Each of 150 newly manufactured items is examined and the number of scratches per item is recorded (theitems are supposed to be free of scratches), yielding the following data:Number ofscratchesper item1346.7Observedfrequency1837423013721Let X be the number of scratches on a randomly chosen item, and assume that X has a Poisson distributionwith parameter u.Q: Knowing the E(X)= µ for the Poisson random variable X, then E(X) would be:1n

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Answered: Each of 150 newly manufactured items is…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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In a survey of 180 females who recently completed high school, 75% were enrolled in college. In a survey of 160 males who recently completed high school, 70% were enrolled in college. At α=0.07, can you reject the claim that there is no difference in the proportion of college enrollees between the two groups? Assume the random samples are independent. Complete parts (a) through (e). Find the critical value(s) and identify the rejection region(s).
In the daily production of a certain kind of rope, the number of defects per foot given by Y is assumed to have a Poisson distribution with mean ? = 4. The profit per foot when the rope is sold is given by X, where X = 70 − 3Y − Y2. Find the expected profit per foot.
For each of the following assertions, state whether it is a legitimate statistical hypothesis and why:a. $\quad H: \sigma>100$b. $H: \tilde{x}=45$c. $H: s \leq 20$d. $H: \sigma_{1} / \sigma_{2}<1$e. $H: \bar{X}-\bar{Y}=5$f. $H: \lambda \leq 01,$ where $\lambda$ is the parameter of an exponential distribution used to model component lifetime
(i) Let X, the number of flaws on the surface of a randomly selected pump of a certain type, have a Poisson distribution with mean 6. a) Compute P(X ≤ 8)b) Compute P(5<X ≤ 8)c) What is the probability that the number of flaws exceeds its mean value by more than one standard deviation?d) If two pumps are independently selected, what is the probability that neither contains a flaw?
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate per hour (alpha = 8), so that the number of arrivals during a time period of t hours is a Poisson rv with parameter (u = 8t) . a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10? b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period? c. What is the probability that at least 20 small aircraft arrive during a 2.5-hour period? That at most 10 arrive during this period?
Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size nfrom a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that thekth order statistic Yk has a beta pdf with parameters α = k and β = n − k + 1.
If X1 and X2 constitute a random sample of size n = 2from a Poisson population, show that the mean of thesample is a sufficient estimator of the parameter λ.
If X follows a binomial distribution with p = 0.1 and n =100, find the approximate value of P(2 ≤ X ≤ 4) using the normal approximation. the Poisson approximation.
Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.
Let X_1, X_2,…, X_n be a random sample of size from a Poisson distribution (θ), with θ unknown. Find the MVUE of P (X = 1).
The number of passengers willing to travel with a certain train is as- sumed to be a Poisson variabel with parameter µ = 300. How many seats should the train have if the probability for an over-booked train should be at most 1%?
A company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?

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