Part b a. Let X and Y be independent exponential random variables with parameters 0g and Oy, respectively.Find the likelihood ratio test statistic A, based on X and Y, to test Ho: 0=20, against Ha: 0»#20y.b. Let X1,..., X25 be a random sample from N(u1, V50) and Y1,..., X2o be a random sample fromN(u2, V60). We want to test the hypothesisНо : Иі — н2 3D 13На : И1M2#13Find the probability of detecting a shift (power) from µi – µ2 = 13 to µi – µ2to accept a Type I error a = 0.05. Draw the supporting graphs to show Type I and Type II errors.15 if we are willingc. Let X1, X2,..., Xn be a random sample from N(0, o). Use the Neyman-Pearson lemma to find thebest critical region for testing Ho : o? = o?Ha : o2 = o%, with o > o3.d. Refer to question (c). Suppose we are testingНо : о?Н, : о2= 416If n =15, find a numerical value of the best critical region of size a = 0.05. Compute the power ofthis test.

b. From the given information, n1=25σ1=50n2=20σ2=60 Hypotheses: H0:μ1-μ2=13Ha:μ1-μ2≠13 Given that…

Answered: Let X1,..., X25 be a random sample from…

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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Related questions

Question

Part b

a. Let X and Y be independent exponential random variables with parameters 0g and Oy, respectively.
Find the likelihood ratio test statistic A, based on X and Y, to test Ho: 0=20, against Ha: 0»#20y.
b. Let X1,..., X25 be a random sample from N(u1, V50) and Y1,..., X2o be a random sample from
N(u2, V60). We want to test the hypothesis
Но : Иі — н2 3D 13
На : И1
M2#13
Find the probability of detecting a shift (power) from µi – µ2 = 13 to µi – µ2
to accept a Type I error a = 0.05. Draw the supporting graphs to show Type I and Type II errors.
15 if we are willing
c. Let X1, X2,..., Xn be a random sample from N(0, o). Use the Neyman-Pearson lemma to find the
best critical region for testing Ho : o? = o?
Ha : o2 = o%, with o > o3.
d. Refer to question (c). Suppose we are testing
Но : о?
Н, : о2
= 4
16
If n =
15, find a numerical value of the best critical region of size a = 0.05. Compute the power of
this test.

Expert Solution

Step 1

From the given information,

$n_{1} = 25 σ_{1} = \sqrt{50} n_{2} = 20 σ_{2} = \sqrt{60}$

Hypotheses:

$H_{0} : μ_{1} - μ_{2} = 13 H_{a} : μ_{1} - μ_{2} \neq 13$

Given that

$α = 0.05 P (R e j e c t H_{0} | H_{0} i s t r u e) = 0.05 P (Z > z) + P (Z \leq - z) = 0.05 1 - P (Z \leq z) + 1 - P (Z \leq z) = 0.05 2 - 2 P (Z \leq z) = 0.05 1 - P (Z \leq z) = 0.025 P (Z \leq z) = 0.975 P (Z \leq z | μ_{1} - μ_{2} = 13) = 0.975 P (Z \leq \frac{{\bar{x}}_{1} - {\bar{x}}_{2} - (μ_{1} - μ_{2})}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}} | μ_{1} - μ_{2} = 13) = 0.975 P (Z \leq \frac{{\bar{x}}_{1} - {\bar{x}}_{2} - 13}{\sqrt{\frac{50}{25} + \frac{60}{20}}}) = 0.975 P (Z \leq \frac{{\bar{x}}_{1} - {\bar{x}}_{2} - 13}{\sqrt{5}}) = 0.975, o b t a i n e d f r o m t h e e x c e l f u n c t i o n, = N O R M . I N V (0.975, 0, 1) \frac{{\bar{x}}_{1} - {\bar{x}}_{2} - 13}{\sqrt{5}} = 1.96 {\bar{x}}_{1} - {\bar{x}}_{2} = 17.3827$