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Unformatted text preview: Chapter 15: The Normal Sampling Theorem and Its First Four Applications A. Revision of Point Estimations The three main areas of statistical inference are: (i) Point estimation (ii) Interval estimation (iii) Test of Hypotheses In chapter 14 , we dealt with some very basic problems of point estimation : From this chapter on , we shall look at interval estimation and test of hypotheses as well as problems such as determination of sample sizes. For these, we need to first look at a basic theory, on which useful concepts and methods can be developed. B1: Normal Sampling Theorem Recall from Chapter 9 Part D: The sum of two independent Normal variables. Theorem (Part 1) For random assembly of and we have ) , ( ~ 2 1 1 N X ) , ( ~ 2 2 2 N Y ) , ( ~ 2 2 2 1 2 1 + + + N Y X Let us extend on this idea. Example 1: A lift has the following specifications: Max Load : 850kg; Capacity 12 persons The weights of passengers have a N(62.5, 5) distribution. In certain trip, 13 passengers squeeze in to the lift. What is the probability that it will be overloaded? Solution : Let = the weight of each passenger (in kg) Are we interested in P(X850) =? NO! We need to find P( sum of 13 passengers 850) =? Let us define another random variable, where We need to find Thus, we need to know the distribution of 13 13 1 13 2 1 ........ T X X X X i i = = + + + = ) 25 5 , 5 . 62 ( ~ 2 2 = = = i i i N X i X 13 T ) 850 ( 13 T P 13 T Now by theorem Normal Sampling theorem Part 1 we have: More simply Thus Using calculator : Mode, 0, 5, FMLA, x=? 2.08 Gives 0.48124 Thus, the probability that the lift will be overloaded is 1.88% ) 325 25 13 , 5 . 812 5 . 62 13 ( ~ ........ 2 13 13 13 13 1 13 2 1 = = = = = = + + + = N T X X X X i i 0188 . 4812 . 5 . ) 08 . 2 ( ) 325 5 . 812 850 ( ) 850 ( 13 = = = = Z P Z P T P ) 325 , 5 . 812 ( ~ 2 13 13 13 = = N T To summarize: Normal Sampling Theorem Part 1 The sum of independent samples of size n, taken from the population, have a distribution, namely, It follows that n n i i n T X X X X = = + + + = 1 2 1 ...........
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 Spring '11
 Gabrille

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