Statistics-STAT 409 Fall 2015 A. Stepanov Homework #9

STAT 409Fall 2015A. StepanovHomework #9( due Friday, November 20, by 4:00 p.m. )Please include your name ( with your last name underlined ), your NetID,and your discussion section number at the top of the first page.No credit will be given without supporting work.1 – 2.Let X have a Binomial distribution with the number of trials n = 15 and withprobability of “success” p. We wish to test H 0 : p = 0.30 vs. H 1 : p ≠ 0.30.ˆRecall that p =1.a)XnFind the values ofis the maximum likelihood estimator of p.Λ(x) =L( p 0 = 0.30 ; x )ˆL( p; x )for x = 0, 1, 2, … , 14, 15.If you are using a computer, include the printout.b)2.Suppose we observe X = 7. Find the p-value of this test.c)Likelihood Ratio Test:Reject H 0 if Λ ( x ) ≤ c.Let c = 0.15. Find(i)the significance level,(ii)power when p = 0.20,(iii)power when p = 0.40for the corresponding rejection region.3.Let X have a Binomial distribution with the number of trials n = 15 and withprobability of “success” p. We wish to test H 0 : p = 0.30 vs. H 1 : p > 0.30.a)Find the best Rejection Rule with the significance level α closest to 0.05.b)Find the power of the test from part (a) at p = 0.40 and at p = 0.50.c)Suppose we observe X = 7. Find the p-value of this test.4 – 6.Let λ > 0 and let X 1 , X 2 , … , X n be independent random variables, each withthe probability density functionf (x; λ ) =λx λ +1x ≥ 1.,nRecallY=∑ ln X iis a sufficient statistic for λ ,∑ ln X ihas a Gamma distribution with α = n and θ = 1 λ ,i =1nY=/i =1nˆthe maximum likelihood estimator of λ is λ =n=∑ ln X inY.i =14.We wish to test H 0 : λ = 3 vs. H 1 : λ ≠ 3.a)Likelihood Ratio Test:Reject H 0 if Λ ( y ) ≤ c.Show that Λ ( y ) ≤ c is equivalent to Yb)Let k = e–3.5.c)n –3Yee≤ k.Then Y = 1 is one ( obvious ) solution of YSuppose n = 7.Yn –3Yn –3Ye=e–3.Find ( to the fourth decimal place ) the value of d such that≤ e– 3⇔Y ≤ 1 or Y ≥ dFind(i)the significance level,(ii)power when λ = 2,(iii)power when λ = 4for the rejection region in part (b).6.We wish to test H 0 : λ = 3 vs. H 1 : λ < 3.a)Suppose n = 7. Find the uniformly most powerful rejection region with a 5%level of significance. Round the critical value for Y to the fourth decimal place.b)Find the power of the test in part (a) when λ = 2 and when λ = 1.7 – 8.7.Crosses of mice will produce either gray, brown, or albino offspring. Mendel’smodel predicts that the probability of a gray offspring is 9/16 , the probabilityof a brown offspring is 3/16 , and the probability of an albino offspring is 4/16 .An experiment to assess the validity of Mendel’s theory produces the following data:36 gray offspring; 21 brown offspring; and 23 albino offspring. TestH 0 : p 1 = 9/16 , p 2 = 3/16 , p 3 = 4/16a)8.at α = 0.05,b)vsH 1 : not H 0at α = 0.10.Suppose the experiment in Problem 7 is repeated, but with twice as many observations.Suppose also that we happened to get the same proportions, namely, 72 gray offspring;42 brown offspring; and 46 albino offspring. Repeat Problem 7 in this case, usingα = 0.01.EXCEL functions you may find helpful:= BINOMDIST( x , n , p , 0 )givesP( X = x )= BINOMDIST( x , n , p , 1 )givesP( X ≤ x )= CHIINV ( α , v )gives2χ α (v ) for χ 2 distribution with v degreesof freedom, x s.t. P ( χ 2 (v ) > x ) = α.= CHIDIST ( x , v )givesχ 2 distributionwith v degrees of freedom, P ( χ 2 (v ) > x ).the upper tail probability for= POISSON( x , λ , 0 )givesP( X = x )= POISSON( x , λ , 1 )givesP( X ≤ x )

 

CLICK HERE TO ORDER A SIMILAR PAPER

We pride ourselves in writing quality essays

CLICK HERE TO CONTACT US

Lets Start Working

Plagiarism Free

We use anti-plagiarism software to ensure you get high-quality, unique papers. Besides, our writers have a zero plagiarism mentality

On Time Delivery

Your essay will be delivered strictly within the deadline.  If you have an urgent order, we can do it!

Money Back Guarantee

We offer warranty service, including free revisions, and a right to request a refund incase your expectations are not met!

THE BEST PAPER WRITER HELPER

Our Advantage

  • Say “NO” to plagiarism – FREE plagiarism report as an addition to your paper
  • The lowest prices that fit excellent quality
  • Authorship – you are the one who possesses the paper. We DO NOT re-sale or re-use any of them.

OUR PAPER WRITER HELPER GOODIES

Our Freebies

  • Free Cover Page
  • Free Revisions
  • Free Reference Page
  • Free 24/7 support

Pin It on Pinterest

Share This