c. Let X1, X2, ..., Xn be a random sample from N(0,ơ). Use the Neyman-Pearson lemma to find the best critical region for testing Họ :o² = o3 Ha :o2 = o%, with o > of.

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Chapter1: Starting With Matlab
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Part c

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.
Transcribed Image Text: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.
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