Solve two quick hypothesis testing problems in R (with explanations)

fpujnwxn
timer Asked: Jun 2nd, 2018

Question Description

For problems 1{4 generate the data as follows: [...] (see attached) where Φ is the cumulative distribution function of N (0; 1) and Φ-1 is its inverse.

In R, you can define a function f calculating ak from k, then apply f to the whole list of k’s to get the list a.data
of ak, and, finally get xk and yk by running qnorm on a.data.

Unformatted Attachment Preview

R.H. Probability and Statistics: Lab Assignment 4 @ UCU Spring 2018 Probability and Statistics Lab assignment 4: Hypothesis testing and Linear regression General comments: • Complete solution will give you 5 points (out of 100 total). Submission deadline — June 03 at 18:00. • The preferred (and strongly advised) language is R (https://www.r-project.org/). It can be installed from the official site; RStudio (https://www.rstudio.com/) is a convenient GUI • You will need just a few basic R commands to complete the task. As a quick reference guide, use the official manual https://cran.r-project.org/doc/manuals/r-release/R-intro.pdf or help section of RStudio • The assignment must be prepared as a Jupyter notebook and submitted to cms. To use R within Jupyter, you will have to install the R kernel (available on https://irkernel.github.io/) • For each task, include the corresponding R code (usually it is just a couple of lines long), the statistics obtained (like sample mean or anything else you use to complete the task) and make conclusions whether to accept the null hypothesis • The id number referred to in tasks is your ordinal number in the student list on cms (see the attached file). Observe that the answers do depend on this id number! Part I: Hypothesis testing For problems 1–4 generate the data as follows: set ak := {k ln (k 2 n + π)}, k ≥ 1, where {x} := x − [x] is the fractional part of a number x and n is your id number. Sample realizations X1 , . . . , X100 and Y1 , . . . , Y50 from the hypothetical normal distributions N (µ1 , σ12 ) and N (µ2 , σ22 ) respectively are obtained as xk = Φ−1 (ak ), −1 yl = Φ k = 1, . . . , 100, (al+100 ), l = 1, . . . , 50, where Φ is the cumulative distribution function of N (0, 1) and Φ−1 is its inverse. In R, you can define a function f calculating ak from k, then apply f to the whole list of k’s to get the list a.data of ak , and, finally get xk and yk by running qnorm on a.data. Instructions: In problems 1–4, test H0 vs H1 . To this end, • point out what standard test you use and why; • indicate the general form of the rejection region of the test H0 vs H1 of level 0.05; • find out if H0 should be rejected on the significance level 0.05; • indicate the p-value of the test and comment whether you would reject H0 for that value of p and why Problem 1. H0 : µ1 = 0 vs. H1 : µ < 0; σ12 is unknown. Problem 2. H0 : µ1 = µ2 vs. H1 : µ1 6= µ2 ; Problem 3. H0 : σ12 = 1 vs. H1 : σ12 6= 1; Problem 4. H0 : σ12 = σ22 vs. H1 : σ12 > σ22 ; σ12 = σ22 = 2. µ1 = 0. µ1 and µ2 are unknown. Hint: this is the f -test; read the details in Ross, p. 321-323 1 R.H. Probability and Statistics: Lab Assignment 4 @ UCU Spring 2018 Part II: Simple linear regression Consider the simple linear regression model Yk = a + bxk + εk , in which ε1 , . . . , ε50 are i.i.d. rv’s with normal distribution N (0, σ 2 ). Generate the data (xk , yk ), k = 1, . . . , 50 as follows:  xk := 10 · 1 + cos(kn) ,    yk := sin n + cos(k 2 ) + cos n + sin(k 2 )/k · 1 + sin(k 2 )/k · xk , where n is your id number. Problem 5. (a) Find estimate â, b̂, σ̂ 2 of the parameters a, b and σ 2 ; (b) test the hypothesis H0 : b = 0 vs the general alternative; (c) find the determination coefficient r2 and comment on whether the linear model is adequate; (d) find the confidence interval for Y at x = 0 and x = 20. Hint: all this can be done with one single command lm 2
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

This question has not been answered.

Create a free account to get help with this and any other question!

Related Tags

Brown University





1271 Tutors

California Institute of Technology




2131 Tutors

Carnegie Mellon University




982 Tutors

Columbia University





1256 Tutors

Dartmouth University





2113 Tutors

Emory University





2279 Tutors

Harvard University





599 Tutors

Massachusetts Institute of Technology



2319 Tutors

New York University





1645 Tutors

Notre Dam University





1911 Tutors

Oklahoma University





2122 Tutors

Pennsylvania State University





932 Tutors

Princeton University





1211 Tutors

Stanford University





983 Tutors

University of California





1282 Tutors

Oxford University





123 Tutors

Yale University





2325 Tutors