This problem explores the question of estimating o(> 0) using a confidence intervalbased on a random sample X1, X2, ..., Xn from the distribution with pdffo(x) ={& exp(-20if x > 0otherwise.(a) Find the pdf of X2 by finding the cdf and differentiating it.(b) Using the result of (a) and techniques we developed in class, propose a pivot forestimating using a confidence interval and find the distribution of your pivot. ReviewSlides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is usedto construct a confidence interval.(c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula forfinding a short 100(1-a)% confidence interval for o.(d) Using the formula obtained in (c), compute the width of the resulting 95% confidenceinterval when n = 1 and ₁ = 1.(e) Is the 95% confidence interval for a you obtained in (d) when n = 1 and ₁ = 1 theshortest possible? If your answer is yes, explain why. If the answer is no, give acounterexample.

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A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

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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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6). A random sample of 90 observations produced a mean of x¯=21 from a population with a normal distribution and a standard deviation σ=4.1.
IF n < 30 AND T= Y- SHOW HOW THE 100(1-2a)% CONFIDENCE INTERVAL FOR μL IS DERIVED 5/√T²
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.
Suppose X1,..., Xn is a random sample from an exponential distribution with mean e. If X = 17.9 with n = 50, find (a) a one-sided 95% confidence interval for 0, and (b) a two-sided 95% confidence interval for 0.
Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
A random sample of 10 observations from population A has sample mean of 152.3 and a sample standard deviation of 1.83. Another random sample of 8 observations from population B has a sample standard deviation of 1.94. Assuming equal variances in those two populations, a 99% confidence interval for μA − μB is (-0.19, 4.99), where μA is the mean in population A and μB is the mean in population B. (a) What is the sample mean of the observations from population B? (b) If we test H0 : μA ≤ μB against Ha : μA > μB, using α = 0.02, what is your conclusion?
(Mathematical Statistics) Let X1,X2,...,X5 be a random sample of the Gamma distribution with parameters a = and ß = 0, e > 0. Determine the 95% confidence interval for the parameter e based on the statistic E(i=1 - 5) Xi. 2
Suppose a simple random sample of size n=10 is obtained from a population with μ=68 and σ=19. Answer a-c (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities regarding the sample mean? Assuming the normal model can be used, describe the sampling distribution x. (b) Assuming the normal model can be used, determine P(x<71.1). (c) Assuming the normal model can be used, determine P(x≥69.2).
Let X equal the length of life of a 60-watt light bulb marketed by a certain manufacturer. We do not know the distribution of X except that Var(X) = 1158. Let u be the mean of X. Suppose a random sample of n = 25 bulbs is tested until they burn out, yielding a sample mean of = 1368 hours. (i) Compute an approximate 95% confidence interval for u. (ii) Compute an approximate 95% one-sided confidence interval for that provides a lower bound for u, i.e. compute & such that ɛ P(X - ≤ μµ)≈ 0.95.
You have taken a random sample of sizen & 95of a normal population that has a population mean ofμ = 140and a population standard deviation ofo = 23. Your sample, which is Sample 1 in the following table, has a mean ofx = 141.3. (In the table, Sample 1 is indicated by "M1", Sample 2 by "M2", and so on.) (to) Based on Sample 1, plot the confidence intervals of80%and95%for the population mean. Use1,282as the critical value for the confidence interval of 80%and use1960 as the critical value for the confidence interval of95%. (If necessary, you can refer to a list of formulas .) • Write the upper limit and the lower limit on the graphs to indicate each confidence interval. Write the answers with one decimal place. • For the points (♦and ◆), write the population mean, μ = 140. 128.0 128.0 80% confidence interval 139.0 X Ś 150.0 150.0 128.0 128.0 95% confidence interval 139.0 X Ś 150.0 150.0
5 How does the dynamic CCE estimator solve the estimation problems of the PMG technique?
IF n < 30 AND T = Ý−µ -H, √n SHOW HOW THE 100(1-2x)% CONFIDENCE INTERVAL FOR U IS DERIVED

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