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Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?

Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 4E
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Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?
Transcribed Image Text:Exercise 4 Show that the Central Limit Theorem, as stated in the form of Theorem 4.1, implies that if X₁, X2,, X12 are independent observations from a U (0, 1) distribution, and S = X₁ + X₂++X12, then S-6 is approximately standard normal. (Work through the details and you will see how the constants 6 and 12 arise!) How well does this method work?
.::: F5 Below is the famous Central Limit Theorem stated in what might be called the "standardized" form: Xn is an Theorem 4.1 (The Central Limit Theorem) Suppose X1, X2, i.i.d. sample of observations of a random variable X with mean u and standard deviation o. Then for all x ER lim P(√n- 84x (X) MacBook Air F6 & AA 6 7 Page (12) σ where denotes the cumulative distribution function of a standard normal ran- dom variable. F7 μ) To put it succinctly, if the sample size is large, for an i.i.d. sample the sample mean (or the sample sum) is approximately normal. The question that always arises is how large is "large"? There are the standard rule-of-thumb answers, e.g. having at least 30 observations, which one should take with more than a couple grains of salt. First of all, the answer depends on how good an approximation is demanded by the nature of the intended application. Secondly, it depends on how close the distribution of the random variable being observed is to a normal distribution. * < x) = Φ(x) To illustrate this latter point, consider that the random variable in question is distributed uniformly on the interval (0, 1). The probability density function is not particularly close to that of a normal random variable; but at least it's a 8 6 of 9 DII F8 ... O F9 ) 0 F10 с 1:
Transcribed Image Text:.::: F5 Below is the famous Central Limit Theorem stated in what might be called the "standardized" form: Xn is an Theorem 4.1 (The Central Limit Theorem) Suppose X1, X2, i.i.d. sample of observations of a random variable X with mean u and standard deviation o. Then for all x ER lim P(√n- 84x (X) MacBook Air F6 & AA 6 7 Page (12) σ where denotes the cumulative distribution function of a standard normal ran- dom variable. F7 μ) To put it succinctly, if the sample size is large, for an i.i.d. sample the sample mean (or the sample sum) is approximately normal. The question that always arises is how large is "large"? There are the standard rule-of-thumb answers, e.g. having at least 30 observations, which one should take with more than a couple grains of salt. First of all, the answer depends on how good an approximation is demanded by the nature of the intended application. Secondly, it depends on how close the distribution of the random variable being observed is to a normal distribution. * < x) = Φ(x) To illustrate this latter point, consider that the random variable in question is distributed uniformly on the interval (0, 1). The probability density function is not particularly close to that of a normal random variable; but at least it's a 8 6 of 9 DII F8 ... O F9 ) 0 F10 с 1:
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