a) Let X₁, X₂, X, be a random sample of size n from population X. Suppose that X has thefollowing probability mass function:f(x; 0) = {0(1(0 (1-0)*-1,x = 1,2,3,...elsewherei) Find the conditional distribution of the likelihood function given the function of the statistic,6 = 1X₁. Hint: Apply f(L(0)|g(0,0)).ii) Based on i) above, is = 1X₁ a sufficient statistic? Justify your answer.iii) Show whether or not X, is a consistent estimator of the parameter

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Answered: i) Find the conditional distribution of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Once an individual has been infected with a certain disease, let X represent the time (days) that elapses before the individual becomes infectious. An article proposes a Weibull distribution with a = 2.6, B = 1.8, and y = 0.5. [Hint: The two-parameter Weibull distribution can be generalized by introducing a third parameter y, called a threshold or location parameter: replace x in the equation below, a -x« – le-(x/B)ª Ba x 2 0 f(x; а, B) %3D x y.] (a) Calculate P(1 1.5). (Round your answer to four decimal places.) 0.8050 (c) What is the 90th percentile of the distribution? (Round your answer to three decimal places.) 2.980 |× days (d) What are the mean and standard deviation of X? (Round your answers to three decimal places.) 1.594 х days mean standard deviation 0.585 X days
The calibration of a scale is to be checked by weighing a 11 kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with o = 0.200 kg. Let u denote the true average weight reading on the scale. (a) What hypotheses should be tested? Ho: H = 11 Ha: u + 11 Ho: H + 11 H3i H 11 a (b) With the sample mean itself as the test statistic, what is the P-value when x = 10.83? (Round your answer to four decimal places.) What would you conclude at significance level 0.01? Conclude that the true mean measured weight differs from 11 kg. Conclude that the true mean measured weight is the same as 11 kg. (c) For a test with a = 0.01, what is the probability that recalibration is judged unnecessary when in fact u = 11.2? (Round your answer to four decimal places.) For a test with a = 0.01, what is the probability that recalibration is judged unnecessary when in fact u = 10.9? (Round your answer to…
3) Explain what is meant by constructing a confidence interval for an unknown parameter 0 from a given sample x, ..., N. Let a family of PDFS fS(x: 0). -o 0, S(x: 0) = 0. x 0. Explain how to construct a 95% confidence interval for a from a sample x, . , x. Justify the claims about the distributions you use in your construction.
i. critical number(s). ii. the maximum and/or minimum value(s) using the second derivativetest.
For 50 randomly selected speed dates, attractiveness ratings by males of their female date partnerS (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield *= 6.3, y = 6.1. r= - 0.283, P.value = 0.046, and g = 8.39 - 0.369x. Find the best predicted value of y (attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x= 4. Use a 0.01 significance level. see score The best predicted value of g when x= 4 is 7. (Round to one decimal place as needed.)
Consider data 5,1,3,5,5,4,3,2 of the Poisson distribution with expectation u. We want to test H0:u=u0 with u0=2. Write the log-likelihood for the data under H0 and the likelihood ratio A(y) for testing H0
For 50 randomly selected speed dates, attractiveness ratings by males of their female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings range from 1 to 10. The 50 paired ratings yield *= 6.4. y = 6.0, r= - 0.279, P-value = 0.050, and g = 8.27 - 0.357x. Find the best predicted value of y (attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x=7. Use a 0.10 significance level. The best predicted value of y when x = 7 is (Round to one decimal place as needed.)
X is a normally normally distributed variable with mean u =10 and standard deviation a =4. Find A) P(x 1) C) P(10
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The average wait time to get seated at a popular restaurant in the city on a Friday night is 12 minutes. Is the mean wait time greater for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 13, 13, 11, 13, 13, 13, 13, 10, 12, 13, 13, 10, 13 What can be concluded at the the αα = 0.05 level of significance level of significance? The null and alternative hypotheses would be: H0 = (p,u) (<,>,=,not equal) to _____ H1 = (p,u) (<,>,=,not equal) to _____ The test statistic (t,z) = ___ The p-value = _____ Thus, the final conclusion is that ... The data suggest that the population mean wait time for men who wear a tie is not significantly more than 12 at αα = 0.05, so there is statistically insignificant evidence to conclude that the population mean wait time for men who wear a tie is more than 12. The data suggest the population mean is not significantly more than 12…
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i) Find the conditional distribution of the likelihood function given the function of the statistic, 6 = ₁X₁. Hint: Apply f(L(0)|g(8,0)). ii) Based on i) above, is = 1X, a sufficient statistic? Justify your answer. iii) Show whether or not X, is a consistent estimator of the parameter