Q9. Let X,, X2, .., X, be a random sample a gamma distribution with a= 3 and B = 0>0, i.e., ..... with population p.d.f. f(x; 0) =x?e. Show that I. the MLE of 0 is ên = X/3. II. ê = X /3 is unbiased estimator of 0. III. ê = X /3 is MVUE of 0. %3D

Q9. Let X,, X2, .., X, be a random sample a gamma distribution with a= 3 and B = 0>0, i.e., ..... with population p.d.f. f(x; 0) =x?e. Show that I. the MLE of 0 is ên = X/3. II. ê = X /3 is unbiased estimator of 0. III. ê = X /3 is MVUE of 0. %3D

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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Suppose an x-distribution has mean μ = 4. Consider two corresponding x distributions, the first based on samples of size n = 49 and the second based on samples of size n = 81. (a) What is the value of the mean of each of the two x distributions? For n = 49, μ x = For n = 81, μ x = (b) For which x distribution is P( x > 5.00) smaller? Explain your answer. The distribution with n = 81 because the standard deviation will be larger.The distribution with n = 49 because the standard deviation will be smaller. The distribution with n = 49 because the standard deviation will be larger.The distribution with n = 81 because the standard deviation will be smaller. (c) For which x distribution is P(3.00 < x < 5.00) greater? Explain your answer. The distribution with n = 49 because the standard deviation will be larger.The distribution with n = 81 because the standard deviation will be larger. The distribution with n = 49 because the standard deviation…
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Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is ? = 7.4.† A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 41 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.5 with sample standard deviation s = 3.0. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (c) Estimate the P-value. Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the…
Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is ? = 7.4.† A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 36 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.6 with sample standard deviation s = 3.0. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood. (A) What is the value of the sample test statistic? (Round your answer to three decimal places.) (B) Find the P-value. (Round your answer to four decimal places.)
7. Let (z,.,) be a random sample from a normal distribution with mean ja and a variance at. Alo let S.. = , - z)* be the sum of squares of the randou sample. State without proof the sampling distribution of the following; (a) t(z) - ạ + b i-, 1,. n, as e positive eal constants. (- ) (b) t(z) - (E) (x) = (e) ti2) - (g) (r) = (e) t(2) = (d) t(a) Er, - E > ( - P (h) 시)- - -1 (-) (e) t(z)
2. Let X, X,.., x, be a random sample from a distribution with p.d.f. f(x;0)=0x, |>x>0 Is the MLE O EE of 0? AEE of 0?
Let x be a random variable that represents the pH of arterial plasma (i.e., acidity of the blood). For healthy adults, the mean of the x distribution is ? = 7.4.† A new drug for arthritis has been developed. However, it is thought that this drug may change blood pH. A random sample of 31 patients with arthritis took the drug for 3 months. Blood tests showed that x = 8.8 with sample standard deviation s = 3.1. Use a 5% level of significance to test the claim that the drug has changed (either way) the mean pH level of the blood. a) What is the level of significance? b) What is the value of the sample test statistic? (Round your answer to three decimal places.)
An individual who has automobile insurance from acertain company is randomly selected. Let Y be thenumber of moving violations for which the individualwas cited during the last 3 years. The pmf of Y is y p(y) 0 0.6 0.25 0.10 0.05 2 4 6 Suppose an individual with Y violations incurs a surcharge of $100. Calculate the standard deviation of the surcharge Y. 1.48 1.72 2.2 1.2
stion 16: nts) A random sample of 200 psychology students was selected from a department's participant pool. A researchers determined each student's score on the NPI-16 and the number of his or her Facebook friends. The marginal distribution of NPI-16 scores was 4, with a standard deviation of 2. The marginal distribution "Facebook Friends" was 450, with a standard deviation of 200. The correlation between the two variables was .15. Assuming that NPI-16 is the predictor variable and Number of Facebook Friends is the outcome variable, compute the following: The lower limit of the 95% confidence interval around b. Note: Enter your answer to the second decimal place.
Q3: The effective life of a component used in a jet-turbine aircraft engine is a random variable with mean 5000 hours and standard deviation 40 hours. The distribution of effective life is fairly close to a normal component that increases the mean life to 5030 hours and decreases the standard deviation to 30 hours. distribution. The engine manufacturer introduces an improvement into the manufacturing process for this Suppose that a random sample of n₁ = 16 components is selected from the "Old" process and a random sample of n₂ = 25 components is selected from the "Improved" process. What is the probability that difference in the two-sample means that the old and improved processes can be regarded as independent populations. 5 X₂X₁ is at least 15 hours? Assume
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…
Q2. There is a claim that the average mileage given by the particular model car is 22 kilometer per liter. A researcher thought that it is less than 22 kilometer per liter. For this he selected a random sample of 16 cars of the same model and found that the sample gave mean mileage 20.5 kilometer per liter with standard deviation 2 kilometer per liter. If you are that researcher, give your conclusion about the claim at a = 0.05?