Let
The expected value (Mean) of is given by
Hence
The variance of is
Given that
The mean of is
The variance of is
The standard deviation of is
Using CLT, we know that has a normal distribution with mean and standard deviation
The probability using CLT, without using the correction factor is
The probability using CLT, with correction factor is
Each of Xs have 1 success, and hence can see that there are 16 successes in . That is is the number of trials required to get 16 successes, with probability of success on any given trial p=0.5.
We can say that has negative binomial distribution with parameters, number of successes =16 and the probability of success p=0.5
The pmf of is
The exact probability is
The CLT estimate with the correction is 0.0918 and without the correction is 0.0793.
Hence we can say that the CLT underestimates the exact probability.
Find the mean of S16. Find the standard deviation of S16. (Round it to one decimal...
Find the mean of S16. Find the standard deviation of S16. (Round it to one decimal place) Find P(S16 > 40) using CLT, without correction factor. (Round it to 4 decimal places) Find P(S16 > 40) using CLT, with correction factor. (Round it to 4 decimal places FIND p0=exact = P(S16 > 40). Note This is negative binomial with number of successes = n. Do not use Mathematica. It gives different answer because its definition of Negative Binomial is slightly...
Find exact value p0 = P(S16 = 16). (round your answer to four decimal places). Use CLT to approximate p0. Assume the answer is equal to p1 (round your answer to four decimal places). Is p1 an over estimate or underestimate or equal up to 4 decimal places? Let S16-Σ Xi where {X1, X2, , X16} iid Poisson each with mean 1 Let S16-Σ Xi where {X1, X2, , X16} iid Poisson each with mean 1
S16 = sigma Xi where (X1,X2 ... X16) iid geometric each with mean 2 Find mean of S16: Find standard deviation of S16: Find P(S16 > 40) using Central Limit Theorem, without correction factor: Find P(S16 > 40) using Central Limit Theorem, with correction factor: Find p0 = exact = P(S16 > 40)
Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 8; σ = 2 P(7 ≤ x ≤ 11) = Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 6.0; σ = 1.4 P(7 ≤ x ≤ 9) = Assume that x has a normal...
Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 49; σ = 15 P(40 ≤ x ≤ 47) =
1. X has a normal distribution with the given mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 41, σ = 20, find P(35 ≤ X ≤ 42) 2. Find the probability that a normal variable takes on values within 0.9 standard deviations of its mean. (Round your decimal to four decimal places.) 3. Suppose X is a normal random variable with mean μ = 100 and standard deviation σ = 10....
Consider a normal distribution with mean 25 and standard deviation 5. What is the probability a value selected at random from this distribution is greater than 25? (Round your answer to two decimal places.) Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 14.9; σ = 3.5 P(10 ≤ x ≤ 26) = Need Help? Read It Assume that x has a...
Let X be normally distributed with mean μ = 2.4 and standard deviation σ = 1.6. a. Find P(X > 6.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places. b. Find P(5.5 ≤ X ≤ 7.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.) c. Find x such that P(X > x) = 0.0869. (Round "z" value and final answer to 3 decimal places.) d. Find x such...
Let X be normally distributed with mean μ = 3.3 and standard deviation σ = 2.3. [You may find it useful to reference the z table.] a. Find P(X > 6.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.) b. Find P(5.5 ≤ X ≤ 7.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.) c. Find x such that P(X > x) = 0.0485. (Round "z" value and...
Let X be normally distributed with mean μ = 2.9 and standard deviation σ = 1.5. Use Table 1. a. Find P(X > 6.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.) P(X > 6.5) b. Find P(5.5 ≤ X ≤ 7.5). (Round "z" value to 2 decimal places and final answer to 4 decimal places.) P(5.5 ≤ X ≤ 7.5) c. Find x such that P(X > x) = 0.0594. (Round "z" value...