2. The random variable $ z $ describes the observation of a physical phenomenon whose statistical description depends on one of two hypotheses. Conditioned on hypothesis $ H_1 $, the random variable $ z $ is Gaussian $(200, 10)$, while conditioned on hypothesis $ H_0 $, the random variable $ z $ is Gaussian $(150, 10)$. If $ \text{Pr}\{H_1\} = 0.1 $, and $ \text{Pr}\{H_0\} = 0.9 $:a) Develop the likelihood ratio to implement a MAP decision test in order to decide between the two hypotheses.b) Obtain the probability of error of the test.c) If $ z = 165 $, what decision results from the test?

The statistical problem involves the observation of physical phenomena described by random variable…

Answered: 2. The random variable z describes the…

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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b) Suppose that X₁, X2, ..., Xn denotes a random samples from a gamma distribution f(x; a, ß). i) What will be the minimal sufficient statistic for ß? Explain. ii) If a = 2, use the appropriate method to show that Y = 2₁1 Xi B is a pivotal quantity. iii) Given that a random sample of heavy machines from Polokwane factory produced the following noise levels 85.4, 86.8, 86.1, 85.3 and 84.8. Use part ii) of question 3 b) to construct a 95% confidence interval for B.
a) Write down the test statistic for the null hypothesis b, = 0 against the alternative that b, 10. (Define all terms!) (b) How would you determine the rejection region? (c) Assume the test statistic is in the nonrejection region. What do you conclude? How would you estimate s', the variance of u,? How does this change your answer to 1.?
Do first three parts (but please explain clearly,I will thumbs up) thanks
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): 684
1.ln a simple model Y = Bo + B1X +u, X is endogenous, and Z is an instrumental variable that satisfy the relevance and exogeneity conditions. Show that the two stage least squares (TSLS) estimator is consistent. 2.In above model, what is the asymptotic distribution of TSLS? How can we do a). hypothesis testing of Bi and b). construct confidence interval for B,?
a) QUESTION 3 Suppose that Y₁, Y2, ..., Yn is a random sample from a normal distribution with parameter mean 0 and variance 0, say N(0,0). If Mr Masela and Mr Sepuru desire to use 5% level of significance to test the null hypothesis Ho: 0 = 1 against alternative hypothesis H₁:0 > 1. i) Show that the uniformly most powerful test of their hypothesis rejects Ho if Σ₁ yc, where c is a constant that solves the probability equation 0.05 = P₁Y≥ c|0 = 1). ii) What will be the value of c in part a) if n = 15.
8) x = 20.5, n = 11,o = 7.1, H0: µ = 18.7; Ha: H = 18.7, a = 0.01 A sample mean, sample standard deviation, and sample size are given. Use the one-mean t-test to perform the required hypothesis test about the mean, µ, of the population from which the sample was drawn. Use the critical-value approach. 9) x = 40.9, s = 6.5, n = 15, Hn :H = 32.6, H: p =32.6, a = 0.05. %3D Find the confidence interval specified. had a of 16 ounces. Assume that o = 3.5 ounces.
4- A gambler plays roulette game and he is insisting to play all his money to black (A roulette wheel has 38 slots: 18 are black, 18 are red, and 2 are green). a) Try to calculate required number of spin for average(µ) loss corresponds to the certain amount of his initial money (M) completely. Around this average loss, what is the confidence interval (95% or u±20 ) of the money (the variance (O? ) of the process should be calculated as well)? b)RV x is money (won (n times) or lost(N-n times) amount where N is number of spin, e.g. 25, 50, 100, 200, 400, 500,..1500...) try to establish a distribution and all moments of this RV. Try to explain whether his decision is sustainable or not. Try to find a winning strategy. Plot the probability density function for various N values.
A company that manufactures aerospace parts is considering the possibility of investing in CMU (Cellular Manufacturing Unit). Top Management desires a good estimates distribution characteristic for the FW. There are three random variables i.e., they have uncertainties in their values. An economic analyst is hired to estimate the desired parameters and FW. He concludes that the scenario presented to him is suitable for Monto Carlo simulation because there are uncertainties in three variables and the direct analytical approach is impossible. The company has done a preliminary economic analysis of the situation and provided the analyst with the following estimates: Capital Investment Normally distributed with mean of $150,000 and standard deviation of $12,000 Useful Life Uniformly distributed with a minimum of 5 years and maximum 10 years Market value $25,000 (single outcome) Annual net cash flow $15,000 probability 0.9 $20,000 probability 0.5 $25,000 probability…
State which test (z-test, t-test, ANOVA, χ 2 test) is most suitable for each of the following cases: (i) In an election, voters may support the Democrat candidate, Republican candidate, or Independent candidate. A researcher would like to conduct a poll to test if this year the supportive rates of the three candidates follow the same supportive rates as in the last election, which was 60% : 25% : 15%. (ii) A vaccine is developed for preventing the infection of a disease. An experiment is developed to test its effectiveness. 300 volunteers are divided into two groups randomly. One group of participants will take the vaccine and the other group of participants will not take the vaccine but only vitamin. The proportion of participants in the two groups being infected by the disease will be compared. (iii) A researcher wants to conduct a test to verify if children’s outside school learning experiences would be affected by the families’ income level. Family income will be classified as high…
A random sample is selected from a normal population with a mean u = 30 and standard deviation sima = 8 . After a treatment is administered to the individuals in the sample, the sample mean is found to be M =33. a) if the sample consist of n=16 scores is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two tailed test with alpha spaceequals space 0.05. b) if a sample consists of n= 64 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two tailed with alpha space equals space 0.05. c) comparing your answers for parts a and b explain how the size of the sample influences the outcome of the hypothesis test?

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