b)Let X₁, X₂, ..., Xn and Y₁, Y₂, ..., Ym be random samples from populations with moment generatingfunctions Mx₁(t) = ³t+t² and My(t) = (₁¹)²5, respectively.i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5.ii) What is the value of the sample size n, if P[₁(X; - X)² > 68.3392] = 0.025?iii) What is the value of the sample size m, if P(|Ỹ - µỹ| ≥ 10) < 0.04?

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Answered: i) Find the sampling distribution of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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b) i. Let Y be the mean of sample of size n from a normal distribution with mean μ and variance 100. Find n so that Pu-5<Ỹ < µ+5)= 0.954
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time, i.e. P(μx-ox ≤ x ≤ μx + 0x). 3. Suppose that people heights are normally distributed with mean of 170 cm and standard deviation of 6 cm. a) What proportion of people are between 165 cm and 175 cm tall? b) Find the minimum ceiling of an airplane such that at most 5% of people walking down the aisle will have to duck. E c) Find the probability that the average height of a random sample of 49 people is greater than 172 cm. 80 F3 4 R % 5 T F6 Y 1 & 7 K ard deviation of the expected U 8 DII
5. Suppose X is the number of successes in a random sample of size n from a large population with a proportion of success p. What are the mean and the standard deviation of the sampling distribution of X in terms of n and p? The mean is p(l-pland the standard deviation is p(1-p) b. The mean is npand the standard deviation is ap(1-p). The mean is p(1-pland the standard deviation is pp. C. d. The mcan is 7pand the standard deviation is rp.
Compute for the mean (??̅) and standard deviation (??̅) of the sampling distribution of the sample means and compare these to the mean and standard deviation of the population.
2.Given a random sample of size n = 29, from a normal distribution, find k such that: a) P(-k < T < 1.313) = 0.88 b) P(-k < T < k) = 0.99 c) P(-2.286 < T < k) = 0.685
It is believed that the mean value of a variable measured for an entire population of individuals has value µ = 17 and that for the population the standard deviation is σ = 5.69. A sample of size n= 43 is to be taken from the population. According to the central limit theorem, the sampling distribution of the mean is Normal( 43 , 0.872 ) Normal( 17 , 5.692 ) Normal( 17 , 0.872 ) Normal( 43 , 5.692 )
Suppose x has a distribution with mean 22 and standard deviation 16. If a random sample of size n=36 is drawn find p(22<×<24)
a) Let X₁, X2, X3,..., X₁ be a random sample of size n from population X. Suppose that X follows an exponential distribution with parameter 0 and Y = ===1 X₁. n b) i) Show that y follows a chi-square distribution with 2n degrees of freedom. ii) What is the value of the sample size n, if P(Y > 20.4831) = 0.025? iii) What is the value of sample size n, if P(|Y - 2n| <40) ≥ 0.025? iv) Use the Central Limit Theorem to compute P(100 < Y < 115), for n = 40. Let X₁, X2, ..., X5 be a random sample of size 5 from the standard normally population. If the statistic Y is given by Y = X1 X2 X3 ₁²+X3² +X5² i) Briefly explain why Y follows a Student's t distribution with 3 degrees of freedom. ii) Compute the probability that Y is between 2.35 and 5.84. iii) What is the mean and variance of the statistic Y?
1.) Suppose 5% of students are veterans and 143 students are involved in sports. How unusual would it be to have no more than 12 veterans involved in sports? (12 veterans is about 8.3916%)When working with samples of size 143, what is the mean of the sampling distribution for the proportion of veterans? When working with samples of size 143, what is the standard error of the sampling distribution for the proportion of veterans? Compute P(ˆp≤p^≤ 0.083916).P(ˆp≤p^≤ 0.083916) = NOTE: Give results accurate to 5 decimal places 2.)Suppose 15% of students are veterans and 138 students are involved in sports. How unusual would it be to have no more than 14 veterans involved in sports? (14 veterans is about 10.1449%)When working with samples of size 138, what is the mean of the sampling distribution for the proportion of veterans? When working with samples of size 138, what is the standard error of the sampling distribution for the proportion of veterans? Compute P(ˆp≤p^≤ 0.101449).P(ˆp≤p^≤…
A simple random sample of size n=49 is obtained from a population with μ=85 and σ=28. (a) What is P x>92.2? (b) What is P x≤75.6? (c) What is P 79.8<x<93?

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i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X₁ + X5. ii) What is the value of the sample size n, if P[Σï=₁(X₁ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - μ| ≥ 10) < 0.04?