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Unformatted text preview: 1 ADM2304: M, N & R January 2012 Statistics for Management II DMS 1130 Section R: Mondays19:0022:00 Section N: Tue 16:0017:30, Thu14:3016:00 Section M: Wed 13:0014:30, Fri 11:3013:00 Text Book: Business Statistics, Canadian Ed. Sharpe et al Prof.: Dr. Suren Phansalker Office : DMS 5142 Office Hours: Tue 14:0015:30 2 Lecture#1 Sample Mean Distributions: ‘t’ & ‘Z’ Distributions Dr. Suren Phansalker • Central Limit Theorem (CLT): As seen before, PS. Laplace proved the main assertion of the CLT. However, in its modern form, it does have different forms. The following three major cases bring out the variations. • Case I of CLT: If a large sample with size, n ≥ 30, is drawn from any much larger population ‘X’, of unknown distribution, then: But, and where μ , σ , and σ 2 are the population parameters. Then simply written: It sometimes is written as: ( 29 ) ( ), ( ~ 2 X X E N X σ μ = ) ( X E n X 2 2 ) ( σ σ = n N X 2 , ~ σ μ n N X σ μ , ~ 3 • Case II of CLT: If the Population RV, ‘X’ is itself Normally Distributed, then if ‘ σ ’ is known, then for any sample size, (even ‘n’ < 30): or • Case III of CLT: If the Population RV, ‘X’ is itself Normally Distributed, then if ‘ σ ’ is unknown and must be estimated by ‘s’, the sample standard deviation, then: or • Special Condition for Case III of CLT: If ‘n’, the sample size is fairly large (n ≥ 30 or n ≥ 120), then: or In other words, the ‘t’ Distribution and ‘Z’ Distribution become almost equal. n N X 2 , ~ σ μ  n s t X n 2 1 , ~ μ n N X σ μ , ~  n s t X n , ~ 1 μ ≈  n s N n s t X n 2 2 1 , , ~ μ μ ≈  n s N n s t X n , , ~ 1 μ μ 4 • Some Examples of the CLT: • Example 1 for Case I: A large sample of size 100 is taken from a population of marks on a...
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 Winter '11
 Phansalker
 Standard Deviation, 175

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