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modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

14. Competing for the title of Miss America are 50 contestants from each of the 50 states plus 1 contestant from the District of Columbia. How many different ways can the contestants be assigned the
b. Of the numher of possible three-person committees indicated in (a), how many involve you?
a. How many different ways can a three-person committee be formed to give a class report?
13. There arc 15 students in an econometrics class that you are attending.
12. A statistics class has 20 students in attendance in a room where 25 desks arc availahle for the students.How many different ways can the students leave 5 desks unoccupied?
11. Define a set function that will assign the appropriate area to all rectangles of the form [XI, xII X [YI, Y21, X2 ::: XI and Y2 ::: YI, contained in R~. Be sure to identify the domain and range
e. What is the invcrse image of 2?
d. What is the range of the set function?
c. What is the image of the set A = [3,31 under P?
b. What is the image of Q under P?
a. What is the image of the set A = [0,21 under P?
10. Define a universal set as Q = (x: a 5 x 5 5), and consider the set function PIA) =.51 xdx + 12.5, xeA where the domain of the set function is all subsets A c Q of the form A = [a, bl, for a 5 a 5
e. What is D(f)? What is R(f)?
d. If you can, define f(2) andf- I (5).
c. Does an inverse function exist?
b. Is S a function?
a. Is 2S8?
9. Let A = (a, (0), and examine the following relation onA:S = {(x, y): y = 2 + 3x, (x, Y) E A 2 )).
a. Let P = ($.01, $.02, ... , $l.OO, $l.Ol, ... ) represent a set of possible prices for a given commodity, and let 0 = [a, (0) represent possible levels of quantity demanded. Define S: P ~ 0 as S =
8. For each relation below, state whether the relation is a function, and state whether an inverse function exists.Explicitly define the inverse function if it exists.
c. A = R;o, B = [a, (0), S = (((XI, X2), y): Y =5XIXi, ((XI, X2), y) E A x BI.
a. A = [a, 101, B = [O,ln(11ll, S = {(x, y): y = In(l +x),(x,y) E A x BI.
7. In each situation below indicate whether the relation is a function. If so, determine the domain and range of the function.
g . ..43 h. A2 - Uie/lAi Problems
f. A4 -A3
d. n~=IAj
c. njehAj
b. nf=IAj
Define the following sets:a. UjeJ,Aj
6. Define the universal set, Q, as Q = {x: 0 :::: x :::: 5 or 10 :::: x :::: 20}, and define the following subsets of Q as Al = {x: 0:::: x < 2.5}, A2 = Ix: IS < x :::: 20}, A3 = {x: 2.5 :::: x ::::
e. Define A4 •
d. Define A4 - AI.
c. Define Al n A2 •
b. Define uf=IAj.
a. Define UjelAj.
5. Let the universal set be defined by Q = [-5,5], and define the following subsets of Q:AI = [-2, 1), A2 = (1,2), A3 = [2,5], A4 = [-5, -2).Also, define an index set I = {I, 3, 4}.
4. Let the universal set be Q = [0,10], and define the following subsets of Q:A = [0,2), B = [2,7], G = [5,6], D = {2}, E = {x: x = y-I, Y is an even positive integer ~ 4}.a. Define the following
d. S = {x: x = 2y, y is a positive integer}.
c. S = {p: p is the price of a quart of milk sold at a retail store in the United States on Friday, September 13,I99I}.
b. S = {(x, y): y :::: x2 , X is a positive integer, y E R~o}.
a. S = {x: X is a U.S. citizen who has purchased a Japanese car during the past year}.
3. For each set below, state whether the set is finite, countably infinite, or uncountably infinite.
2. Label the sets you have identified in Problem 1. as being either finite, countably infinite, or uncountably infinite, and explain your choice.
d. the set of all two-tuples (XI, X2) where XI is any real number and X2 is related to XI by raising the number e to the power XI.
c. the set of all possible outcomes resulting from rolling a red and a green die and calculating the values of y - x, where y = number of dots on the red die and x = number of dots on the green diej
b. the set of all positive numbers that are positive integer powers of the number 10 (Le., 101, 102, etc.);
a. the set of all senior citizens receiving social security payments in the United States;
1. Using either an exhaustive listing, verbal rule, or mathematical rule, define the following sets:
c. Calculate an outcome of a .90-level confidence interval for 8 based on the expression derived in (b).Note: For large degrees of freedom> 3D, X~;a::::; v [1- :v +z" (92v y/2r, where f"" N(x; 0,
b. Define a general expression for a level-y confidence interval for 8 based on n iid observations from f(x;8).
a. Define a pivotal quantity for the parameter 8 (Hint:Theorem 10.13 and Problem 10.21 might be useful 675 here.)
23. The population distribution of income in a populous developing country is assumed to be given (approximately)by the continuous PDF fIx; 8) = 8(1 +x)-le+1JI10.""J(x), where 8 > o. A summary
a .95-level confidence interval for a2 •·d. A colleague claims that mean daily gasoline demand is only 37 million gallons. Is your answer to(b) consistent with this claim? Explain.
c. Define a pivotal quantity for a2, and use it to define
a .95-level confidence interval for /-t. Also, define a.95 lower confidence bound for /-t.Problems
b. Define a pivotal quantity for /-t, and use it to define
a. Show that N(/-t, (72) is a location-scale parameter family of PDFs (recall Theorem 10.13).
22. Regional daily demand for gasoline in the summer driving months is assumed to be the outcome of an N(/-t, (72) random variable. Assume you have 40 iid daily observations on daily demand for
21. Follow the proof of Theorem 10.14 to demonstrate that when random sampling from the continuous PDF I(z; 8) with scalar parameter 8, n-2 Lln[1 - F(Xi ; 8n - X~n i=l is a pivotal quantity for 8.
e. Would the use of an LM test be tractable here? If so, perform an LM test of the hypothesis.
d. Test that the observations can be interpreted as a random sample from a Beta(8, 1) population distribution.Use a size .05 test.
c. Test that the observations might be interpreted as a random sample from some population distribution.Use a size .05 runs test.
b. Test the hypothesis that the expected weekly storage tank capacity utilized is equal to .5 at a significance level of .05. If the exact distribution of the GLR test appears to be intractable, then
a. Define a GLR size ex test of Ho: 8 = 8 0 versus Ho: 8 =I 8 0 •
A random sample of a year's worth of observations of capacity utilization at the site produced the following in sequence rowwise:0.148 0.501 0.394 0.257 0.759 0.763 0.092 0.155 0.409 0.257 0.586
b. Testing hypotheses about the mean of a Poisson distribution based on a random sample of size n from a Poisson population distribution.20. The weekly proportion of storage tank capacity that is
a. Testing hypotheses about the mean of a binomial distribution based on a random sample of size 1 from the binomial distribution.
19. Describe how you would test one-sided and twosided hypotheses, using both finite sample and asymptotically valid testing procedures, in each of the following cases. Use whatever procedures you
Does this die appear to be fair? What size test will you use? Why?
b. The Nevada Gaming Commission has been called in to investigate whether the infamous Dewey, Cheatum & Howe Casino is using fair dice in its crap games. One of the alleged crooked dice is tossed 240
a. Describe how you would use the X2 goodness-of-fit size ex test to assess the hypothesis that the population distribution was a discrete uniform distribution with support 1,2, ... , rn.
18. Testing Whether Random Sample Is from a Discrete Uniform Population Distribution Let Xi, i = 1, ... , n be a random sample from some discrete integer-valued population distribution. Let the range
c. Define an asymptotically valid, size GLR-type test of Ho versus Ha. Repeat part (b).
b. If nl == 45, n2 == 34, XI == 19, and X2 == 24, is the hypothesis of equality of proportions rejected if Q( = .10?674 Chapter 1 0 Hypothesis-Testing Methods
a. Define an asymptotically valid size a Wald-type test of Ho versus H".
17. Test of the Equality of Proportions Let XI and X2 be independent random variables with binomial distributions Binomial(n;,p;L i == 1,2, where the n;'s are assumed known. Consider testing the null
c. Show that you can transform the GLR test into a test involving a critical region for the test statistic w == SXy/[(s~ + s~)/2).'d. Derive the sampling distribution of the test statistic W defined
b. In a sample of SO observations, it was found that s~ == 5.3 7, s~ == 3.62, and Sxy == .98. Is this sufficient to reject the hypothesis of independence based on the asymptotically valid test above?
a. Define a size .05 GLR test of the independence of the two random variables. You may use the limiting distribution of the GLR statistic to define the test.
16. Testing for Independence in a Bivariate Normal Population Distribution Let (X" Y,), i == I, ... , n, be a random sample from some bivariate normal population distribution N(tL, ~)where af = ai-
d. Define and test an hypothesis that will assess whether the expected return of the firm is greater than market return for any Rmt :::: O. Use a size .05 test.
c. Assuming normality, test whether the expected daily return of the firm is proportional to market return. Use a size .05 test.
b. Test to see whether the abnormal returns can be interpreted as a random sample from some normal population distribution of returns. Use a size .05 level of test.
a. Use a runs test to provide an indication of whether the residuals of the returns-generating equation, and hence the abnormal returns, can be viewed as iid from some population distribution. Use a
15. An analyst is investigating the effect of certain policy events on common stock prices in a given industry.In an attempt to isolate abnormal from normal returns of firms in the industry, the
d. Test the hypothesis that the level of advertising expenditure that maximizes expected sales:::: IS."e. Calculate a confidence interval with confidence coefficient.95 for the level of advertising
c. Define a confidence interval with confidence coefficient.95 for the expected level of sales expressed Problems as a function of advertising level at. Plot the confidence interval as a function of
b. Test whether the relationship between advertising expenditures and sales is actually a linear as opposed to a quadratic relationship.
a. Test the hypothesis that advertising expenditure has a significant impact on the expected sales level.Use any significance level you feel is appropriate.
14. The relationship between sales and level of advertising expenditure is hypothesized to be quadratic over the relevant range of expenditure levels being examined, i.e., YI = /31 + /3zal + /33a; +
d. Test the hypothesis that the expected number of daily breakdowns for this equipment ~ 8, using a significance level of .10. Use whatever test procedure you feel is appropriate. Based on this test
a uniform population distribution. Use a size .10 test.
c. Test the hypothesis that the observations are from
b. Test the hypothesis that the observations are from a Poisson population distribution. Use a size .10 test.
a. Test whether the observations might be a randomsample outcome from some population distribution.Use a .1O-level runs test.
13. The production department of a paper products manufacturer is analyzing the frequency of breakdowns in a certain type of envelope machine as it contemplates future machinery repair and
g. Test the hypothesis that the variance of the proQ.uction process ~ 6 at a .10 level of significance.Assume normality in conducting this test. What information would you need to conduct an
e. Calculate a confidence-interval outcome for the expected level of production at the input levels indicated in (d). Use a .95 confidence coefficient. What is the meaning of this confidence interval?
d. Test the hypothesis that expected output ::: 35 when labor, energy, and capital are applied at levels 8, 4, and 3, respectively. Use a size .05 test.
c. Define confidence-interval outcomes for each of the marginal products of the inputs. Use .95 confidence coefficients. What do these confidence intervals mean?

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