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ECON20003 – QUANTITATIVE METHODS 2

First Semester, 2023

Assignment 1

Due date and time: Monday 27 March, 10:00AM

Please read the following instructions carefully before starting to work on the assignment.

• There is a total of 43 marks for this assignment. It is worth 5% of the final grade for QM2.

• This assignment must be submitted online via the LMS by 10:00AM on Monday 27 March. Any assignment not submitted by this due date and time will be given a mark of zero.

• Students may work alone and submit their own assignment answers, if they wish to do so, or they can work on the assignment in pairs. In the latter case, each assignment pair must submit only one set of assignment answers and both students of the pair will receive the same mark for their assignment. You are not required to be in the same tutorial to form a pair. It is not allowed to form assignment groups of more than two students.

• Please note that the assignment submission process has two stages:

1. Registering your assignment group (only if you work in a pair), and

2. Submitting the assignment online via the LMS.

Students who intend to work on the assignment in pairs must register their groups. To do so, click the “People” link and then the Groups tab in the Canvas course navigation menu. The group names (set by default) are A1 1, A1 2, A1 3 etc. Every assignment pair MUST register as one of these created groups for submitting the assignment and not create a new group. The deadline for registering your group is 5:00PM on Friday 17 March. If a pair fails to register their group before the deadline for group registration, both students will need to make an individual, i.e., sufficiently different, submission.

Students making an individual submission do not need to register.

• Answer the assignment questions using Microsoft Word. Make sure to include a cover page in the document with the student ID, the name, and the tutorial group of each group member.

• If a task involves some manual calculations, use a calculator (not R, Excel, or any other software), the relevant statistical table(s), and show the major steps, including the formulas in the document. If a task involves using R / RStudio, paste your relevant scripts, screenshots, and printouts (graphs, output tables etc.) in the document.

• Convert the whole Microsoft Word file to PDF before submitting it online via the LMS.

• Do not forget to preview your assignment after uploading on the LMS to ensure that you have indeed uploaded the correct and complete assignment, and that its formatting is in order as in their original document. Submissions that are late because of formatting issues or because a version is incomplete, will not be accepted.

Assignment Tasks and Questions

Exercise 1 (4 marks = 1 + 1 + 1 + 1)

A large population is not normally distributed with a mean and a variance of 4 and 16, respectively. Consider the sampling distribution of the sample mean based on simple random samples of size 100. Answer the following questions by performing all required calculations manually.

(a)  (1 mark)

What can you tell about the shape of this sampling distribution? Explain why this is the case.

(b)  (1 mark)

What are the mean and the standard deviation of this sampling distribution?

(c)  (1 mark)

What is the probability that the mean of a single sample is at most 5.5?

(d)  (1 mark)

What proportion of the sample means are between 3 and 5?

(d)  (6 marks)

Perform an appropriate test at the 5% significance level on the sample data to see whether  the  new  shipping  method  is  a  success.  Specify  meaningful  null  and alternative  hypotheses,  calculate  the  observed  test  statistic,  make  a  statistical decision based on the p-value, and draw an appropriate conclusion (following the 6 steps of hypothesis testing mentioned in lectures).

(e)  (6 marks)

What conditions are required to validate the confidence interval and the test in parts (c) and (d)? Are they likely satisfied? Use as much evidence as you can to support your answers.

Exercise 3 (16 marks = 2 + 6 + 8)

A coffee shop owner wants to determine if the median customer satisfaction score for their coffee is higher than the industry median of 80 on a scale of 1 to 100. The owner randomly selects 100 customers and asks them to rate the coffee on this satisfaction scale. These scores  are  available  in the a1e3.xlsx Excel file.  Perform all  required calculations and tasks with RStudio / R.

(a)  (2 marks)

Granted that the  required  conditions are satisfied, which tests can  be  used to answer  the research question? Explain your answer.

(b)  (6 marks)

Perform the tests you nominated in part (a) at the 1% significance level. For each test specify the null and alternative hypotheses, the observed test statistic, make a statistical decision based on the p-value, and draw an appropriate conclusion (following the 6 steps of hypothesis testing mentioned in lectures).

(c)  (8 marks)

What conditions are required to validate the tests you performed in part (b)? Are they likely satisfied this time? Which test do you think is the best this time? Explain your answers and provide as much evidence as you can.