Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X; are i.i.d. random variableswith P(X₂ = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observationsx1, x2, x3 = {0, 1}.-Assume we already know that either p = 0.3 or p = 0.8, and other values of p are not possible.In this case we want to decide whether the hypothesis Ho: p = 0.3 or the alternative H₁: p = 0.8is more likely. Consider the test which rejects Ho if and only if s := ³-₁ ; ≥ 2. For this test,determine the probabilities of type I and type II errors.

Given that We have to find probability of type 1 and type 2 error:

Answered: Exercise 14. Let X₁, X2, X3 be…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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6
2. We wish to test the hypothesis Ho: = 0.20 against the alternative hypothesis Ha: #0.20 using the following region of rejection: y in (0,1,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}. a. What is the probability of a Type I error for this experiment? b. What is the probability of a Type II error for this experiment for a= 0.25? c. What is the power for this experiment for a = 0.25? d. The experiment was performed and y obs was equal to 6. What statistical conclusion would you draw based on this observed number of successes?
8.STATISTICAL INFERENCEQuestion 3 b ii)
In random samples, 74 of 250 persons who watched a certain television program on a small TV set and 92 of 250 persons who watched the same program on a large set remembered 2 hours later what products were advertised. Use the x² statistic to test the null hypothesis 01 = 02 against the alternative hypothesis 01 02 at the 0.01 level of significance. 2
A hypothesis was tested twice using a two - tailed z- test. The first test was done with a 5% significance level while the second test using a 10% significance level. If the test stastic for both tests is - 1.75, which of the following is true? A At 5% and 10% alpha level, the null hypothesis is accepted B The null hypothesis is rejected at 5% and accepted at 10% C The null hypothesis rejected for both tests At 5% the null hypothesis is accepted and rejected at 10%
a) What is Fcrit for a one way ANOVA with 5 treatments and 6 replicates per treatment at α=0.05? b)What is Fcrit for a one way ANOVA with 3 treatments and 5 replicates per treatment at α=0.01?
3. Let X, X2, .. X5, be a random sample from a population having mean u and variance o?, and let ô, =(x, + X, + X, + X,) and ô, =(2x, + X, + X, - X,) be two estimates of u. * Which of these two estimators is unbiased? * Which is the better estimator, and why? %3D
6. A transportation department is considering placing stop lights at a busy 4-way-stop intersection. Let X1, X2, X3, X4 represent the number of drivers arriving per minute at the respective stop signs. Assume- these to be four mutually independent Poisson random variables with respective means of 2, 5, 4 and 2. (a) Find the mean and standard deviation of S = Xị. (b) Confirm that S ~ Poisson(13) (c) Compute P(S = 0).
6.STATISTICAL INFERENCEQuestion 3 a ii)
3. The paper "Effects of Caffeine on Repeated Sprint Ability, Reactive Agility Time, Sleep and Next Day Performance"† describes an experiment in which male athlete volunteers who were considered low caffeine consumers were assigned at random to one of two experimental groups. Those assigned to the caffeine group drank a beverage which contained caffeine 1 hour before an exercise session. Those in the no-caffeine group drank a beverage that did not contain caffeine 1 hour before an exercise session. That night, participants wore a device that measures sleep activity. The researchers reported that there was no significant difference in mean sleep duration for the two experimental groups. In the context of this experiment, explain what it means to say that there is no significant difference in the group means. A) The difference in mean sleep duration for the caffeine and no-caffeine groups could have occurred by chance just due to the random assignment, assuming there is a real difference…
a) Suppose that Y₁, Y₂, ..., Y₁ is a random sample from a normal distribution with parameter mean 0 and variance e, 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 H, if Σyc, where c is a constant that solves the probability equation 0.05=PY≥c|0 = 1). ii) What will be the value of c in part a) if n = 15. b) An academic Professor conducted a research aimed at exploring the experience and challenges faced by people who are diagnosed with dementia and their families. If the professor perform a hypothesis test to decide whether the mean age of all people diagnosis with initial occurrence of dementia is less than 55 years old. Based on the statement, explain, what each of the following would mean. i) Type I error. ii) Correct decision. iii) Type II error. c) Suppose that the results of carrying out the…
1. Consider a single random variable X and the following null hypotheses. State if each is a simple or composite hypothesis and explain, (a) Ho : X~ gamma distribution with a = 3, 8 +2 (b) Họ : X- gamma distribution with a = 3, B = 2 (c) Ho : X - exponential distribution

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