Homework Help is Here – Start Your Trial Now!
Math
Statistics
Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X, are i.i.d. random variables with P(X; = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observations F1, F2, F3 € {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=0.8 is more likely. Consider the test which rejects Ho if and only if s:=1 ≥ 2. For this test, determine the probabilities of type I and type II errors.

Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X, are i.i.d. random variables with P(X; = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observations F1, F2, F3 € {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=0.8 is more likely. Consider the test which rejects Ho if and only if s:=1 ≥ 2. For this test, determine the probabilities of type I and type II errors.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question

Exercise 14 pls

Exercise 12. Let X, Y, Z ~ N(0,1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X+Y+Z b) x² + y² + Z² Exercise 13. Let X₁,..., X₁~ N(2,02) i.i.d. and Y₁,...,Y₁ ~ N(Hy, ²) i.i.d. with known o². We assume that for each i the random variables X, and Y; are correlated, with Corr(X₁,Y)= 1/2. Define D₂ = X₁ - Y; for all i € {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Y; and D₁, respectively. a) Show that Cov(X₁,Y₁) = 0²/2. b) Determine E(D;) and Var(D₂). c) Consider the test for Ho: z = Hy which uses the test statistic Z= and which rejects Ho, if and only if |Z| > 1.96. Show that P(type I error) = 5%. d) In two or three sentences, discuss the differences between the test from part (c) on the one hand, and the non-paired test for comparing the means of two populations on the other hand. 2 Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X, are i.i.d. random variables with P(X; = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observations #1, #2, #3 € {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 = 0.8 is more likely. Consider the test which rejects Ho if and only if s = ₁1₁ ≥ 2. For this test, determine the probabilities of type I and type II errors.
Transcribed Image Text:Exercise 12. Let X, Y, Z ~ N(0,1) be independent. Determine the distribution, expectation and variance of the following random variables: a) X+Y+Z b) x² + y² + Z² Exercise 13. Let X₁,..., X₁~ N(2,02) i.i.d. and Y₁,...,Y₁ ~ N(Hy, ²) i.i.d. with known o². We assume that for each i the random variables X, and Y; are correlated, with Corr(X₁,Y)= 1/2. Define D₂ = X₁ - Y; for all i € {1,...,n}. Finally, let X, Y and D be the averages of the X₁, Y; and D₁, respectively. a) Show that Cov(X₁,Y₁) = 0²/2. b) Determine E(D;) and Var(D₂). c) Consider the test for Ho: z = Hy which uses the test statistic Z= and which rejects Ho, if and only if |Z| > 1.96. Show that P(type I error) = 5%. d) In two or three sentences, discuss the differences between the test from part (c) on the one hand, and the non-paired test for comparing the means of two populations on the other hand. 2 Exercise 14. Let X₁, X2, X3 be independent Bernoulli trials, i.e. the X, are i.i.d. random variables with P(X; = 1) = p and P(X; = 0) = 1-p. We want to make inference about p, from observations #1, #2, #3 € {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 = 0.8 is more likely. Consider the test which rejects Ho if and only if s = ₁1₁ ≥ 2. For this test, determine the probabilities of type I and type II errors.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS