6. Let X₁, X2, ..., X6 be a random sample from a distribution with densityfunction0-1for 0 0f(x; 0) ==0otherwise.The null hypothesis Ho : 0 = 1 is to be rejected in favor of the alternativeHa> 1 if and only if at least 5 of the sample observations are larger than0.7. What is the significance level of the test?

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Answered: 6. Let X₁, X2, ..., X6 be a random…

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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Let X1 and X2 constitute a random sample from a nor-mal population with σ2 = 1. If the null hypothesis μ = μ0 is to be rejected in favor of the alternative hypothesis μ = μ1 > μ0 when x > μ0 + 1, what is the size of the criti-cal region?
Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?
7 Let X1,...Xn be iid Normal( θ+ c, σ^2), where c and σ ^2 are known constants (i.e., E(Xi) = θ + c). Find a sufficient statistic forθ then obtain the minimum-variance unbiased estimator for θ.
Let x and y be random variable such that the mean and variance of X are 2 and 4. respectively, while the mean and variance of y are 6 and k, respectively. A sample of size 4 is taken from the x-distribution and a sample of size 9 is taken from the y-distribution .If p[(x-y)>8]=0.0228,then what is the value of the constant k?
Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]
2.Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of a random sample of 10 containers are 10.1, 9.9, 10.2, 10.3, 10.3, 9.7, 9.8, 10.5, 10.2, and 9.8 liters. Use alpha 0.01 and assume that the distribution of content is normal.
A random sample of a size n = 25 is obtained from a population with a variance of σ2 = 400, and the sample mean is computed to be x̅ c = 70. Consider the null hypothesis H0: μ = 80 versus the alternative hypothesis H1: μ = 80. Compute the p-value for this test
Let X1, · · · , Xn be drawn independently from a distribution with density 1. f(x|θ) = 2/β, θ − 0.25β < x < θ + 0.25β, where β is a positive constant. 2. f(x|θ) = 2x/θ 2 , 0 < x < θ, θ > 0. 3. f(x|θ) = e θ−x for x > θ. For each of the above three cases, construct a 95% confidence interval for θ.
Let X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.
To test H0: μ = 50 versus H1: μ < 50, a random sample of size n = 24 is obtained from a population that is known to be normally distributed with σ = 12a) If the sample is determined to be x = 47.1, compute the test statistic.b) If the researcher decides to test this hypothesis at α = 0.05 level of significance, determine the P-value.
An electrical engineer wishes to determine if, among two specific municipal buildings in town, Building “North” and Building “South”, whether the tensile strength of pipes (in psi) is not the same in each of these two buildings. A sample of pipes was chosen at random from both Building “North” and Building “South”, respectively. Using α = 0.05, which of the following statistical test, or parameter, would be best for determining whether tensile strength of pipes (in psi) is not the same in each of these two buildings? (Assume all statistical assumptions met.) a) Binomial Distribution b) Population Difference in Means (i.e., Unpaired Data) c) The Chi-Squared Test of Independence d) Population Mean Difference (i.e., Paired Data)
Which of the following can cause the usual OLS t statistics to be invalid (that is, not to have t distributions under H0)?(i) Heteroskedasticity.(ii) A sample correlation coefficient of .95 between two independent variables that are in the model.(iii) Omitting an important explanatory variable.

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