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- Let X1, X2, ... , Xn be a random sample from N(μ, σ2). Find the Moment Generating Function of X̅. If n = 16 and σ = 2, compute P(-1 ≤ X̅ - μ ≤ 1).Each of 14 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 9 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. (I have figured out part "a" but need help with "b" and P(X ≤ 3) in "c") (a) Calculate P(X = 4) and P(X ≤ 4). (Round your answers to four decimal places.) P(X = 4) = P(X ≤ 4) = (b) Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.) (c) Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 25 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately)…A screening test for a particular infectious disease shows a positive test result in 90% of all cases when the disease is actually present, and in 10% of all cases when it is not. Assume that t% of the population is infected, where t is = 1. For a randomly chosen from the population person, find the probability that the person is actually infected if the test shows positive.
- Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?Each of 13 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 10 of these refrigerators have a defective compressor and the other 3 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 9 examined that have a defective compressor. (a) Calculate P(X = 7) and P(X ≤ 7). (Round your answers to four decimal places.) (b) Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.)Each of 14 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 9 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. (a) Calculate P(X = 4) and P(X ≤ 4). (Round your answers to four decimal places.) P(X = 4)= ? P(X ≤ 4)= ? (b) Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.) (c) Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 25 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X ≤ 3) than to use the hypergeometric pmf. We can approximate the…
- An entomologist writes an article in a scientific journal that claims that fewer than 16 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Use the parameter p, the true proportion of fireflies unable to produce light. A) H 0 : p > 0.0016 H 1: p ≤ 0.0016 B) H 0 : p = 0.0016 H 1: p > 0.0016 C) H 0 : p = 0.0016 H 1: p < 0.0016 D) H 0 : p < 0.0016 H 1: p ≥ 0.0016Theorem: Suppose (Xn)n≥1 is a sequence of random variables with corresponding momentgenerating functions M_Xn , and X is a random variable with moment generating functionM_X such that for some δ > 0 we have M_X (t) < ∞ for all t ∈ (−δ, δ). If lim n→∞ MXn (t) = MX (t) for all t, then lim n→∞ F_Xn (x) = F_X (x) for all x where F_X is continuous. That is, if the moment generating functions of X_n converge to the moment generating function of X, then the distribution of X_n converges to the distribution of X. Use this to show that if Sn ∼ Binomial(n, λ/n ), then the distribution of Sn converges to Poisson(λ) as n → ∞.A time series with a periodic component can be constructed fromxt = U1 sin(2πω0t) + U2 cos(2πω0t),where U1 and U2 are independent random variables with zero means andE(U21 ) = E(U22 ) = σ2. The constant ω0 determines the period or time ittakes the process to make one complete cycle. Show that this series is weaklystationary with autocovariance functionγ(h) = σ2 cos(2πω0h
- Workers at a large toxic cleanup project are concerned that their white blood cell counts may have been reduced. Let x be a random variable that represents white blood cell count per cubic millimeter of whole blood in a healthy adult. Then μ = 7500 andσ ≈ 1750.† A random sample ofn = 70 workersfrom the toxic cleanup site were given a blood test that showedx = 6920.What is the probability that, for healthy adults,xwill be this low or lower?(a) How does the central limit theorem apply? Explain. The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈1750.0.The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈209.2. The central limit theorem does not apply because the sample size is too small.The central limit theorem describes the distribution of x as normal with mean μ x = 7500 and σ x≈25.0. (b) ComputeP(x ≤ 6920).(Round your answer to four decimal places.) P(x ≤ 6920)= (c) Based on your answer to…Show that the following series is divergentA doctor is called to see a sick child. Thedoctor has prior information that 90% of sick children in that neighborhoodhave the flu, while the other 10% are sick with 1 measles. Let F stand for anevent of a child being sick with flu and M stand for an event of a child beingsick with measles. Assume for simplicity that F ∪ M = Ω,i.e., that there no other maladies in that neighborhood. A well-known symptomof measles is a rash (the event of having which we denote R). Assume that theprobability of having a rash if one has measles is P(R | M) = 0.95. However,occasionally children with flu also develop rash, and the probability of havinga rash if one has flu is P(R | F) = 0.08. Upon examining the child, the doctorfinds a rash. What is the probability that the child has measles?