University of Sydney Business School Discipline of Business Analytics

QBUS5001 Quantitative Methods for Business Individual Assignment (2018 Semester 1)Notes to Students

1. [50 points] You have recently commenced a graduate role at a financial services firm. Your manager has asked you to investigate the performance of two of its biggest rivals; Australia New Zealand Banking Group (ANZ) and the Commonwealth Bank of Australia (CBA). She is also interested in testing your skills as a graduate, and has assigned you a third stock to investigate that she has randomly allocated to you from the Australian stock market (see sheet Q1 of the dataset file). Your analysis will involve monthly adjusted closing prices of all three stocks for the five years from 01 November 2011 to 01 November 2016.

(i) [2 points] Find your allocated stock by typing your student ID (SID) number into the data spreadsheet.IN YOUR OWN WORDS, write a short paragraph (3-4 sentences) describing the company whose stock you have been assigned. Answers that copy sentences from Wikipedia will receive 0 marks.

Using the adjusted closing prices, create a series of percentage monthly returns for each stock. The percentage monthly return for ANZ, for example, can be found using the following formula;

, = 1 0 0 × −   − 1−1

QBUS5001 Quantitative Methods for Business (2018, S1) p.1 of 7

where at is the adjusted closing price of ANZ at time t.

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2. (ii)   [2 points] For each of the three stocks, find the mean percentage return and the standard deviation of the percetage returns. Present this information in a neat table, using 4 decimal places for each value presented (for example, a monthly return of 1.0341% and a standard deviation of 0.2954% are each presented to 4 decimal places).

3. (iii)   [6 points] Create two line charts. The first illustrating the monthly closing prices of each of the three stocks of interest (one line for each stock), the second comparing the monthly percentage returns of each stock. By using your charts only, can you tell which stock has the highest return? Which stock seems to have the highest variance? Explain.

(iv) [4 points] Using a parametric test at the 5% level of significance, test whether the mean monthly percentage return of your allocated stock is positive. Would your answer change if you used a 1% level of significance?

(v) [4 points] Using a parametric test at the 1% level of significance, test whether the mean monthly percentage return of CBA is greater than 0.4%. What is the p-value associated with this test?

(vi) [4 points] Using a parametric test at the 10% level of significance, test whether the mean monthly percentage return of your allocated stock differs from ANZ assuming (a) equal variance and (b) unequal variances. What assumptions have you made by conducting this parametric test?

(vii) [4 points] Test, at the 5% level of significance, whether or not the variances of the monthly percentage return of your allocated stock and ANZ are equal. Hence, comment on the conclusion you make in (vi).

(viii) [6 points] Redo (vi) using non-parametric methods. Here we can make the assumption that the variances are equal.

Consider a new Australian stock called “Gaussian Bank Ltd” (GBL). The monthly returns for GBL are distributed normally with a mean of 0.75% and a standard deviation of 1.1%.

Discipline of Business Analytics

Individual Assignment

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2.

(i)

(ii)

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9. (ix)   [6 points] What is the probability of GBL achieving a negative return in any given month? What is the probability of GBL achieving 2 or more negative monthly returns in a given year?

10. (x)   [4 points] Simulate 60 random monthly returns for GBL. GBL’s share price at 01 November 2011 is $8.50. Use the random percentage returns you’ve created to generate a series of monthly share prices for GBL over the period 01 November 2011 to 01 November 2016. Re-produce the line plots from part (iii) with GBL included.

11. (xi)   [4 Points] Produce a correlation matrix of returns for the four stocks (CBA, ANZ, GBL and your allocated stock). Comment on whether the correlations are as you would expect.

(xii) [4 Points] Minh-Ngoc wants to invest in both CBA and ANZ stocks. What proportion of the portfolio should be invested in CBA such that the resulting portfolio has a minimum variance? What is this minimum variance?

[10 points] In 2016, the Discipline of Business Analytics organised a Python programming workshop. It was facilitated by Dr Marcel Scharth and aimed to improve students’ knowledge of Python for Business Analytics. The students were given a Python test (min. 0 mark, max. 100 marks) both before and after the workshop to assess their improvement. The facilitator randomly chose the scores of 40 attendees from all the attendees to analyse. The facilitator believed that the workshop can improve students’ Python programming skills by at least 10 marks.

[4 points] Do you think that the facilitator’s claim is supported by the data, which is in Sheet Q2 of the dataset file? Use Excel to answer this question and use α = 0.05 .

[6 points] State the assumption that you have made above and test, at the 5% level of significance, whether or not this assumption is correct.

[12 points] A business analyst of a large Australian café chain wanted to compare the weekly sales (in $’000) of the cafés in Sydney, Melbourne, Brisbane and Adelaide. In each city, he randomly selected 20 cafés and their weekly sales amounts were recorded. The data is in Sheet Q3 of the dataset file. To analyse the data, he considered the following one-way ANOVA model

yij =μi +εij , =1,...,4;=1,...,20
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3.

4.

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Use α = 0.05 for all hypothesis tests below.

1. (i)   [4 points] Perform a formal parametric test for the equality of the weekly sales in the four cities.

   State any assumptions required for this test.

2. (ii)   [4 points] Perform a multiple comparison test. State all hypotheses and conclusions to fully explain the output.

3. (iii)   [4 marks] Perform a formal non-parametric test for the equality of the weekly sales in the four cities.

[16 points] The National Broadband Network (NBN) is an Australian national wholesale-only, open- access data network. The NBN project started in 2011 and the Fibre to the Premises (FTTP) rollout is planned to reach approximately 93% of premises in Australia by June 2021. In November 2016, the rollout in The Ponds, a northwest suburb of Sydney was completed. In December 2016, A telco Company, Exetel Pty Ltd, was promoting an NBN connection to their existing ADSL customers in the area. The number of NBN connection online applications received by Exetel on a particular day (24 hours, counting from the mid-night 00:00 to the next midnight) is assumed to follow a Poissondistribution with mean μ = 16.

1. (i)   [4 points] Find the probability that at least 10 applications are received on a day. Verify your answer using MS Excel.

2. (ii)   [4 points] Find the probability that more than 80 applications are received on 5 days using MS Excel. You do not need to calculate the individual probabilities manually.

3. (iii)   [4 points] Find the probability that Exetel does not receive any application in the two hours between 00.00 and 02:00am on a particular night. What additional assumption is required to get this answer?

4. (iv)   [4 points] What is the distribution of the time between arrival of consecutive NBN connection online applications and what are the mean and variance of that distribution?

[10 points] Perform an analysis to solve the case provided below. Note that there are 10 points awarded here, so we are expecting a professional report drawing on various parts of the QBUS5001 course

5.

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material. In particular you should conduct statistical tests, present graphs and consider any assumptions you make. The data is in Sheet Q5 of the dataset file.
Case 17.8 of Textbook:
Most Australians purchase life insurance to help their families in case of unexpected events, such as accidents, that could cost them their lives. From the insurance companies’ point of view, they would like to see their customers live longer so that the chances of their making a pay claim will be smaller. In light of this, life insurance companies are keenly interested in predicting how long their customers will live, because their premiums and profitability will depend on such information. It is a common belief among actuarial researchers that a person’s lifetime is very much related to the lifetime of his or her mother, father and grandparents.

An actuary for an insurance company, however, doesn’t believe that these variables are significantly correlated. He gathered data from 100 recently deceased male customers in order to investigate it further. He recorded the age at death of these male customers and the age at death of the customers’ fathers. Can he conclude that a significant relationship exists between male customers’ age at death and their fathers’ age at death?

6. [20 points] Complete the following tasks:

1. (i)   [5 points] A study is trying to estimate the difference between the annual salaries paid to

   female and male employees working for the same large company. They take a random sample of 60 females and find that their average annual salary is $76 500 with a standard deviation of $1 250. They take a random sample of 65 males and find that their average annual salary is $79 500 with a standard deviation of $2 500. Find and interpret a 95% confidence interval for the difference in annual salaries for female and male employees of this large company.

2. (ii)   [5 points] A marketing researcher wishes to determine the sample size needed to estimate the proportion of mobile phone users who prefer an Apple iPhone. How many mobile phone users should be surveyed if the researcher wants to be within 0.02 with 96% confidence?

3. (iii)   [5 points] The plant manager of a company that manufactures office equipment believes that worker productivity is a function of the design of the job. Two designs are being considered for the production of a new type of ergonomic computer desk. Ten workers are randomly selected to assemble the desk using Design-One, and another fifteen workers are randomly selected to assemble the desk using Design-Two. Assume that the assembly times for Design-

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One and Design-Two follow two normal population, respectively. Some of the summary statistics are given below:

10, 40.

15, 20.
Is there enough evidence at the 5% significance level to infer that the variance of assembly time for Design-One is larger than the variance of the assembly time for Design-Two?

(iv) [5 points] A statistician investigating the relationship between the amount of precipitation (in inches) and the number of car accidents gathered data for 10 randomly selected days. The results are presented below. Calculate the Spearman rank correlation coefficient, and test to determine at the 5% significance level whether we can infer that a linear relationship exists between the number of accidents and the amount of precipitation.

7. [20 points] A production line that assembles computer keyboards has been experiencing problems since a new process was introduced. The supervisor notes that there has been an increase in defective units and occasional backlogs when one station’s productivity is not matched by that of the other stations. Upon reviewing the process, the supervisor discovers that the management scientists who developed the production process assumed that the amount of time to complete a critical part of the process is normally distributed with a mean of 1100 seconds and a SD of 150 seconds. He is satisfied with the normality assumption and the value of the SD but he is unsu̅re about the mean time. In order to examine

the problem, he measures the times for 20̅assemblies. Let denote the mean of these times.
(i) [3 points] For which values of can the supervisor conclude at the 10% level of significance

that the assumption of mean assembly time of 1050 is incorrect?
(ii) [4 points] Calculate , the probability that the supervisor makes a Type II error. Use the

assumption that =1100.
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3. (iii)   [2 points] Calculate the power of the test.

4. (iv)   [3 points] Interpret the meaning of the power in the previous question.

5. (v)   [4 points] Recalculate if n is increased from 20 to 40. What is the effect of increasing the sample size on the value of ?

6. (vi)   [4 points] Following question (v), recalculate if α is lowered from 0.10 to 0.05. What is the effect of decreasing the significance level on the value of ?

The end of Questions