代写STAT601、代写R设计、R程序调试、代做AUTonline留学生

日期：2019-03-17 10:41

STAT601 Statistical methods – Assignment 1 Semester 1, 2019

STAT601 Statistical methods

Semester 1, 2019

Assignment 1

Instructions

Submit to Blackboard (AUTonline) by the due date

This assignment is worth 10% of your final grade

Assignments should be submitted as a single pdf or word document.

Your submission should contain relevant explanations, mathematical notation, r code,

and workings. Answers which do not include appropriate notation and/or workings will

be penalised.

Handwritten answers should be scanned and included in the word or pdf file. Ensure

that the scanned images are of sufficiently high quality that they can be easily read when

printed.

Your assignment file should include the Individual Assignment Coversheet:

Blackboard/ Assessment Policies, Regulations, Guides and Forms/ Forms and Coversheets/

Individual Assignment Coversheet

Late Assignments:

Failure to submit the assignment on time will result in a mark of 0 for the assignment. If

extenuating circumstances (e.g. illness) prevent the timely submission of your assignment

you can apply for special consideration. You may also apply for special consideration if

such circumstances result in your submission being incomplete. The required form is

available on Blackboard:

Blackboard/ Assessment Policies, Regulations, Guides and Forms/ Forms and Coversheets/

Special Consideration and Extensions

Originality: This assignment is an individual piece of work. You are encouraged to

discuss the assignment with your lecturers and classmates, however, the work you submit

must be your own. Assignments that show similarities to work submitted by other students

will be investigated for plagiarism and treated very seriously. Plagiarism software, such as

TurnItIn, may be used to electronically compare submissions to those of other students

and to documents on the internet. Talk to the lecturer if you have any questions about

this requirement. Before you submit this assignment, you should complete the Academic

Integrity module on Blackboard.

Question: 1 2 3 Total

Marks: 70 40 40 150

Score:

Page 1 of 2

STAT601 Statistical methods – Assignment 1 Semester 1, 2019

1. In this question we will evaluate type I and type II error probabilities for one-sided tests. We

will consider normally distributed data, with unit variance and independent obervations.

We will use H0 : μ = 0 for the null and H1 : μ = 1 for the alternative, unless otherwise stated.

(a) Suppose we have n = 6 observations x1,...,x6. What is the sampling distribution of the (10 marks)

sample mean (that is, of x =16(x1 +··· + x6))

(b) We want a test with size α = 0.05. This test is to be of the form “reject H0 if the sample (10 marks)

mean x exceeds T ” (where T is a value to be determined).

You will recall that α is the probability of rejecting H0 when true. Find an appropriate

value of T .

(c) Calculate β, the probability of failing to reject the null hypothesis when the alternative (10 marks)

is true, and state the power of the test.

(d) Consider a test of size α = 0.01. Calculate the power of this test. (10 marks)

(e) How many observations would it take to have a size of at most 0.01 and a power of at (10 marks)

least 0.99?

(f ) (harder) Now we will consider the case where the null and alternative hypotheses are (20 marks)

very close. We will have H0 : μ = 0 but now H1 : μ = 0.02. Now how many observations

are needed to ensure α is at most 0.01 and the power is at least 0.99?

2. Consider the following dataset:

fuel <- c(0.95, 0.52, 0.82, 0.89, 0.81)

The numbers correspond to the amount of fuel burnt by a new type of high-efficiency

engine under a randomised test load. A value of 1 corresponds to the same fuel efficiency

as the old engine, values greater than one correspond to more fuel burned (hence lower

efficiency) and values less than one correspond to greater efficiency.

(a) One-sided or two-sided test Justify (10 marks)

(b) State a sensible null hypothesis (care!) (10 marks)

(c) Test your hypothesis using Student t test and interpret (10 marks)

(d) Interpret the -Inf in the confidence interval reported by R in such a way that a nonstatistician (10 marks)

could understand it

3. Here we consider the amount of data needed to perform hypothesis testing.

(a) Suppose we are testing a coin using observations of tosses. We wish to test H0 : p = 0.5 (20 marks)

against an alternative of HA : p = 0.6 (in this question use one-sided tests only). How

many tosses are needed to guarantee a size α ≤ 0.05 and β ≤ 0.2?

(b) Now generalize to consider HA : p = 0.5+δ. Choose sensible values for δ and quantify (20 marks)

the number of observations needed as a function of δ.