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- Let Y be a random variable with an exponential distribution with E(Y) = 0, and supposed a single observation of Y was taken from the population. a) Show that the pivotal 2Y has a chi-square distribution using the moment generating function technique. What are the degrees of freedom for this distribution? b) Using the pivotal quantity derive a 95% confidence interval for 0. c) Imagine an observation of Y = 1.5 was collected, compute the 95% confidence interval. Does this interval contain the population parameter? Explain your answer.Let Y₁, Y2₂,..., Yn be a random sample whose probability density function is given by Sape et. 0, 23 0 0 elsewhere and suppose that n = 200, y = 20, y = 100, 20y = 250 and 8 = 0.025. 200 200 200,3 i) Derive the standard error of ß, se() = 0.0009, using MLE approach. ii) Find an approximate 95% Confidence interval for p. f(y; B): 684For each of the following distributions, compute the maximum likelihood estimator based on 7 i.i.d. observations X₁,..., X and the Fisher information, if defined. If it is not, enter DNE in each applicable input box. (a) (Enter barX_n for the sample average X Maximum likelihood estimator = X~ Ber (p), p = (0,1) Xn = 1-7 n T X₁. Hint: Use the definition of Fisher information that leads to the shorter computation. (If the Fisher information is not defined, enter DNE.) Fisher information I (p) = Use Fisher Information to find the asymptotic variance V (p) of the MLE >>. V (B) = 9 Save
- Let X1, X2, *. Xn be a random sample of size n from norm population with mean and unknown a. Let X and s be the sample mean and sample standard deviation. Then for any 0Assume we have a sample of size n=16 from independent and identically distributed random variables from a normal distribution with mean u and unknwo standard deviation estimated by the sample standard deviation. We observe x =8 and s=4. We let e denote the length of the confidence interval with confidence level 1-a=95%. Which of the following statements are true? > qnorm(c (0.8,0.9,0.95,0.975,0.99,0.999)) [1] 0.8416212 1.2815516 1.6448536 1.9599640 2.3263479 3.0902323 > qt(c(0.8,0.9,0.95,0.975,0.99,0.999), df=15) [1] 0.866245 1.340606 1.753050 2.131450 2.602480 3.732834 Veuillez choisir au moins une réponse: O a. The confidence interval equals [6.25,9.75]. O b. The confidence interval equals [5.87,10.13]. The confidence interval equals [6.36,9.64]. O c. O d. The confidence interval equals [6.04,9.96].Assume that we want to construct a confidence interval. Which letter is correct? The confidence level is 99%, σ=3624 thousand dollars, and the histogram of 60 player salaries (in thousands of dollars) of footbal players on a team is as shown. a) find the critical value tα/2 b) find the critical value zα/2 c) neither the normal distribution nor the t distribution applies.Suppose you have 36 observations from a Poisson distribution with parameter λ. Suppose the sample mean of the 36 observations is 3.7. (a) Calculate an upper confidence bound for λ using a confidence level of 95%. ( b) Calculate a lower confidence bound for λ using a confidence level of 95%. (c) Calculate a two sided confidence interval for λ using a confidence level of 95%.A sample of 66 night-school students' ages is obtained in order to estimate the mean age of night-school students. x = 24.7 years. The population variance is 18. (a) Give a point estimate for u. (Give your answer correct to one decimal place.) (b) Find the 95% confidence interval for u. (Give your answer correct to two decimal places.) Lower Limit Upper Limit (c) Find the 99% confidence interval for u. (Give your answer correct to two decimal places.) Lower Limit Upper LimitA new method has been developed in the treatment of a disease. 12 randomly selected patients were treated with this method and the time until recovery was calculated as in the picture. Establish a confidence interval for the mass mean µ. (α=0.05)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.Let X1,., X, be iid from Normal(e,0) population (0>0). Use the pivotal quantity to find (1-a) Confidence Interval for 0. ...A sample of size n=79 Is drawn from a population whose standard deviation is o = 29, Part 1 of 2 (a) Find the margin of error for a 90% confidence interval for l. Round the answer to at least three decimal places. The margin of error for a 90% confidence interval for u is Part 2 of 2 (b) If the sample size weren=44, would the margin of error be larger or smaller? (Choose one) ▼ because the sample size is (Choose one) Larger SmallerSEE MORE QUESTIONS
