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8. (15 points) Let X ~ Binomial(30,0.6). (a) (5 points) Using the Central Limit Theorem (CLT),...
(b) (5 points) Using CLT, approximate the probability that P(X = 18). (c) (5 points) Calculate...
(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).
8. Using Minitab to illustrate the Central Limit Theorem (CLT), the CLT tells us about the...
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...
(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...
(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 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
(Using Central Limit Theorem) Let S100 sum of 100 independent Bernoulli (toss a coin) random variables....
(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.
Why is the Central Limit Theorem useful? [Q8P5.3] a. Because when the conditions for the CLT...
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...
(d) Explain briefly the Central Limit Theorem. Using this theorem, how can you approximate the Binomial...
(d) Explain briefly the Central Limit Theorem. Using this theorem, how can you approximate the Binomial distribution?
6. In this question, you are going to study the approximation to binomial probabilities using the...
6. In this question, you are going to study the approximation to binomial probabilities using the nor mal distribution. The binomial distribution is discrete while the normal distribution is continuous Therefore, we would expect some issues with approximating the binomial with the normal. (a) (2 points) Suppose X ~ Bin (25,04). Calculate E (N) and Var . (b) (4 points) Use the central lit theorem along with (a) to approximate Pr (X 8). Compare this with your result in #4(a)....
Please use the central limit theorem. 9.14 A baseball player has a batting average of 0.328....
Please use the central limit
theorem.
9.14 A baseball player has a batting average of 0.328. Let X be the number of hits the player gets during 20 times at bat. Use the central limit theorem to find the approximate probability P(X<k) for k = 1, 3, 6. Compare with the exact probability for each k. Problem 9.14. The problem assumes that the batter's probability of getting a hit stays constant at p=0.328 while he comes up to bat 20...
Central Limit Theorem (CLT) 1. The CLT states: draw all possible samples of size _____________ from...
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 ______________-
(2) (a) Argue using the Central Limit Theorem that one may approxímate X ~ Poisson(n) by...
(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.
(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).
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...
(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 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
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...
(d) Explain briefly the Central Limit Theorem. Using this theorem, how can you approximate the Binomial distribution?
6. In this question, you are going to study the approximation to binomial probabilities using the nor mal distribution. The binomial distribution is discrete while the normal distribution is continuous Therefore, we would expect some issues with approximating the binomial with the normal. (a) (2 points) Suppose X ~ Bin (25,04). Calculate E (N) and Var . (b) (4 points) Use the central lit theorem along with (a) to approximate Pr (X 8). Compare this with your result in #4(a)....
Please use the central limit
theorem.
9.14 A baseball player has a batting average of 0.328. Let X be the number of hits the player gets during 20 times at bat. Use the central limit theorem to find the approximate probability P(X<k) for k = 1, 3, 6. Compare with the exact probability for each k. Problem 9.14. The problem assumes that the batter's probability of getting a hit stays constant at p=0.328 while he comes up to bat 20...
(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.
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