www.downloadslide.com Supplementary Exercises 557 composite null and alternative hypotheses if the underlying distribution is specified except for the value of a single parameter. The likelihood ratio procedure provides a general method for developing a statistical test. Likelihood ratio tests can be derived whether or not nuisance parameters are present. In general, likelihood ratio tests possess desirable properties. The Neyman–Pearson and likelihood ratio procedures both require that the distribution of the sampled population(s) must be known, except for the values of some parameters. Otherwise, the likelihood functions cannot be determined and the methods cannot be applied. References and Further Readings Casella, G., and R. L. Berger. 2002. Statistical Inference, 2d ed. Pacific Grove, Calif.: Duxbury. Cramer, H. 1963. Mathematical Methods of Statistics. Princeton, N.J.: Princeton University Press. Hoel, P. G. 1984. Introduction to MathematicalStatistics, 5th ed. New York: Wiley. Hogg, R. V., A. T. Craig, and J. W. McKean. 2005. Introduction to MathematicalStatistics, 6th ed. Upper Saddle River, N.J.: Pearson Prentice Hall. Lehmann, E. L., and J. P. Romano. 2006. Testing Statistical Hypotheses, 3d ed. New York: Springer. Miller, I., and M. Miller. 2003. John E. Freund’s MathematicalStatisticswithApplications, 7th ed. Upper Saddle River, N.J.: Pearson Prentice Hall. Mood, A. M., F. A. Graybill, and D. Boes. 1974. Introduction to the Theory of Statistics, 3d ed. New York: McGraw-Hill. Supplementary Exercises 10.115 True or False. a If the p-value for a test is .036, the null hypothesis can be rejected at the α = .05 level of significance. b In a formal test of hypothesis, α is the probability that the null hypothesis is incorrect. c If the p-value is very small for a test to compare two population means, the difference between the means must be large. d Power(θ ∗ ) is the probability that the null hypothesis is rejected when θ = θ ∗ . e Power(θ) is always computed by assuming that the null hypothesis is true. f If .01 < p-value L( ˆ). i −2 ln(λ) is always positive. 10.116 Refer to Exercise 10.6. Find power(p), for p = .2,.3,.4,.5,.6, .7, and .8 and draw a rough sketch of the power function.
www.downloadslide.com 558 Chapter 10 Hypothesis Testing 10.117 Lord Rayleigh was one of the earliest scientists to study the density of nitrogen. In his studies, he noticed something peculiar. The nitrogen densities produced from chemical compounds tended to be smaller than the densities of nitrogen produced from the air. Lord Rayleigh’s measurements 18 are given in the following table. These measurements correspond to the mass of nitrogen filling a flask of specified volume under specified temperature and pressure. Compound Chemical Atmosphere 2.30143 2.31017 2.29890 2.30986 2.29816 2.31010 2.30182 2.31001 2.29869 2.31024 2.29940 2.31010 2.29849 2.31028 2.29889 2.31163 2.30074 2.30956 2.30054 a b c d For the measurements from the chemical compound, y = 2.29971 and s = .001310; for the measurements from the atmosphere, y = 2.310217 and s = .000574. Is there sufficient evidence to indicate a difference in the mean mass of nitrogen per flask for chemical compounds and air? What can be said about the p-value associated with your test? Find a 95% confidence interval for the difference in mean mass of nitrogen per flask for chemical compounds and air. Based on your answer to part (b), at the α = .05 level of significance, is there sufficient evidence to indicate a difference in mean mass of nitrogen per flask for measurements from chemical compounds and air? Is there any conflict between your conclusions in parts (a) and (b)? Although the difference in these mean nitrogen masses is small, Lord Rayleigh emphasized this difference rather than ignoring it, and this led to the discovery of inert gases in the atmosphere. 10.118 The effect of alcohol consumption on the body appears to be much greater at higher altitudes. To test this theory, a scientist randomly selected 12 subjects and divided them into two groups of 6 each. One group was transported to an altitude of 12,000 feet, and each member in the group ingested 100 cubic centimeters (cm 3 ) of alcohol. The members of the second group were taken to sea level and given the same amount of alcohol. After 2 hours, the amount of alcohol in the blood of each subject was measured (measurements in grams/100 cm 3 ). The data are given in the following table. Is there sufficient evidence to indicate that retention of alcohol is greater at 12,000 feet than at sea level? Test at the α = .10 level of significance. Sea Level 12,000 feet .07 .13 .10 .17 .09 .15 .12 .14 .09 .10 .13 .14 18. Source: Proceedings, Royal Society (London) 55 (1894): 340–344.