(1) (a) Argue using the Central Limit Theorem that one may approximate X ~ by a normal law when n is large. (b) Under the CLT approximation, find a so that P(X > a)- 10 (1) (a) Argue using...
(2) (a) Argue using the Central Limit Theorem that one may approxímate X ~ Poisson(n) by a normal law when the integer n is large. (b) Compare the true values of E(X3) and E(X4) with those based on the CLT approximation. Note: the requisite moments of normal and Poisson random variables can be extracted from their MGFs.
8. (15 points) Let X ~ Binomial(30,0.6). (a) (5 points) Using the Central Limit Theorem (CLT), approximate the probability that P(X > 20). (b) (5 points) Using CLT, approximate the probability that P(X = 18). (c) (5 points) Calculate P(X = 18) exactly and compare to part(b).
(b) (5 points) Using CLT, approximate the probability that P(X = 18). (c) (5 points) Calculate P(X = 18) exactly and compare to part(b). 8. (15 points) Let X~Binomial(30,0.6). (a) (5 points) Using the Central Limit Theorem (CLT), approximate the probability that P(X > 20).
Why is the Central Limit Theorem useful? [Q8P5.3] a. Because when the conditions for the CLT are met, it allows us to use a Normal distribution to approximate the distribution of the whole population, even if we don't know whether the population follows a Normal distribution. Because when the conditions of the CLT are met, it allows us to calculate the area in the tails of the population distribution and therefore the probability of obtaining an observation as or more...
8. Using Minitab to illustrate the Central Limit Theorem (CLT), the CLT tells us about the sampling distribution of the sample mean. With Minitab we can easily "sample" from a population with known properties (4,0 , shape). a. Our population consists of integer values X from 1 through 8, all equally likely P(x) = 1/8; x = 1, 2, 3, 4, 5, 6, 7, 8 o = 2.29 Using methods from the beginning of Chapter 4 in the textbook, find...
Using Central Limit Theorem) Let S10 sum of 10 Poisson random variables each with mean = 1 1. Find P(S 10 > 10) exactly using Minitab CDF command (Poisson mean =10). 2. Approximate above probability using bell curve approximation -- Normal mean = 0 and standard deviation 1. 3. Show Minitab Command line output
Central Limit Theorem (CLT) 1. The CLT states: draw all possible samples of size _____________ from a population. The result will be the sampling distribution of the means will approach the ___________________- as the sample size, n, increases. 2. The CLT tells us we can make probability statements about the mean using the normal distribution even though we know nothing about the ______________-
(Using Central Limit Theorem) Let S100 sum of 100 independent Bernoulli (toss a coin) random variables. 1. Find P(S 100 > 55) exactly using Minitab CDF command (Binomial n=100, p=0.5). 2. Approximate this probability using bell curve approximation--Normal mean = 0 and standard deviation 1.
Q5 (please also show the steps): CLT = Central Limit Theorem Q5 Consider a problem of estimating the difference of proportions for two populations. In sample 1, out of n subjects, Si of them are "successes" and the rest are "failures". In sample 2, out of n2 subjects, S2 of them are "successes" and the rest are "failures". It is known that Si~ B(ni,P) and S2 ~ B(n2, p). We are interested in estimating P1 - P2. 1. Denote fi =...
The Central Limit Theorem says A) When n<30 , the sampling distribution of x¯¯¯ will be approximately a normal distribution. B) When n<30 , the original population will be approximately a normal distribution. C) When n>30 , the original population will be approximately a normal distribution. D) When n>30 , the sampling distribution of x¯¯¯ will be approximately a normal distribution.