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STATS 191 Quantitative Methods for Business

Assignment 2, First Semester 2006

Due: 4pm Wednesday 12th April

Instructions for handing assignment in:

Use (standard) A4 sized paper. Number each page in the top centre and print your name legibly at

the top right hand corner of each page with the surname or family name underlined. Attach aDepartment of Statistics Assignment Cover Sheet to the front of the assignment. Staple or clip all

pages together in the extreme top left hand corner. Fold thepaper lengthwise so that the side of the

cover sheet with your name faces out. Print your name, course number and assignment number on

the outside of the cover sheet. Print your ID number on the inside of the cover sheet. Hand the

assignment in to the appropriate hand-in box.

Assignments handed in to the wrong place or received after the due time will not be marked.

To make your marked assignment easier to find when they are returned, you could draw a

pattern along the edges of the cover sheet using coloured pens or put some sort of small sticker on

the cover sheet.

Notes: Statistics is about summarising, analysing and communicating information. Communication is an

important part of statistics. For this reason you will be expected to write answers which clearly

communicate your thoughts. The mark you receive will be based on your written English as well

as your statistical/technical work.

Assignment 2 will be marked out of 40 marks, 38 marks for the questions as shown below and 2

marks for communication and presentation. (Refer to the Worked Examples under assignment

resources on the Course Resource CD-ROM for examples for examples of how to set out your

assignment answers.) Your final mark will be converted to a mark out of 10 which will be recorded

towards your course work.

This assignment is worth 4% of your final mark. Do not leave it until the last day.

- Attempt questions 1 and 2 when chapter 4 has been covered.

- Attempt question 3 when chapter 6 has been covered.

- Attempt question 4 when chapter 7 has been covered.

- Attempt questions 5 and 6 when chapter 8 has been covered.

We encourage working together. Working together is discussing assignments and methods of

solution with other students or getting help in understanding from staff and tutors. If you work with

other students, you must write up your final assignment individually, in your own words.

We view cheating on assignment work seriously! Cheating is: copying all or part of another

students assignment or allowing another student to copy all or part of your assignment. Studentswho hand in substantially similar work will receive 0 marks for the assignment and, in addition, willalso lose ALL marks from CECIL tests. This means a loss of 8% from the final mark. A student whoallows someone else to copy their work is treated identically to the student who did the copying.Taking a copy of another student's work without their knowledge is theft and will be referred to theUniversity Discipline Committee.

COMPUTER USE IN THIS ASSIGNMENT:

Question 3 and 4 will require use of eitherExcel, SPSS or a graphics calculator for calculatingprobabilities from Normal distributions. Do not hand in any computer output for these questions.

Simply use the computer package (or graphics calculator) to find the solutions. DO NOT USE

TABLES.

Department of Statistics STATS 10x - Assignment 1 Page 1

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Question 1. [5 marks] [Chapter 4]

The New Zealand Department of Labour regularly collects data on employment in NewZealand. The following table is based on the regional labour market summary, July 2003 June2004.

Working Age Population

Region

Not inLabour Force Employed Unemployed Total

Auckland 315700 604100 24200 944000

Wellington 118300 245700 12700 376700

Rest NI 336400 593200 29500 959100Canterbury 142900 301600 12900 457400

Rest SI 124200 236600 9900 370700

Total 1037500 1981200 89200 3107900

Use the table above to answer the following:

(a) What percentage of the New Zealand working age population was in the Auckland region?

(b) Given that a working age person was from Wellington, what is the probability they wereunemployed?

(c) Of working age people who were unemployed, what proportion were from the Wellingtonregion?

(d) What proportion of working age people were in Wellington and unemployed?

(e) What is the probability that a working age person is from Wellington given that they were

unemployed?Question 2. [6 marks] [Chapter 4]

Auditors developing systems to check the accuracy of regular tax returns for such taxes as GSTlook at changes in the firms returns between tax periods. If the change is greater than somethreshold the firms return is tagged to be subject to rigorous audit. To check the accuracy of onesuch system a large sample of returns were all audited. It was found that 23% of returns taggedfor audit by the system revealed tax evasion while only 1 out of 200 returns that were not taggedfor audit by the system revealed tax evasion.

The system was implemented at a tax department and run on a sample of 10,000 tax returns. Of

these, 600 were tagged for audit.

(a) (i) How many of the 600 returns tagged are estimated to be for firms trying to evade tax?

(ii) How many of the remaining returns are estimated to be for firms trying to evade tax?

(b) Use your answer from (a) to help construct a 22 table of counts displaying the results for thissample. Complete the table.

(c) What is the estimated number of tax returns in this sample where the firm is trying to evade tax?

(d) What is the estimated proportion of firms that are trying to evade tax which have their return

tagged for audit?


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Reminder: When calculating Normal probabilities, use SPSS, Excel or a graphics calculator.

Do not use tables. Report any probabilities to 4 decimal places.

A lecturer in a large university was interested in how heavy students backpacks were. A sampleof students was recruited and the weights of their backpacks were recorded. The results were:

Male students backpacks had a mean weight of 5.4 kg and a standard deviation of 2.4 kg.

Female students backpacks had a mean weight of 4.9 kg and a standard deviation of 2.2 kg.

Assume that backpack weights for students in general are Normally distributed with means andstandard deviations as stated above.

(a) What proportion of :

(i) female students carry backpacks weighing more than 10 kg?

(ii) male students carry backpacks weighing less than 4 kg?

(iii) female students carry backpacks weighing between 5 and 7 kg?

(b) Between what two weights do the central 80% of female backpack weights fall?

(c) What weight do only the heaviest 5% of backpacks carried by male students exceed?

(d) State whether each of the statements below is true or false. You must justify each answer using

only z-scores, diagrams of Normal curves and/or the 68-95-99.7 rule. DO NOT calculate theprobabilities. All the marks are for the justifications.

Statement 1: Approximately 68% of male students have backpacks weighing between 2.7 and7.1 kg.

Statement 2: The proportion of male students with backpacks lighter than 4 kg is greater thanthe proportion of female students with backpacks heavier than 7 kg.

Statement 3: About 5% of male students have backpacks heavier than 10.2 kg.

Statement 4: The proportion of male students with backpacks lighter than 3 kg is greater thanthe proportion of male students with backpacks heavier than 9 kg.


Specifications required the nickel content of manufactured stainless steel hydraulic valves to be12.0g. The manufacturing plant is set up so that the nickel content of the valve is randomlydistributed with mean 12.8g and standard deviation 0.5g.

The quality controller of the plant sets up a monitoring process. Once each day, a sample of 48valves is taken and the mean nickel content of the sample is measured.

(a) Can we calculate the probability that one randomly selected valve has less than 12.0g of nickel?If you can, then calculate the probability (showing any working). If you cant calculate it, thenexplain why you cant.

(b) Can we calculate the probability that a randomly selected sample of 48 valves has a mean ofless than 12.6g of nickel? If you can, then calculate the probability (showing any working). Ifyou cant calculate it, then explain why you cant.


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Notes for question 5 and 6:

When calculating confidence intervals by hand, follow the step-by-step guide to calculating a

confidence interval by hand given on page 2 of Chapter 8 of the Lecture Workbook. Show ALL

eight steps of the process.


A statistics student was interested in investigating how long it takes to get a pizza deliveredfrom the local pizzeria. Over a few weeks, a random sample of 9 delivery times (in minutes)were recorded. The data are displayed below:

28.5, 36.7, 42.2, 31.8, 25.9, 30.8, 23.4, 42.2, 37.1, 23.6

(a) What is the sample mean and standard deviation of these 10 observations?

(b) By hand, calculate a 95% confidence interval for the mean delivery time. Interpret your results.


Just before Christmas in 2002, a Herald-DigiPoll was carried out asking about Christmas giftsand Christmas shopping. A sample of 400 New Zealanders was questioned. Some of thequestions with number of responses to each answer are given below:

Question 1: Which one of the followingwould you most like as a surprise gift?

Question 2: Have you given someone a giftthat you had received as a gift?

Answer Male Female Total

Clothing 24 10 34

All respondents: Yes (84) No (316)

Tools 36 13 49 Question 3: Do you like Christmas shopping?

Book 37 41 78

Jewellery 4 56 60Male respondents: Yes (41) No (151)

CD 17 10 27Female respondents: Yes (74) No (134)

Perfume 5 22 27 Question 4: Are you going to give presents tosomeone this Christmas?Cash 39 31 70

Other 30 25 55

Total 192 208 400All respondents: Yes (370) No (30)

(a) State the sampling situation for calculating the standard error of the difference in:

(i) estimating the difference between the proportion of New Zealanders that have given

someone a gift that they had received as a gift and the proportion of New Zealanders thatare going to give presents to someone this Christmas.

(ii) estimating the difference between the proportion of New Zealanders that would most likejewellery as a surprise gift and the proportion of New Zealanders that would most likecash as a surprise gift.

(iii) estimating the difference between the proportion of New Zealand males that would mostlike clothing as a surprise gift and the proportion of New Zealand females that would mostlike clothing as a surprise gift.

(b) By hand, calculate a 95% confidence interval for the difference between the proportion of New

Zealand males that like Christmas shopping and the proportion of New Zealand females thatlike Christmas shopping.


191 a2 s1 2006

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