st371-22 fns of rv 02

# st371-22 fns of rv 02 - ST371 Introduction to Probability...

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(c) Tom Gerig ST371-22 fns of rv 02 1 ST371 Introduction to Probability and Distribution Theory Central Limit Theorem

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(c) Tom Gerig ST371-22 fns of rv 02 2 / n X U n 1 22 Assume that , ..., are with mean and variance . Suppose that . Then as increases to the distribution of n X X iidrv n     approaches that of the standard normal, (0,1). N Central Limit Theorem
(c) Tom Gerig ST371-22 fns of rv 02 3 ( ) ( ) / X P t P Z t n Central Limit Theorem Thus, for large we may use the approximation: n where is a standard normal . The approximation improves as increases. Z rv n

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(c) Tom Gerig ST371-22 fns of rv 02 4 Central Limit Theorem When the Central Limit Theorem ( ) holds, we say CLT 2 " is normally distributed with mean and asymptoti variance call / y " X n  or that 1 asymptotically " is normally n i i X 2 distributed with mean and variance " nn
(c) Tom Gerig ST371-22 fns of rv 02 5 Example 2,716 with standard deviation 72.8   USDA publications state that the mean daily caloric intake by males ages 20-39 is For a random sample of 35 of males ages 20-39, what is the probability that the sample mean daily intake is larger than 2750 calories? n Ref: Michael Sullivan, III. (2004) Statistics: Informed Decisions Using Data.

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(c) Tom Gerig ST371-22 fns of rv 02 6 12 Denote the random sample by , , ..., .
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## st371-22 fns of rv 02 - ST371 Introduction to Probability...

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