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modern mathematical statistics with applications

Mathematical Statistics For Economics And Business 1st Edition Ron C. Mittelhammer - Solutions

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?
b. Test the significance of the input variables individually.Which input variables contribute significantly to the explanation of the expected level of output? Be sure to explain the basis for your conclusion.
a. Test the joint significance of the input variables for explaining the expected level of production using a size .05 test.
what is the
f12. The production function for a commodity can be approximated over a restricted range of relevant input levels by a linear relationship of the form Y = x{3 + €.The columns of x include, in order, a column of l's and three column vectors representing observations on labor, energy, and capital
d. Suppose that normality of the observations is not assumed. Can you define another test of the effectiveness of the meal-planning program? If so, use the test to assess the effectiveness of the mealplanning program. Discuss any differences that are required in the interpretation of the outcome of
c. Describe how you might test the hypothesis that the observations on paired differences are from a normal population distribution. What information would you need to test the hypothesis?
b. Can you define an LM test of the null hypothesis in(a)? Can you test the hypothesis with the information available? If so, perform the test-if not, what other information would you need?
a. Assuming that the pairs of observations are iid outcomes from some bivariate normal population distribution with before and after means /J-b and /J-a, define the appropriate likelihood function for the mean /J- = /J-a - /J-b and variance a2 of the population distribution of differences in pairs
e. Consider using the WALD test procedure for this problem. What is the test rule? Can you perform the test with the information provided?
d. Consider using the LM test procedure for this problem.What is the test rule? Can you perform the test with the information provided?
c. How would your test rule change if you wanted to test a one-sided hypothesis concerning the means of the population distributions?
Conduct the test of the null hypothesis. Does the test outcome help you decide from which supplier to purchase the chips?
b. The respective sample means of operating lives for the two samples were 3h = 24.23 and X2 = 18.23.
a. Define a size .10 GLR test of the equality of the means of the two population distributions, i.e., a test of Ho: 01 = O2 versus Ha: not Ho.
a. Attempt to find a GLR test of the hypothesis Ho: J.tl = J.t2 versus H,,: not Ho based on at-statistic similar to what you found in Problem lO.3. Is this approach valid? The problem of testing hypotheses concerning the means of two normal population distributions when it is not assumed that the
"i. Test the least-squares residuals for normality, preferably using the Shapiro-Wilks test Iyou'll need tables for this), or else use the X2 square goodnessof-fit test.
·h. Calculate a confidence region having confidence coefficient .90 for the income and price elasticities.With the aid of a computer, graph the confidence region. Superimpose the intervals you calculated in If) and Ig) on this graph. The regions are differentinterpret the difference.
g. Calculate a confidence interval that has a .95 confidence coefficient for the price elasticity. Interpret the meaning of this confidence interval.
f. Calculate a confidence interval that has a .95 confidence coefficient for the income elasticity. Interpret the meaning of this confidence interval.
e. Is the demand equation homogeneous degree zero in price and income? 'Plot the power function of the test, and interpret its meaning.
d. Test whether the price elasticity is inelastic. Test whether the income elasticity is inelastic.
c. Test whether price is significant in explaining textile consumption.
b. Test whether income is significant in explaining textile consumption. "Plot the power function of the test, and interpret its meaning.
a. Test the hypothesis that the matrix of explanatory variables is significant in explaining changes in the logarithm of textile demand.
"d. Define the power function for each of the hypothesis-testing procedures you defined above. With the aid of a computer, plot the power functions and interpret their meanings.
c. Define a GLR size .05 test of Ho: ILl = IL2 versus Ha: not Ho. Test the hypothesis at the .10 level of significance.
b. Repeat (a) for the hypothesis Ho: ILl 2: IL2 versus Ha: notHo.
a. Define a GLR size .05 test of Ho: J-l,l ::: IL2 versus Ha: not Ho, where J-l" and J-l,2 refer to the means of the populations from which the first and second samples were taken. Test the hypothesis.
"d. Define the power function for each of the hypothesis-testing procedures you defined above. With the aid of a computer, plot the power functions and interpret their meanings.
c. Define a GLR size .05 test of Ho: ar = ai versus Ha: not Ho. Either go to the computer and perform this test, or else define an approximation to this test based on "equal tails" and perform the test.
b. Repeat (a) for the hypothesis Ho: ar 2: ai versus Hn: not Ho.
a. Define a GLR size .05 test of Ho: ar ::: ai versus Ha: not Ho, where ar and ai refer to the variances of the populations from which the first and second samples were taken. Test the hypothesis.
·d. Can you define a Wald test for the hypothesis in (a)?If so, perform a Wald test of the joint null hypothesis at significance level .10.
c. Test the two hypotheses Ho: J-l, = 16.03 and Ho: a = .01 individually, using size .05 tests. Use whatever test procedures you feel are appropriate.Interpret the outcomes of the tests individually.Interpret the tests jointly using a Bonferroni approach.
b. Define a size .10 LM test of the same hypothesis as in (a), and test the hypothesis at significance level.10. Are the two tests different? Are the two test decisions in agreement?
a. Define a size .10 GLR test of the hypothesis Ho: IL = J-l,o, a = ao, versus Ha: not Ho. Test the hypothesis that the filling process is under control.You may use the asymptotic distribution of the GLR if you wish.
b. Test the hypothesis with a UMP size .10 test. Does the disgruntled employee have a legitimate concern?
a. Define a UMP level-a test of the hypothesis Ho: P = .05 versus Ha: P < .05. You may use an asymptotic normal distribution for the test statistic, if it has one.
·20. A certain business uses national telephone solicitation to sell its product. Its sales staff have individual weekly sales quotas of 10 sales that they must meet or else their job performance is considered to be unsatisfactory and they receive only base pay and no sales commission.In hiring
c. Supposing you rejected the hypothesis, what would be your conclusion? Is further statistical analysis warranted?
b. Test the null hypothesis using a size .05 UMPU test.Does the outcome contradict classifying the vehicle in the average risk class? Why or why not?
a. Design a UMPU level-a test of the null hypothesis Ho: tL E [2,4J versus Ha: tL ¢ [2,4J. You can base your test on the asymptotic normal distribution of the test statistic.
19. The Gibralter Insurance Co. is reevaluating the premiums it charges on car insurance and is analyzing classifications of cars into high risk, average risk, and low risk on the basis of frequency of accidents. It is currently examining an imported mid-size four-door sedan and wishes to examine
b. Plot the isopower contour in the (tL, a)-plane for a power level of .90. (An isopower contour is the set of(tL,a) points that result in the same level of power, which in the case at hand is equivalent to the set of (tL,a) points that result in the same value of the noncentrality parameter A.)
a. Plot the power surface in three dimensions, with the axes referring to power, the value of tL, and the value ofa. (This is probably best done with the aid of a computer!)
18. Revisit Ex. 9.27 and the power function graph in Figure 9.16 and consider the implications of the power function graph in the two-dimensional parameter space(tL,a).
f. American Statistical Assoc., March, pp. 6-24.)
c. After quality-control training and closer monitoring of clerical workers, a daily random sample resulted in x = 2.6. Is there evidence that quality has increased over what it has been in the past? Why or why not?(The mail-order company of Alden's Inc. was one of the earliest companies to use
b. Conduct a size .05 UMPU test of the null hypothesis on a day where x = 3.4. Is the order process under control?
a. Define a UMPU level-a test of the hypothesis Ho: P = .032 versus Ha : pol .032. You may use the asymptotic normal distribution of the test statistic in defining the critical region.
17. Control Charting A large mail-order house has initiated a quality-control program. It randomly samples 100 of each day's orders and monitors whether or not the mail order-taking process is "under control" in the sense that errors in order taking are at minimum levels.The number of orders per
16. Suppose that a random sample of size n is drawn from a normal population distribution for which fL is assumed to be known and equal to the given value fL •.Define UMPU level-a tests of the following null hypotheses:
c. Ho: fL = fLo versus Ha: fL i= fLo
b. Ho: fL ::: fLo versus Ha: fL < fLo
a. Ho: fL ::5 fLo versus Ha: fL > fLo
15. Suppose that a random sample of size n is drawn from a normal population distribution for which a2 is assumed to be known and equal to the given value a;.Define UMPU level-a tests of the following null hypotheses:
d. Test the hypothesis. Are university professors paid more than white collar, middle management workers in this state?
c. Define a UMP size .05 test of the null hypothesis Ho: fL ::5 62.471 versus Ha: fL > 62.471. (Hint: Use a test statistic for which you can apply an asymptotic normal distribution, and use the normal approximation.)
b. Define a test statistic on which you can base a UMP level-a test of the null hypothesis Ho: fL ::5 62.471 versus Ha: fL > 62.471.
a. Express the mean level of income as a function of the parameter e.
c. Conduct the hypothesis test. Should your company get the contract? Why or why not?*14. The Pareto distribution f(x;e, c) = cllex-ll+liJllc.ooJ(X), for e > 0 and c > 0, has been used to model the distribution of incomes in a given population of individuals, where c represents the minimum level of
b. Design a UMP size a test of whichever null hypothesis you feel is appropriate based on your discussion 591 in• (a). Choose whatever size test you feel is appropriate, and discuss your choice of size.
a. In designing a UMP level-a test in this situation, should the null hypothesis be defined as e ::: 7.5 or e ::5 7.5? Base your discussion on the characteristics of the power function of each of the tests.
13. Your company supplies an electronic component that is critical to the navigational systems of large jet aircraft. The operating life of the component has an exponential distribution with some mean value v, where operating life is measured in 100,OOO-hour units. You are seeking a contract to
12. Referring to Def. 9.15, state the form of the UMPU level-a test of the null hypothesis Ho: fL ::: fLo versus Ho: fL < fLo. Justify your answer.
11. Referring to Def. 9.14, state the form of the UMPU level-a test of the null hypothesis Ho: a2 ::: a5 versus Ha: a2 < a5' Justify your answer.
c. A nondestructive test of the disc players that determines their operating life until failure is applied to 50 players that are randomly chosen from the assembly line. The measurements resulted in x = 4.27. Test the null hypothesis in (b).d. Plot the power curve of this test, and interpret its
b. Based on a random sample of size 50, design a uniformly most powerfullevel-.1O test of the null hypothesis that ,8 will be in the set of values you identified in (a).
P(x ::5 2; ,8) = f; f(x; ,8)dx ::5 .05.
a. The manufacturer wants its exposure to warranty claims to be, on the average, no more than 5 percent of the units sold. Find the values of ,8 for which
10. Being both quality and cost conscious, a major foreign manufacturer of compact disc players is contemplating its warranty policy. The standard warranty for compact disc players sold by competing producers is one year. The manufacturer is considering a two-year warranty.The operating life until
b. Test the null hypothesis using the test you defined in part (a). What can you say about the effectiveness of the safety program?
a. Design a uniformly most powerfullevel-.1O test of the null hypothesis Ho: A ~ 3 versus the alternative hypothesis Ha: A < 3 having size as close as possible to, without exceeding, .10.
9. The number of work-related injuries per week that occur at the manufacturing plant of the Excelsior Corporation is a Poisson-distributed random variable with mean A ~ 3, according to company analysts. In an attempt to lower insurance costs, the corporation institutes a program of intensive
"d. Is the test you defined in (a) a most powerful size.05 test of the null hypothesis?
c. Is the test you defined in (a) an unbiased size .05 test of the null hypothesis?
b. Is it possible that two analysts, using exactly the same random sample outcome and exactly the same test rule could come to different conclusions regarding the validity of the null hypothesis? Explain.(This feature of randomized tests has discouraged their use.)
that defines a size·.Os test of the null hypothesis.
a. Find a value of
To implement the rule when x = 7 occurs, a uniform random number z with range (0,1) could be drawn, and if z :5 ., Ho would be rejected, and if z > ., Ho would be accepted.
8. Randomized Test It was demonstrated in Ex. 9.10 that the choices of size for most powerful tests of the hypothesis Ho: p = .2 versus Ha: p = .8 were quite limited. Suppose that a .05 level test of the null hypothesis was desired and that you were willing to utilize a randomized test. In
7. In a random sample of size 10 from a Bernoulli population distribution, how many (nonrandomized) critical regions can you define that have size :5 .10 and that are also unbiased for testing the null hypothesis Ho: p = .4 versus Ha: p f .4?
"e. Repeat (a)-(c) using a pooled sample of 500 observations.(Hint: In (c), it might be useful to consider Bonferroni's inequality for placing an upper bound on the probability of type I error.)
a :5 .05 when using the outcome of the two test statistics above to determine acceptance or rejection of the joint null hypothesis? Does the complaint against the company appear to be valid?
d. Treating the hypotheses in (a) and (b) as a joint hypothesis on the parameter vector of the normal population distribution, what is the probability of type I error for the joint hypothesis Ho: J.t = 16.1 and
c. Calculate and interpret the p values of the tests in(a) and (b).
b. Define a UMPU level-.Os test of Ho: a :5 .05 versus Ha: a > .05 based on a random sample of size 250.Test the hypothesis using the statistics associated with the second random-sample outcome. Plot and interpret the power function of this test.
a. Define a UMPU level-.Os test of Ho: J.t = 16.1 versus Ha: J.t f 16.1 based on a random sample of size 250. Test the hypothesis using the statistics associated with the first random-sample outcome. Plot and interpret the power function of this test.
16.1 ounces and a standard deviation of :5 .05, so that over 97 percent of its product has a weight of ~ 16 oz.It suggests that its product be randomly sampled and its claims be tested for accuracy. Two independent random samples of observations on fill weights, each of size 250, resulted in the
6. A complaint has been lodged against a major domestic manufacturer of potato chips stating that its 16-oz. bags of chips are being underfilled. The manufacturer claims that its filling process produces fill weights that are normally distributed with a mean of
c. Calculate and interpret the p value for the test.
b. Plot the power function for the test. Interpret the power function both from the standpoint of a potential investor in a restaurant and from the perspective of the managing director of a chamber of commerce.
a. Define a UMP level-.OS test of the hypothesiS that less than three-quarters of new restaurants are expected to survive at least one year in business in U.S. cities of size 2: 500,000. Test the hypothesis.
5. The annual proportion of new restaurants that survive in business for at least one year in a U.S. city with population 2: 500,000 people is assumed to be the outcome of some Beta population distribution. Part of the maintained hypothesis is that b = 1 in the Beta distribution, so that the
4. A large metropolitan branch of a savings and loan is examining staffing issues and wants to test the hypothesis that the expected number of customers requiring the services of bank personnel during the midweek(Tuesday-Thursday) noon hour :5 50. The bank has obtained the outcome of a random
d. You are random sampling from a Poisson population distribution, and you are testing Ho: A = 2 versus Ha: A = 3.
c. The joint density of the random sample Y = xfJ + £is N(xfJ, (J"2 I), and you are testing whether fJ = [0].
b. You are random sampling from a geometric population distribution, and you are testing Ho: P = .01 versus Ha: p> .Ol.

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