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QUESTIONS IN MATHEMATICS 264 - FIRST SET - 1940 .1. Give the definitions of (a) the probability ?l B!A) of an object A possessing a property B; (b) the relative probability of B given C, P( BlAC \ ; (c) the independence of the property B from another property C. Illustrate by appropriate examples. 2. When do we say that two properties B anc C are exclusive? If B and C are exclusive, must they be independent? What is the actual state of affairs? f 3. Define the logical sum of two properties B and C and state the addition theorem on probabilities. Prove the theorem. 4. Define the logical product of £wo properties p and C and state the multiplication theorem on probabilities. Prove the theorem. 5. State the appropriate conditions and prove the binomial probability formula. 6. Explain the meaning of the statement that the theory of probability provides us with a model of repeated random trials. (Empirical Law of Large Numbers) 7. Describe the sampling experiments which prove that the binomial probability formula may be used for predictions of relative frequencies of various outcomes of repeated random trials. 8. Give the definitions of (a) the mathematical expectation of a random variable x, (b) the standard error of a random variable x. Illustrate these definitions by examples. 9. State and prove the addition theorem on mathematical expectations. 10. State land prove the multiplication theorem on mathematical expectations. 11. Deduce the formula for the standard error of a linear function of n random variables. Show how this formula is simplified when (a) all the variables are mutually independent and besides, (b) when the linear function reduces itself to the arithmetic mean. 12. Give a simplified statement of the theorem of Liapounoff and use it to explain the bearing of the concept of the standard error on the practical problems of sampling. 13. Describe the sampling experiment conducted in the laboratory, illustrating the working of the theory of Liapounoff. 14. Describe the arrangement of the table of 'normal 1 probability law as given in the book used in the laboratory. 15. Give the definition of a confidence interval for a given parameter A, corresponding tb the confidence coefficient, say a = .99. 2 16. Denote by U the arithmetic mean of a characteristic of some products. Suppose that a sample x t , * 4 ,..,, * of n such products is drawn at random and that it is desired ot estimate U. Give the formulas for the confidence interval for U under the assumptions that (a) the S.E. of the x»s is known and (b) the S.E. of the x's is unknown. 17. Describe the table necessary to solve the problem of estimating U in case (b) of question 16. What is meant by the number of degrees of freedom in S * ? Assume that in one case the number of degrees of freedom is f» * 1 and in the other fx - 100. What can you say about the precision in estimating U in both of these cases? 18. Give the definition of cases where (a) the quality of products of mass production is 'under statistical control 1 and (b) the accuracy of routine analyses is 'under statistical control. 1 20. Explain how it is possible to test whether the accuracy of routine analyses is actually under statistical control or not. 21. What ismeant by the X distribution? Describe the tables of that distribution known to you. 22. Give the description of the % test for goodness of fit, 23. What are the semiconstant errors of routine analyses and how can one correct for them? 24. Give the definition of (a) a statistical hypothesis, (b) a test of a statistical hypothesis. 25. Describe what is-meant by the errors of the first and second kind in testing a statistical hypothesis and also the convention concerning them. Give an example. 26. Describe the arrangement of the Incomplete Beta Function Tables and their relation to the Binomial Probability formula. , 27. Assume that the specification concerning some product implies that a consignment is unsatisfactory when the proportion p of products having a specified characteristic does not exceed a specified value p, (such as p , = .5). Explain how it is possible to arrange sampling inspection so that (a) consignments with p ^ p, will be passed with a relative frequency not exceeding some specified £ (e.g., £=.01) and (b) so that consignments with p 2. px (e.g., px ».7) have a fair chance of passing the test. * * * * ***** * * *