Question

Let ?1 and ?2 be a random sample of size 2 from a normal distribution ?(?, 1). Find the likelihood ratio critical region of size 0.005 for testing the null hypothesis ??: ? = 0 against the composite alternative ?1: ? ≠ 0?

Answer #1

Given a random sample of size ? from a normal population with ?
= 0, use the Neyman-Pearson lemma to construct the most powerful
critical region of size ? to test the null hypothesis ? =
?0 against the alternative ? = ?1>
?0 .

Let Y_1, … , Y_n be a random sample from a normal distribution
with unknown mu and unknown variance sigma^2. We want to test H_0 :
mu=0 versus H_a : mu !=0. Find the rejection region for the
likelihood ratio test with level alpha.

Using a random sample of size 77 from a normal population with
variance of 1 but unknown mean, we wish to test the null hypothesis
that H0: μ ≥ 1 against the alternative that Ha: μ <1.
Choose a test statistic.
Identify a non-crappy rejection region such that size of the
test (α) is 5%. If π(-1,000) ≈ 0, then the rejection region is
crappy.
Find π(0).

Let X1, X2, . . . , Xn be a random sample from the normal
distribution N(µ, 36). (a) Show that a uniformly most powerful
critical region for testing H0 : µ = 50 against H1 : µ < 50 is
given by C2 = {x : x ≤ c}. Find the values of c for α = 0.10.

Let X1, X2 be a random sample of size 2 from the standard normal
distribution N (0, 1). find the distribution of {min(X1, X2)}^2

Let X1, X2, . . . , X12 denote a random sample of size 12 from
Poisson distribution with mean θ.
a) Use Neyman-Pearson Lemma to show that the critical region
defined by
(12∑i=1) Xi, ≤2
is a best critical region for testing H0 :θ=1/2 against H1
:θ=1/3.
b.) If K(θ) is the power function of this test, find K(1/2) and
K(1/3). What is the significance level, the probability of the 1st
type error, the probability of the 2nd type...

Let ?1, ?2, … , ?? denote a random sample from a normal
population distribution with a known value of ?.
(a) For testing the hypotheses ?0: ? = ?0 versus ??: ? > ?0
where ?0 is a fixed number, show that the test with test statistic
?̅ and rejection region ?̅≥ ?0 + 2.33(?⁄√?) has a significance
level 0.01.
(b) Suppose the procedure of part (a) is used to test ?0: ? ≤ ?0
versus ??: ? >...

Let X1, X2, · · · , Xn be a random sample from the distribution,
f(x; θ) = (θ + 1)x^ −θ−2 , x > 1, θ > 0. Find the maximum
likelihood estimator of θ based on a random sample of size n
above

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

1. Let X1, X2, . . . , Xn be a random sample from a distribution
with pdf f(x, θ) = 1 3θ 4 x 3 e −x/θ , where 0 < x < ∞ and 0
< θ < ∞. Find the maximum likelihood estimator of ˆθ.

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