Study Sheet for Exam 2 - Probability and Statistics I | MTH 445, Exams of Probability and Statistics

Download Study Sheet for Exam 2 - Probability and Statistics I | MTH 445 and more Exams Probability and Statistics in PDF only on Docsity! 03/11/2009 Statistics 445/545 STUDY SHEET for EXAM 2 (Wednesday, March 18) The test will cover: - Ch. 9 (sections 9.2 – 9.4, 9.6, 9.7) - Ch. 10 (sections 10.2 – 10.6, 10.8, 10.9) The best way to prepare to the midterm is to read the book and to do the homework exercises. Please, take time to go over the material. I. Chapter 9 Properties of Point Estimators and Methods of Estimation 1. Properties of point estimators: a) Relative efficiency; b) Consistency; c) Sufficiency. 2. Law of large numbers. 3. Convergence in probability. Properties. 4. Factorization criterion. 5. Methods for funding estimators: a) The method of moments; b) The method of maximum likelihood. II. Chapter 10 1. Elements of a statistical test: a) Null hypotheses; b) Alternative hypothesis; c) Test statistic; d) Significance level; e) Rejection region; f) Decision; g) Conclusion in words of the problem; h) Type I and Type II errors. 2. Common large-sample tests for a) Mean; b) Difference of two means; c) Proportion; d) Difference of two proportions. 3. Sample size for one-tailed Z test. 4. Relationship between hypothesis testing and confidence intervals. 5. Attained significance level or P-value. 6. Small-sample hypothesis testing for mean and difference of two means; 7. Hypothesis testing for variance. Problems: 1. Let nXXX ,..., 21 constitute a random sample from an exponential distribution with parameter  . Show that X is a consistent estimator of the parameter  . 2. For 10  consider a random sample from a uniform distribution on the interval from  to  1 . Identify a sufficient statistics for  . 3. Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose that    1,10,)1();( xxxf Use the method of moments to obtain an estimate for  . 4. Let nXXX ,..., 21 be a random sample from a distribution with probability density function     0,10,);( 1 xxxf a) Obtain the method of moments estimator for  . b) Find the sufficient statistics for  . c) Show that the maximum likelihood estimator ̂ is a function of the sufficient statistics from part (a). d) Argue that ̂ is also sufficient for  . 5. Let nXXX ,..., 21 be a random sample from a distribution with density function       0,0, 1 );( 2 xxexf x Find the maximum likelihood estimator for  . 6. Suppose that a manufacturer of a new medication wants to test the null hypothesis that the proportion of successes (recoveries) is at least 0.90. His test statistic is X = the observed number of successes in 20 trials, and he will accept the null hypothesis if x > 14; otherwise, he will reject it. a) Find , the probability of committing a type I error; b) Find , the probability of committing a type II error if true proportion is actually 0.60. 7. The Rockwell hardness index for steel is determined by pressing a diamond point into the steel and measuring the depth of penetration. For 50 specimens of an alloy of steel, the Rockwell hardness index averaged 62 with standard deviation 8. The manufacturer claims that this alloy has an average hardness index of at least 64. Is there sufficient evidence to refute the manufacturer’s claim at 1% significance level?