F_{x}(x; θ)=1/ θ for 0<x< θ. We want to test H_{0}: θ=1 and H_{1} : θ>1. To do this, we take a single observation X1 of X and we reject H_{0} if X1> 0.9 .
1. what is the probabilty of Type I error? (false positive, probability = α)
2. what is the value of ß ( type 2 erros ) if θ=1.5?
error type ressources: https://en.wikipedia.org/wiki/Type_I_and_type_II_errors
Fx(x; θ)=1/ θ for 0<x< θ. We want to test H0: θ=1 and H1 : θ>1. To do...
Let X1, . . . , Xn ∼ Geo(θ), f(x)= θ(1-θ)^x, and we wish to test H0 : θ ≤ 1/3 vs H1 : θ > 1/3. a) Using the full sample, X1....Xn, find the form of the UMP test for the hypotheses H0: θ=1/3 vs H1: θ=1/2. b)If n=15 and α = 0.1, what is the rejection region and the size of test in (a)?
Let X1, . . . , Xn ∼ Geo(θ), f(x)= θ(1-θ)^x, and we wish to test H0 : θ ≤ 1/3 vs H1 : θ > 1/3. a) Using the full sample, X1....Xn, find the form of the UMP test for the hypotheses H0: θ=1/3 vs H1: θ=1/2. b)If n=15 and α = 0.1, what is the rejection region and the size of test in (a)?
Suppose you want to test the following hypotheses: H0: p ≥ 0.4 vs. H1: p < 0.4. A random sample of 1000 observations was taken from the population. Answer the following questions and show your Excel calculation for each question clearly: (a) Let p ̂ be the sample proportion. What is the standard error of sample proportion (i.e., σ_p ̂ ) if H0 is true? (b) If the sample proportion obtained were 0.38 (i.e., p ̂=0.38), what is its p-value?...
H0: theta <= theta0 vs. H1: theta > theta0. Let’s say we are testing this based on one observation of X from a Beta distribution with PARAMETERS (1,theta). Reject H0 if X<= c for some c. Write an expression for the power function of the test and sketch a plot of it. Find c so that the test has size alpha0. alpha0=0.05 and theta0=. Find c, the probability of Type 1 error if theta = 0.9, and the power of...
Let X1, . . . , Xn ∼ Exp(θ) and we wish to test H0 : θ = θ_0 vs H1 : θ not= θ_0. Find the asymptotic LRT for this scenario.
Calculate the MP and UMP test of H0:θ≤1 versus H1:θ >1 for a random sample of 40 from Log-normal(0,θ), at the significance level α=.05.
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A hypothesis test is as follows: H0: θ = 2 H1: θ ≠ 2 It has been decided to reject H0 if the observed value of x is x ≤ 0.1 or x ≥ 1.9. Part a: Find the probability of committing a Type I error. Part b: Suppose the true value of θ is 3. Find the probability of committing a Type II error....
Suppose X1, . . . , Xn be a random sample from the Beta(θ, 1) distribution. Find the P-value for the LRT test of the hypotheses H0 : θ ≥ 1 vs H1 : θ < 1.
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...