 7.3 5.0 7.8 8.2 6.1 8.4 Study the effect of
14.12.1 Twelve overweight subjects participated in a study comparing the effectiveness of three weight-reducing diets. The subjects were grouped according to their initial weight, and each of three subjects from each initial weight group was randomly assigned to a diet. The weight loss (in pounds)
14.11.4 Show that the test developed in this chapter is equivalent to the Student's t test for the case when the factor operates at two levels only.
14.11.3 Suppose that we take a random sample of size n from a normal distribution N(p, a'). We divide the observations into k groups of sizes n1, . . . , n k , where ni 2 2 for i = 1, . . . , k and n1 + ' . + nk = n. Let S; be the sample variance in the ith group. Find: (i) The distributionof [(nl
14.11.2 To test the milage achieved by cars produced by different companies, but of comparable price, size, and so on, one make of cars was selected from among the three major American companies and two foreign companies. For each make selected, a number of new cars was chosen and their mpg (miles
14.11.1 Verify that the cross-products in (14.24) do indeed equal zero
14.8.4 Given data ( x i , y i ) ,i = 1,.. . ,n, find the LS estimators of the quadratic regression a + bx + cx’.
14.8.3 Provide the same argument as in Problem 14.8.2 for the LS estimator of b.
14.8.2 Carry out the calculations in the following direct proof of the Gauss-Markov theorem showing that the LS estimators of a and b are BLUE. For a you need to determine the constants a{, i = 1,. . . , n such that the statistic T = cZl aiY, satisfies the conditions: (1) E ( T ) =a. (2) Variance
14.8.1 Suppose that the number of errors in a text of length x is known to be a Poisson random variable with unknown mean A. We observe n texts of lengths x1 , 2 2 , . . . , x , and find the numbers of errors they contain, Y1, . . . , Y,, satisfy E(Y,) =Var(Yj) = Xxj. Find the BLUE of A.
14.7.2 Using data from Problems 14.4.2 and 14.5.2, estimate the score on an entrance exam of a student who graduated with a GPA equal to 3.95.
14.7.1 Five measurements of Y at z = 10 are 10.5, 10.6, 9.7, 11.1, and 12.3. Six measurements of Y at x = 20 are 3.1, 3.6, 3.1,4.0, 5.2, and 2.9. Assuming that the regression of Y on X is linear, estimate the value of x if two observations made at this value are 6.3 and 7.1.
14.6.4 Under condition of Problem 14.6.3 find the prediction interval for the mean of k observations taken for value 20 of the independent variable.
14.6.3 Find the prediction interval with a probability 1 - y of coverage for an observation to be taken at the value 50 by an independent variable, given the data(xi,y i), i = 1, . . . , n, and assuming the normal model of the form Y, = bx + E with E "(0, a?).
14.6.2 Combine data points in Problem 14.4.2 and 14.5.2, and find the 95% prediction interval for the GPA of a student who scored 400 on an entrance exam.
14.6.1 At harvest, the weight of a certain fruit grown in a greenhouse has the N( a +bt, 02) distribution, where t is the average temperature. Weights in a sample of five fruits for t = 80°F are 1.02, 1.03,0.98, 1.05, 1.02, while a sample of seven fruits for t = 86°F (other conditions being
14.5.4 Derive, if possible, a test of linearity of regressiop under the assumptions of this section, if the individual observations for values x,+~,. . . , x, are now unknown, and instead we have the data on averages Y:+~, . . . , y b , and the corresponding numbers of observations TL:+, ~. . . ,
14.5.3 Check the identity (14.17).
14.5.2 Suppose that we have six data points in addition to those in Problem 14.4.2:x I 355 402 402 309 375 375 y 1 3.44 3.91 3.95 3.24 3.52 3.31 Test the hypothesis (using all 17 data values) that the regression of GPA's on the scores from the entrance exam is linear.
14.5.1 The output of a certain device is suspected to decrease linearly with the temperature.Two observations were taken for each temperature, and the data (in appropriate units) are as follows:Temperature 55 65 75 85 95 105 Observation 1 2.01 2.01 2.02 1.48 1.93 1.90 Observation2 2.03 2.02 2.00
14.4.5 Suppose it is known that the true regression (assuming a normal case) has the form E(YZ)= bx. Derive the MLE’s for b and for 02
14.4.4 Using the ideas given on deriving the confidence set for ( a , b ) , derive the testing procedure for the null hypothesis HO : a = ao, b = bo against the general alternative H1 : Ho is false. Consider two cases: (i) u known. (ii) u unknown.
14.4.3 Suppose that observations are taken only at two values, X I and X Z , of an independent variable. Let B1 and ?jzb e the average observed responses for x = X I and for z = 5 2 , respectively. Show that the estimated regression line passes through points (z1,Bi) and ( x 2 , i J 2 ) .
14.4.2 The scores on an entrance exam ( x ) and the GPA’s upon graduation (y) for 10 randomly selected students of a certain university are x 355 361 402 365 375 404 349 380 420 395 y 3.66 3.49 3.86 3.24 3.55 3.92 3.11 3.19 3.16 3.15 Assume normality and homoscedasticity. (i) Compute the MLE’s
14.4.1 Derive the test for the null hypothesis HO : a = a0 against the one- or two-sided alternative.
14.3.5 A still more realistic model is that when all eggs are laid, a disaster occurs, and with probability Q all eggs are destroyed. If there is no disaster, eggs hatch independently, each with probability p . Find the regression E( Y IX) ,t he likelihood of the data, and (where possible) the
14.3.4 Answer Problem 14.3.3 assuming a more realistic model, where Xi's have a Poisson distribution conditional on a positive value, that is, for k = 1 , 2 , . . . , P{X;=k} = (A/k!)e-A 1-e-A
14.3.3 Assume that the total number X of eggs laid by a bird has a Poisson distribution with parameter A, and that each egg hatches with probability p , independent of other eggs. Find the regression of the number Y of eggs hatched on X. Let Xi, Y, be the number of eggs laid and hatched by the ith
14.3.2 Let ( X , Y) have the distribution uniform on the quadrangle with vertices( O , O ) , (1, l), (0,2), and (0.5,l). Find: (i) The true regression of Y on X and of X on Y . (ii)Var(X/Y) andVar(Y1X).
14.3.1 The random variable Yx has the density 1 for a , + b x - ~ < y < a + b x + ~ f ( Y b ) = { 0 otherwise.Find the MLE's of a and b given the sample: (-1,1.3), (0,1.4), ( 1 , O . l), (2, -0.4), (3, -0.1 Compare the MLE's with the LS estimates.
14.2.7 The number of eggs in nests of a certain species of birds is one, two, or three, with two eggs found in about 80% of nests, and one or three eggs in about 10% of nests each. In one-egg nests, the egg hatches successfully in 75% of cases.In two-egg nests the probabilities for the number of
14.2.6 Find true regression Y on X if X and Y have a joint trinomial distribution forrc = 0 , l . . . n , y = 0,1,. . .,n,andO 5 z + y 5 n,
14.2.5 Suppose that X = ( X I ,X 2 ) . Find coefficientsa, b l , b2 such that Yp =a + blXl + bzX2 is the best linear predictor of Y given X.
14.2.4 Let (xi,y j), i = 1, . . . , n be the data where at least one value of zi is not zero. Find an estimate of the slope parameter b if it is known that (I. = 0 [i.e., find the best fit of the model E(Y Iz) = bz].
14.2.3 Suppose that the true regression of Y on X is not linear in X. Is it possible that the marginal distribution of X is such that the expected square error of the best linear predictor of Y is the same as the expected square error of predictor based on the true regression u?
14.2.2 Find an example of random variables ( X , Y ) such that their true and linear regressions of Y on X coincide, but ( X , Y ) do not have a joint normal distribution.
14.2.1 Let X and Y have a joint distribution uniform on a parallelogram with vertices at points (-1, -l), (0, -l), (1,l) and (0,l). Find the true and linear regression of: bf (i) X on Y.(i i) Y on X .
13.10.7 Two random samples 4.49, 7.68, 5.97, 0.97, 6.88, 6.07, 3.08, 4.02, 3.83, 6.35, and 4.59, 3.39, 3.79, 6.89, 5.07, 7.41, 0.44, 2.47, 4.80, 7.23 were obtained independently from distributions with the same mean. Perform a permutation test to test the hypothesis that the variability in both
13.10.6 A study of pollution was carried out in two lakes, A and B. The level of a specific pollutant was measured using a certain instrument, and the results for lake A were 3.17,4.22,2.58,4.01, and 3.79. In lake B the measurements (made with the same instrument) were 4.04,4.32, and 4.12. Test the
13.10.5 (Shoshoni Rectangles) The following problem is taken from Larsen and Mam (1986). Since antiquity, societies have expressed esthetic preferences for rectangles having a certain width (w) to length (1) ratio. For instance, Plato wrote that rectangles formed of two halves of an equilateral
13.10.4 Use the bootstrap test for testing Ho : /I = 2 against H I : p > 2, based on a random sample: 3.49,2.21, 1.07,3.04, 2.57,2.43, 2.18, 1.10, 1.04, 1.92,0.99, 3.13,0.92, 2.72,4.03.
13.10.3 Assume that the hypothesis Ho : 6 = 0 is to be tested against H I : B > 0, where 6 is a parameter in the U[6,0 + 11 distribution. (i) Derive the test of size cr = 0.05, based on one observation only, and obtain its power function for 6 =0.1,0.2, . . . ,0.9. (ii) Use Monte Carlo simulations
13.10.2 Arandom sample 1.138,1.103,3.007,1.307,1.885,1.153waos btainedfrom a distribution with density f ( z l 0) = 282z-3 for z 2 0 and 0 otherwise. Find the MLE of 6, and use it to test Ho : 6 = 1 against H I : 6 > 1. Find the p-value and compare it with the p-value based on 2000 Monte Carlo
13.10.1 A random sample 0.38,-0.49,0.03,0.21,0.12,0.14,-0.18,-0.34,0.46,-0.01 was selected from a Laplace distributionwith density f(z,b) = exp{ -212 -p i } . Use Monte Carlo simulations to estimate the p-value for testing hypothesis Ho : p = 0 against the alternative H1 : p > 0.
13.7.2 A company A that produces batteries claims that their product is “at least 50% better” than the batteries produced by company B. To test the claim, batteries A and B are used one after another in two analogue devices. That is, one device has a battery A installed and is left running
13.7.1 Suppose that in a group of 10 randomly sampled Democrats only 2 favor a certain issue, whereas in a sample of 12 Republicans the same issue is favored by 5 persons. At the level cy = 0.05, does this result indicate that the fractions p~ and p~of Democrats and Republicans favoring the issue
13.6.8 Let two independent samples XI,. .. , X, and Y1,. . . , X, be selected from the N(p1,of) and N(p2, ug) distribution, respectively. Obtain the GRL test for HO :p1 = p2 and of = ui against H I : p1 # p2 and of # ui.
13.6.7 Two independent samples XI,. .. , X , and Y1,. . . , X, were selected from the EXP(O1) and EXP(O2) distributions, respectively. Derive the GLR test of size 0.05 for Ho : 01 = 6'2 against HI : O1 # 82.
13.6.6 Derive the GLR test for Ho : 6' = 6'0 against HI : 6' # 6'0 based on a random sample XI,. . . , X, selected from the BETA(1,O) distribution. Determine an approximate critical value for a size a.
13.6.5 A random sample of size n was selected from the EXP(6') distribution. Perform the GLR test of HO : i3 = 2 against the alternative HI : 0 # 2 if the actual observations are 0.57,0.21,2.18,0.85, 1.44. Use a = 0.05.
13.6.4 Let XI, . . , , X, be a random sample from the GAM(3,O) distribution. Derive the GLR test for HO : 6' = 6'0 against HI : O # 6'0.
13.6.3 The times for the diagnosis and repair of a car with a certain type of problem are assumed to be normally distributed with mean p and standard deviation u = 15 minutes. A mechanic serviced five cars in one day, and it took him a total of 340 minutes. (i) Test, at the level a = 0.05,
13.6.2 The following data concerning accidents on various types of highways were obtained from the Ohio Department of Transportation, September 1990 (see Al-Ghamdi, 1991):Highway Number of Annual Million Accident Type Accidents Vehicle Miles Rate~~ ~ ~ ~~Scenic 3,621 1,021 3.55 Other 2-lane 36,752
13.6.1 Suppose that we test the null hypothesis that p = 100 against the alternative that p # 100. The distribution is normal with its variance unknown. We have just two observations, XI = 105 and X2 = 105 f a . Find a such that the null hypothesis is rejected at the significance level Q: = 0.05?
13.5.4 For testing hypothesis HO : o = C$ against the alternative H I : o2 #ui at level Q, find an unbiased test with a critical region of the form "reject Ho if El"=X,?/ o," < C1 or Cb, X?log > C2." Assume that the observations X1 . . . X, form a random sample from: (i) N(0, g2) distribution. (ii)
13.5.3 Let XI, . . . X, be a random sample from the POI(X) distribution. Find the(approximate) UMP unbiased test for the hypothesis Ho : X = XO against the twosided alternative H I : X # XO, where XO is assumed to be large. [Hint: Use the fact that if X has Poisson distribution with mean A, then (X

Showing 1 - 100 of 5397

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Last

Probability And Statistical Inference 2nd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

Step by Step Answers