[This problem is about (1) UMP hypothesis tests and (2) confidence intervals obtained by inversion. And a Bayes credible interval (set).] Let X1,...,Xn be a random sample from the density f(x|✓) = ✓2 xe✓xI(0,1)(x), where ⌦ = {✓ : ✓ > 0} is the parameter space. a) Consider the problem of testing H0 : ✓ = ✓0 versus H1 : ✓ 6= ✓0. Derive the size ↵ likelihood ratio test of these hypotheses. Show that the rejection region can be written in terms of the statistic Pn i=1 Xi. Give an explicit form for the rejection region in terms of percentiles of a 2 distribution. Hint: The random variable 2✓ Pn i=1 Xi has a chi-square distribution with 4n degrees of freedom. b) Now consider testing H0 : ✓  ✓0 versus H1 : ✓ > ✓0. Doe

For this option, you will work individually. the pythagorean theorem can be used in many real-world scenarios. part 1 write your own real-world scenario where the pythagorean theorem can be applied to find a missing piece. you may choose to write a problem that is two- or three-dimensional in nature. be sure that you will be able to draw a diagram of your scenario. write out your problem and submit it for part 1. be sure to end your scenario with a question. part 2 draw a diagram of the scenario you created in part 1. you may draw by hand and scan and upload your drawing or create a computer-generated drawing for submission. be sure to label all parts and dimensions of the drawing. part 3 solve the question that you posed in part 1. show all of your steps in answering the question. for this option, you will need to submit all three parts for full credit—your real-world problem and question, the diagram that you created, and your work solving the problem, showing all steps. * note that your instructor is looking for your own original idea. while it is acceptable to use the internet for research and inspiration, academic integrity policies apply.

Desmond wants to sell his car that he paid $8,000 for 2 years ago. the car depreciated, or decreased in value, at a constant rate each month over a 2-year period. if x represents the monthly depreciation amount, which expression shows how much desmond can sell his car for today? \ 8,000 + 24x 8,000 - 24x 8,000 + 2x 8,000 - 2x

Aaron invested a total of $4,100, some in an account earning 8% simple interest, and the rest in an account earning 5% simple interest. how much did he invest in each account if after one year he earned $211 in interest? select one: a. $200 at 8%, $3,900 at 5% b. $3,200 at 8%, $900 at 5% c. $900 at 8%, $3,200 at 5% d. $3,900 at 8%, $200 at 5%