a)P(type I error )= P(X>3 |λ =1) =1-(P(X<=3|λ =1)
=1-(e-110/0!+e-111/1!+e-112/2!+e-113/3!)
=1-0.9810
=0.0190
b)
P(type II error )=P(X<=3 |λ =2)
=e-220/0!+e-221/1!+e-222/2!+e-223/3!)
=0.8571
c)
Power =P(X>3 |λ =2)
=1-P(X<=3 |λ =2)
=1-0.8571 =0.1429
from above power =1-type II error
=1 against H :1 2. consider the 2. A sample of size 1 is taken from...
A study is designed to test Ho: P-0.50 against H: p>0.50, taking a random sample of size n-100, using a significance level of 0.05. Show that the rejection region consists of values of p> 0.582 a. Sketch a single picture that shows (i) the sampling distribution of p when Ho is true and (ii) the sampling distribution of p when p-0.60. Label each sampling distribution with its mean and standard error and highlight the rejection region. b. c. Find P(Type...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
2.Let Xj,X,, Xj, X4, Xj be a random sample of size n-5 from a Poisson distribution with mean ?. Consider the test Ho : ?-2.6 vs. H 1 : ? < 2.6. a)Find the best rejection region with the significance level a closest to 0.10 b) Find the power of the test from part (a) at ?= 2.0 and at ?=1.4. c) Suppose x1-1, x2-2, x3 -0, x4-1, x5-2. Find the p-value of the test.
gore the lecture in your streann r 10 minutes. Random variable X follows Poisson umber of calls torecepti,:2-A, against alternative H: λ > , the following Let X' is a of against a distribution with mean A. To test null hypothesis Ho decision rule is used: Null hypothesis is rejected if x, +x,+х,2c.(X,X,X,is a Let 4, 0.2 and c 2 (a) Find the significance level of the test. (b) Find the probability of type II error if the alternative is...
(10 pts) One generates a number z from a uniform distribution on the interval [0.0]. 2 by rejecting Ho if r 0.1 or z > 19. : θ One decides to test Ho : θ 2 against 10 (a) Compute the probability of to have type I error. (b) Find the power for this test if the true value of θ is 2.5.
Let X 1, X 2, X 3, X 4 be a random sample of size n=4 from a Poisson distribution with mean . We wish to test Ho: I = 3 vs. H1: \<3. a) Find the best rejection region with the significance level a closest to 0.05. Hint 1: Since H1: X< 3, Reject Ho if X 1+X 2 +X 3 +X 4<= 0 Hint 2: X 1+X 2 +X 3 + X 4 ~ Poisson (4) Hint 3:...
2. A randon sample XI, X. is drawn frotn Normal(μ, σ2), where-oo < μ < oo and 0 < σ2 < x. To test the null hypothesis Ho : σ2-1 against the alternative H1: σ2 > 1, we have designed the following test Reject Ho if S>k where S2 = "LE:-1(x,-X)2, k ís a constant. Noticed that (n-1) distribution with degree of freedom 1 has a (a) Determine k so that the test will have size a. (b) Use k...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
(18.52) You have an SRS of size n = 11 from a Normal distribution with s = 0.9. You wish to test Ha: μ > 0 You decide to reject Ho if z > 0.03 and to accept Ho otherwise Find the probability (±0.1) of a Type 1 error. That is, find the probability that the test rejects Ho when in fact μ = 0: Find the probability (±0.001) of a Type II error when μ 0.37. This is the...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7. A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...