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nonparametric statistical inference

Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

Use the data from Problem 15.3.5 to test if high school GPA’s are positively related to college GPA’s.
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”; II, “between 3.0 and 3.5”; and III, “above 3.5”) and their attitude toward the
The data on the incidence of a certain disease, classified by age and gender, are as follows: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 high incidence of this disease among women over 60.
Find a decomposition of G2 into independent components by decomposing: (i) A 2 × k table. (ii) A 3 × 4 table.
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?
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 of job-related stress as low, moderate, or high. The
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 as follows:Second exam First exam Below
For each 3 × 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) kkkk (ii) 5555 k k 0 k 5 5 k 5 .kkkk 5555
Show that in the case of a 2 × 2 contingency table, the statistic Q2 given by (15.9) is proportional to (N11N22 − N21N12)2, and find the proportionality constant.
Given the data on numbers of hits of various areas of London by V2 rockets(see Example 8.14), test the hypothesis that the numbers of hits have a Poisson distribution.
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.
Assume that the sexes 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 01 2 34 Count 7 21 40 27 5 Test the hypothesis that the distribution of
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 0123 Count 51 92 40 17 Test the hypothesis that the number of defective batteries in a toy has a binomial distribution.
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 follows:0 1 2
STAT 102 can only taken 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 Z. Ordered according to their performance, the students of the three instructors are
Prove relations (14.18).
Assume that n is odd and that all m 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. (iv)
Use the runs test for the data of Problem 14.5.2.
Two samples of sizes m = 20 and n = 10, respectively, are selected from two populations. Let r1, ...,r10, denoting the numbers of observations from the first sample that do not exceed the kth (k = 1, ..., 10) element in the ordered second sample, be 1, 1, 2, 4, 4, 6, 8, 9, 11,d. (i) Find the value
Prove formula (14.14) showing first that for i = j Var(IAi ) = mn(m + n)2 and Cov(IAi , IAj ) = − mn(m + n)2(m + n − 1).
A machine is set up to produce items, each with a diameter above 1 inch. The diameters of 15 consecutive items produced are 1.11, 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 a runs test, taking 1 inch as a threshold to test the hypothesis that the
Solve Problems 14.3.6 and 14.3.7 using a runs test. Compare the results obtained by different methods (use the same α = 0.05). Explain the differences, if they exist, for m = 10 and k = 5.
Some texts define the Wilcoxon signed rank statistic as S˜n = η˜iRi, where η˜i = 1 if Xi > θ0 and 0 otherwise. Determine the mean and variance of S˜n, and show that tests based on Sn and on S˜n are equivalent in the following sense: under a null hypothesis, Sn = S˜n − S˜∗n, where
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 5 weeks are as follows:Pair 1 2 3 4 5 6 7 8 9 10 11 12 Diet A 15 33 21 17 14 25 25 31 18 5 46 11 Diet B 18 17 10 10 32 11
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.
Prove the asymptotic normality of the Wilcoxon signed rank statistic Sn using the Liapunov theorem.
Assume that each of two samples contains 2m + k elements with the following ordering:Y X ··· Y X 2m Y Y ··· Y k XY ··· XY 2m XX ··· X k(14.10)For given m find k such that the Kolmogorov–Smirnov test will reject the hypothesis that both samples are drawn
Suppose that one sample contains 2m data points while the other contains 2m + k data points. The first 2m and the last 2m data points in the joint sample alternate between samples. Thus the ordered data has the form
Suppose that out of 30 fires in Example 14.2 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
What is the minimal possible value of the statistic Dm,n if k values of Xi precede the third in the magnitude value Yj ? If m = 100, k = 30, and n = 200, is there enough evidence to reject (at the level α = 0.05) the null hypothesis that the distributions of X’s and Y ’s are the same?
Find the joint density of (X, Y ) resulting from the “time-saving scheme” of Example 14.3. Find also the expected number of random variables ξi necessary to sample in order to obtain one pair (X, Y ) under both schemes.
Suppose that the data are as in Example 14.2, except that there were only 25 fires, none of them in November. Test that the fires occur according to the Poisson process.
Let the observed values x1, ...,xn be such that 1/3 ≤ xi ≤ 2/3 for all i. What can be said about n, if the null hypothesis that the underlying distribution is U[0, 1] cannot be rejected by the Kolmogorov–Smirnov test at α = 0.05 level?
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 X1, X2, ...,X200 be a random sample from the distribution of X. Suppose that among the first 100 observation of Xi, 55 were equal to a and 38 were equal tob. Among the next 100 observations, 51 are equal
Let F(t)=0 for t < 0, F(t) = p for 0 ≤ t < 1, and F(t)=1 otherwise. Use the central limit theorem to evaluate directly the distribution of supt|Fn(t) − F(t)|, and show that Fn(t) tends to F(t) almost surely and uniformly in t.
To test the mileage 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 per
Verify that the cross-products in (13.24) do indeed equal zero.
Given data (xi,yi),i = 1, ...,n, find the LS estimators of the quadratic regression a + bx + cx2.
for the LS estimator of b.
Provide the same argument as in Problem
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 ai,i = 1, ...,n such that the statistic T = n i=1 aiYi satisfies the conditions: (1) E(T) =a. (2) Variance of T is
Suppose that the number of errors in a text of length x is known to be a Poisson random variable with unknown mean λ. We observe n texts of lengths x1,x2, ...,xn and find the numbers of errors they contain, Y1, ...,Yn, satisfy E(Yj ) = Var(Yj ) =λxj . Find the BLUE of λ.
Under condition of Problem 13.6.3, find the prediction interval for the mean of k observations taken for value x0 of the independent variable.
Find the prediction interval with a probability 1 − γ of coverage for an observation to be taken at the value x0 by an independent variable, given the data (xi,yi),i =1, ...,n, and assuming the normal model of the form Yx = bx + with ∼N(0,σ2).
and 13.5.2, and find the 95% prediction interval for the GPA of a student who scored 400 on an entrance exam.
Combine data points in Problems
At harvest, the weight of a certain fruit grown in a greenhouse has the N(a + bt,σ2)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 equal) are
Derive, if possible, a test of linearity of regression under the assumptions of this section, if the individual observations for values x r+1, ...,x m are now unknown, and instead we have the data on averages y r+1, ...,y m, and the corresponding numbers of observations n r+1, ...,n m.
Check the identity (13.17).
Suppose that we have six data points in addition to those in Problem 13.4.2:x 355 402 402 309 375 375 y 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.
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:
Suppose it is known that the true regression (assuming a normal case) has the form E(Yx) = bx. Derive the MLE’s for b and for σ2.
Using the ideas given on deriving the confidence set for (a,b), derive the testing procedure for the null hypothesis H0 : a = a0,b = b0 against the general alternative H1 : H0 is false. Consider two cases: (i) σ known. (ii) σ unknown.
Suppose that observations are taken only at two values, x1 and x2, of an independent variable. Let y1 and y2 be the average observed responses for x = x1 and for x = x2, respectively. Show that the estimated regression line passes through points (x1, y1)and (x2, y2).
The scores on an entrance exam (x) and the grade point average (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 3.76 3.75 Assume normality and homoscedasticity. (i) Compute
Derive the test for the null hypothesis H0 : a = a0 against the one- or two-sided alternative.
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
Find true regression Y on X if X and Y have a joint trinomial distribution P{X = x,Y = y} = n!x!y!(n − x − y)!px 1 py 2(1 − p1 − p2)n−x−y for x = 0, 1 ...,n,y = 0, 1, ...,n, and 0 ≤ x + y ≤ n.
Suppose that X = (X1,X2). Find coefficients a,b1,b2 such that Yp = a + b1X1 +b2X2 is the best linear predictor of Y given X.
Let (xi,yj ), i = 1, ...,n be the data where at least one value of xi is not zero. Find an estimate of the slope parameter b if it is known that a = 0 [i.e., find the best fit of the model E(Y |x) = bx].
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?
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.
Let X and Y have a joint distribution uniform on a parallelogram with vertices at points (−1, −1), (0, −1), (1, 1), and (0, 1). Find the true and linear regression of: (i)X on Y . (ii) Y on X.
The following bivariate data show 16 independent observation of variables (X,Y ).
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 populations
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 the lake A were 3.17, 4.22, 2.58, 4.01, and 3.79. In the lake B, the measurements (made with the same instrument) were 4.04, 4.32, and 4.12.
(Shoshoni Rectangles) The following problem is taken from Larsen and Marx (1986).Since antiquity, societies have expressed esthetic preferences for rectangles having a certain width (w) to length (l) ratio. For instance, Plato wrote that rectangles formed of two halves of an equilateral triangle
Use the bootstrap test for testing H0 : μ = 2 against H1 : μ > 2, based on the 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.
Assume that the hypothesis H0 : θ = 0 is to be tested against H1 : θ > 0, where θ is a parameter in the U[θ,θ + 1] distribution. (i) Derive the test of size α = 0.05, based on one observation only, and obtain its power function for θ = 0.1, 0.2, ..., 0.9.(ii) Use Monte Carlo simulations to
A random sample 1.138, 1.103, 3.007, 1.307, 1.885, 1.153 was obtained from a distribution with density f(x,θ)=2θ2x−3 for x ≥ θ and 0 otherwise. Find the MLE of θ, and use it to test H0 : θ = 1 against H1 : θ > 1. Find the p-value and compare it with the p-value based on 2000 Monte Carlo
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 distribution with density f(x,μ) = exp{−2|x − μ|}. Use Monte Carlo simulations to estimate the p-value for testing the hypothesis H0 : μ =0 against the alternative H1 : μ > 0.
A company A that produces batteries claims that their product lasts “at least 50%longer” 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 until the
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 α = 0.05, does this result indicate that the fractions pD and pR of Democrats and Republicans favoring the issue in
A survey on a proportion of population using a certain product was conducted in four different cities, in each 200 people were interviewed. Test the hypothesis that the proportion of users of the product is the same in each city if the numbers of people using the product were 34, 52, 41, 45,
Let X1, ...,Xm and Y1, ...,Xn be two independent random samples selected from distributions with P(X = x) = p1(1 − p1)x,x = 0, 1, ... and P(Y = y) = p2(1 − p2)y,y = 0, 1, ..., respectively. Test H0 : p1 = p2 against H1 : p1= p2 at 0.1 significance level. Assume that m = 5,n = 10 and the actual
Two independent samples X1, ...,Xm and Y1, ...,Xn were selected from the EXP(θ1) and EXP(θ2) distributions, respectively. (i) Derive the GLR test of size 0.05 for H0 : θ1 = θ2 against H1 : θ1= θ2. (ii) Perform the test at 0.05 significance level based on observations of X : 0.81, 1.33, 2.10,
Derive the GLR test for H0 : θ = θ0 against H1 : θ= θ0 based on a random sample X1, ...,Xn selected from the BETA(1,θ) distribution. Determine an approximate critical value for a size α.
A random sample of size n was selected from the EXP(θ) distribution. Perform the GLR test of H0 : θ = 2 against the alternative H1 : θ= 2 if the actual observations are 0.57, 0.21, 2.18, 0.85, 1.44. Use α = 0.05.
Let X1, ...,Xn be a random sample from the GAM(3,θ) distribution. Derive the GLR test for H0 : θ = θ0 against H1 : θ= θ0.
The times for the diagnosis and repair of a car with a certain type of problem are assumed to be normally distributed with mean μ and standard deviation σ = 15 minutes. A mechanic serviced five cars in one day, and it took him a total of 340 minutes. (i) Test, at the level α = 0.05, hypothesis
The following data concerning accidents on various types of highways were obtained from the Ohio Department of Transportation, September 1990 (see Al-Ghamdi, 1991):
Suppose that we test the null hypothesis that μ = 100 against the alternative thatμ= 100. The distribution is normal with its variance unknown. We have just two observations, X1 = 105 and X2 = 105 +a. Find a such that the null hypothesis is rejected at the significance level α = 0.05?
Let X1, ...,XN be a random sample from a multinomial distribution with three possible outcomes: O1,O2,O3 and their corresponding probabilities: p1,p2, 1 − p1 − p2.Obtain a UMPU test of a H0 : p1 ≤ p2 against the alternative H1 : p1 > p2 knowing that among N observation, outcomes O1,O2, and O3
Let X1, ...,Xn be a random sample from the N(μ,σ2) distribution with both parameters unknown. Find an unbiased test with a critical region of the form “reject H0 if n i=1 (Xi − X)2/σ2 0 < C1 or n i=1 (Xi − X)2/σ2 0 > C2” to test the hypothesis H0 : σ = σ2 0 against the alternative
Assume that X1, ...,Xn is a random sample from the N(0, σ2) distribution. For testing hypothesis H0 : σ = σ2 0 against the alternative H1 : σ2= σ2 0 at significance levelα, find an unbiased test with a critical region of the form “reject H0 if n i=1 X2 i /σ2 0 C2.”
Let X1, ...,Xn be a random sample from the POI(λ) distribution. Find the (approximate) UMPU test for the hypothesis H0 : λ = λ0 against the two-sided alternative H1 : λ= λ0, where λ0 is assumed to be large. [Hint: Use the fact that if X has Poisson distribution with mean λ, then (X −
Let X be a single observation from a distribution with density f(x,θ) =1 − θ2(x − 0.5)3 for 0
Suppose that X1, ...,Xn is a random sample from the U[0,θ] distribution. Test hypothesis H0 : θ = θ0 against the two-sided alternative H1 : θ= θ0 using an unbiased test that rejects H0 if Xn:n < c1 or Xn:n > c2,c1 < c2. Find c1 and c2 if θ0 =5,n = 10, and α = 0.05.
A reaction time to a certain stimulus (e.g., time until solving some problem) is modeled as a time of completion of r processes, running one after another in a specified order. The times τ1, ...,τr of completion of these processes are assumed to be iid exponential with mean 1/λ. If r = 3 and the
Let X1, ...,Xn be a random sample from the distribution with f(x; θ) =θx−1/(θ + 1)x,θ> 0,x = 1, 2, ... . Determine a UMP test of the hypothesis H0 : θ = θ0 against the alternative H1 : θ>θ0, at significance level α.
Recall Problem 12.3.8. Assume that n = 50 and that a student with a score at most 30 will fail. Does there exist a UMP test for the hypothesis H0 : θ ≤ 30? If yes, find the test; if no, justify your answer.
Let X1, ...,Xn be a random sample from the GAM(α,λ) distribution. (i) Derive a UMP test for the hypothesis H0 : α ≤ α0 against the alternative H1 : α>α0 if λ is known. (ii) Derive a UMP test for the hypothesis H0 : λ ≤ λ0 against the alternative H1 : λ>λ0 if α is known.
Suppose that X1, ...,Xn is a random sample from the U[0,θ] distribution.(i) Hypothesis H0 : θ ≤ θ0 is to be tested against the alternative H1 : θ>θ0. Argue that the UMP test rejects H0 if Xn:n >c. Find c for θ0 = 5,n = 10, and α = 0.05.(ii) If the hypothesis H0 : θ ≥ θ0 is tested
The effectiveness of a standard drug in treating specific illness is 60%. A new drug was tested and found to be effective in 48 out of 70 cases when it was used. Specify an appropriate alternative hypothesis and perform the test at the 0.01 level of significance. Find the p-value.
Suppose that the number of defects in magnetic tape of length t (yards) has POI(λt)distribution. Assume that 2 defects were found in a piece of tape of length 500 yards.(i) Test the hypothesis H0 : λ ≥ 0.02 against the alternative H1 : λ < 0.02. Use a UMP test and the significance level α =
Let X1, ...,Xn be a random sample from a folded normal distribution with density f(x; θ) = 2/πθ exp{−θ2x2/2}, for x > 0,θ> 0. (i) Derive the UMP test for H0 : θ = θ0 against H1 : θ>θ0. (ii) Show that the power function is increasing.
Let X1, ...,X16 be a random sample from Laplace distribution with density f(x; λ)=(λ/2) exp{−λ|x|},λ> 0. Find a UMP test for testing H0 : λ = 1 against H1 : λ < 1 at the significance level 0.01.
Check whether the following families of distributions have a monotone likelihood ratio: (i) Poisson. (ii) Exponential. (iii) Gamma, for each parameter separately. (iv)Beta, for each parameter separately.
Let X1, ...,X10 be a random sample from a POI(λ) distribution. Find: (i) The critical region for the most powerful test of hypothesis that θ = 0.7 against the alternative that θ = 0.2. Use significance level α = 0.05. (ii) The size of the test in(i) and the power for θ = 0.2. (iii) The p-value
Let X1, ...,Xn have a joint density f(x; θ), and let U be a sufficient statistic for θ. Show that the most powerful test of H0 : θ = θ0 against H1 : θ = θ1 can be expressed in terms of U.

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