Xn from Consider the testing problem where we have observed a random sample £₁, the model X₁,..., Xn ~ N (µ, o²), where the X₁ are independent, and we want to test the hypothesis H₁: o² = o² against the alternative H₁: 0² ‡ 0². For the test we use the test statistic y = n i=1 (x₁ - x)² 0² where is the sample mean of the xi. (a) What are the type I errors and type II errors in this testing problem? (b) If Ho is true, what is the distribution of the random variable Y corresponding to y? (c) Assume that n = 20 and we reject Ho if and only if y > 32.852. What is the probability of a type I error in this case?
Xn from Consider the testing problem where we have observed a random sample £₁, the model X₁,..., Xn ~ N (µ, o²), where the X₁ are independent, and we want to test the hypothesis H₁: o² = o² against the alternative H₁: 0² ‡ 0². For the test we use the test statistic y = n i=1 (x₁ - x)² 0² where is the sample mean of the xi. (a) What are the type I errors and type II errors in this testing problem? (b) If Ho is true, what is the distribution of the random variable Y corresponding to y? (c) Assume that n = 20 and we reject Ho if and only if y > 32.852. What is the probability of a type I error in this case?
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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could you please do a to c
Transcribed Image Text:B2. Consider the testing problem where we have observed a random sample x₁, ...,
Xn from
the model X₁,..., Xn ~ N(µ, 0²), where the X, are independent, and we want to test
the hypothesis Ho: o² = o against the alternative H₁: 0² ‡ 0². For the test we use
the test statistic
y =
n
i=1
-
2
(x₂ − x)²
0²/
where is the sample mean of the xi.
(a) What are the type I errors and type II errors in this testing problem?
(b) If Ho is true, what is the distribution of the random variable Y corresponding to y?
(c) Assume that n = 20 and we reject Ho if and only if y > 32.852. What is the
probability of a type I error in this case?
(d) Assume now that H₁ is true for the test in part (c). Draw a sketch of the type II
error as a function of o2. (There is no need to calculate explicit values.)
(e) Explain why the test described in part (c) is not a good test for testing the hypoth-
esis Ho: 0² o against the alternative H₁: 0² # of. How would you improve
the test?
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