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- Prove that if a sequence of random variables X_1,X_2,... converges in probability to a random variable X, the sequence also converges in distributionWorkers 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…The distance between two subsequent rust attacks is exponentially distributed with parameter λ (the distancebetween subsequent events in a Poisson process is exponentially distributed), and we still assumethat λ = 5 per kilometer.What is the probability that the distance between two consecutive rust attacks is between 200and 300 meters ?
- 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 ? = 7500 and ? ≈ 1750.† A random sample of n = 50 workers from the toxic cleanup site were given a blood test that showed x = 6760. What is the probability that, for healthy adults, x will 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 ? x = 7500 and ? x ≈ 247.49.The central limit theorem describes the distribution of x as normal with mean ? x = 7500 and ? x ≈ 1750.00. The central limit theorem describes the distribution of x as normal with mean ? x = 7500 and ? x ≈ 35.00.The central limit theorem does not apply because the sample size is too small. (b) Compute P(x ≤ 6760). (Round your answer to four decimal places.) P(x ≤…The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places. Then, Apply the central limit theorem for the binomial distribution to find the probability that the number of individuals in Lance's sample who regularly skip breakfast is less than 9898. Express the result as a decimal precise to three places.The United States Department of Agriculture (USDA) found that the proportion of young adults ages 20–39 who regularly skip eating breakfast is 0.2380.238. Suppose that Lance, a nutritionist, surveys the dietary habits of a random sample of size ?=500n=500 of young adults ages 20–39 in the United States. Apply the central limit theorem to find the probability that the number of individuals, ?,X, in Lance's sample who regularly skip breakfast is greater than 126126. You may find table of critical values helpful. Express the result as a decimal precise to three places.
- Consider a 20-year-old male. Suppose the survival rate is reduced. 2% per year. That means survival rate at age 21 is 98%, and that at age 22 is 96%, etc. a) According to the above assumption, what is his maximum age? b) Compute the probability that he can celebrate his 60th birthday. Using R to slove it.In a certain desert, the probability that it rains on any given day is 1/500 (i.e., 0.2 percent). Assume that whether it rains or not on a given day does not depend on what happens on other days(a) Write a formula for the probability that it will rain exactly twice during the next 1000 days.(b) Use the Poisson Limit Theorem to approximate this probability.In continuous Probability Distributions such as Uniform, Normal, and Exponential is the range of x both finite or infinite −∞≤x≤∞ ?
- Suppose an insect lays a very large number of eggs Y, where Y ~ Po(lambda), and suppose that each egg survives with probability p. Assuming that the egg's survival is independent, on the average, how many eggs will survive?Suppose that historically, 35% of applicants that are offered admittance to Texas Tech actually enroll, while the others take offers somewhere else. If Texas Tech will accept 9300 this coming year, what is the probability that less than 3250 will actually enroll? Use the normal approximation to the binomial. P(X < 3250)The time between successive calls to a corporate office is exponentially distributed with a mean of 10 minutes. Assume that the times between successive calls are independent of each other. Thus, the calls arrive according to a Poisson process. Determine x such that the probability that there are no calls within x minutes is 0.005.