QUESTION 1 Probability and Statistical Quality Control

Show all calculations/reasoning

(a)

An unbiased coin is tossed twice. Calculate the probability of each of the following:

1. A head on the first toss

2. A tail on the second toss given that the first toss was a head

3. Two tails

4. A tail on the first and a head on the second, or a head on the first and a tail on the second

5. At least one head on the two tosses

(b) 2 marks

Consider the following record of sales for a product for the last 100 days.

SALES UNITS NUMBER OF DAYS

0 15

1 20

2 30

3 30

4 5

100

1. What was the probability of selling 1 or 2 units on any one day? (1/2 mark)

2. What were the average daily sales units? (1/2 mark)

3. What was the probability of selling 3 units or more? (1/2 mark)

4. What was the probability of selling 2 units or less? (1/2 mark)

(c) 3 marks, one for each part

The lifetime of a certain type of colour television picture tube is known to follow a normal distribution with a mean of 4600 hours and a standard deviation of 400 hours.

Calculate the probability that a single randomly chosen tube will last

1. more than 5000 hours

2. less than 4500 hours

3. between 4700 and 4900 hours

(d) 4 marks

A company wishes to set control limits for monitoring the direct labour time to produce an important product. Over the past the mean time has been 20 hours with a standard deviation of 9 hours and is believed to be normally distributed. The company proposes to collect random samples of 36 observations to monitor labour time.

1. If management wishes to establish x ¯ control limits covering the 95% confidence interval, calculate the appropriate UCL and LCL. (1 mark)

2. If management wishes to use smaller samples of 9 observations calculate the control limits covering the 95% confidence interval. (1 mark)

3. Management is considering three alternative procedures in order to maintain tighter control over labour time:

o Sampling more frequently using 9 observations and setting confidence intervals of 80%

o Maintaining 95% confidence intervals and increasing sample size to 64 observations

o Setting 95% confidence intervals and using sample sizes of 100 observations.

Which procedure will provide the narrowest control limits? What are they? (2 marks)

(e) 6 marks (2 for each part)

(a) Search the Internet for the latest figures you can find on the age and sex of the Australian population.

(b) Then using Excel, prepare a table of population numbers (not percentages) by sex (in the columns) and age (in the rows). Break age into about 5 groups, eg, 0-14, 15-24, 24-54, 55-64, 65 and over. Insert total of each row and each column. Paste the table into Word as a picture.

Give the table a title, and below the table quote the source of the figures.

(c) Calculate from the table and explain:

o The marginal probability that any person selected at random from the population is a male.

o The marginal probability that any person selected at random from the population is aged between 25 and 54 (or similar age group if you do not have the same age groupings).

o The joint probability that any person selected at random from the population is a female and aged between 25 and 54 (or similar).

o The conditional probability that any person selected at random from the population is 65 or over given that the person is a male.

QUESTION 2 Decision Analysis

Show all calculations to support your answers. Round all probability calculations to 2 decimal places.

John Carpenter runs a timber company. John is considering an expansion to his product line by manufacturing a new product, garden sheds. He would need to construct either a large new plant to manufacture the sheds, or a small plant. He decides that it is equally likely that the market for this product would be favourable or unfavourable. Given a favourable market he expects a profit of $200,000 if he builds a large plant, or $100,000 from a small plant. An unfavourable market would lead to losses of $180,000 or $20,000 from a large or small plant respectively.

(a) 2 marks

Construct a payoff matrix for John's problem. If John were to follow the EMV criterion, show calculations to indicate what should he do, and why?

(b) 2 marks

What is the expected value of perfect information and explain the reason for such a calculation?

John has the option of conducting a market research survey for a cost of $10,000. He has learned that of all new favourably marketed products, market surveys were positive 70% of the time but falsely predicted negative results 30% of the time. When there was actually an unfavourable market, however, 80% of surveys correctly predicted negative results while 20% of surveys incorrectly predicted positive results.

(c) 4 marks

Using the market research experience, calculate the revised probabilities of a favourable and an unfavourable market for John's product given positive and negative survey predictions.

(d) 4 marks

Based on these revised probabilities what should John do? Support your answer with EVSI and ENGSI calculations.

(e) 8 marks

The decision making literature mostly adopts a rational approach. However, Tversky and Kahneman (T&K) (Reading 3.1) adopt a different approach, arguing that often people use other methods to make decisions, relying on heuristics.

What do they mean by the term heuristics? (2 marks)

Describe three types of heuristics that T&K suggest people use in judgments under uncertainty. (3 marks)

Give one example from your own experience of a bias that might result from each of these heuristics. (3 marks)

QUESTION 3 Simulation

Dr Catscan is an ophthalmologist who, in addition to prescribing glasses and contact lenses, performs laser surgery to correct myopia. This laser surgery is fairly easy and inexpensive to perform.

To inform the public about this procedure Dr Catscan advertises in the local paper and holds information sessions in her office one night a week at which she shows a videotape about the procedure and answers any questions potential patients might have.

The room where these meetings are held can seat 10 people, and reservations are required. The number of people attending each session varies from week to week. Dr Catscan cancels the meeting if 2 or fewer people have made reservations.

Using data from the previous year Dr Catscan determined that reservations follow this pattern:

Number of reservations 0 1 2 3 4 5 6 7 8 9 10

Probability 0.02 0.05 0.08 0.16 0.26 0.18 0.11 0.07 0.05 0.01 0.01

Using the data from last year Dr Catscan determined that 25% of the people who attended information sessions elected to have the surgery performed. Of those who do not, most cite the cost of the procedure ($2,000 per eye, $4,000 in total as almost everyone has both eyes done) as their major concern. The surgery is regarded as cosmetic so that the cost is not covered by Medicare or private hospital insurance funds.

Dr Catscan has now hired you as a consultant to analyse her financial returns from this surgery. In particular, she would like answers to the following questions, which you are going to answer by building an Excel model to simulate 20 weeks of the practice. Random numbers must be generated in Excel and used with the VLOOKUP command to determine the number of reservations,0 and there must be no data in the model itself. The same set of random numbers should be used for all three parts. An IF statement is required for part (a) to determine attendance each week, given cancellation of meetings.

(a) On average, how much revenue does Dr Catscan's practice in laser surgery earn each week? If your simulation shows a fractional number of people electing surgery use such fractional values in determining revenue. Paste your model results into Word including a copy of formulas with row and column headings.

(b)Adjust your model to determine on average, how much revenue would be generated each week if Dr Catscan did not cancel sessions with 2 or fewer reservations? Paste results into Word.

(c) Dr Catscan believes that 35% of people attending the information sessions would have the surgery if she reduced the price to $1,500 per eye or $3,000 in total. Under this scenario how much revenue per week could Dr Catscan expect from laser surgery? Modify your Excel model to answer this and paste results into Word.

(d)Write a brief report with your recommendations to Dr Catscan on the most appropriate strategy.

QUESTION 4 Regression Analysis and Cost Estimation

The CEO of Carson Company has asked you to develop a cost equation to predict monthly overhead costs in the production department. You have collected actual overhead costs for the last 12 months, together with data for three possible cost drivers, number of Indirect Workers, number of Machine Hours worked in the department and the Number of Jobs worked on in each of the last 12 months:

Overhead Costs Indirect Workers Machine Hours Number of Jobs

$2,200 30 350 1,000

4,000 35 500 1,200

3,300 50 250 900

4,400 52 450 1,000

4,200 55 380 1,500

5,400 58 490 1,100

5,600 90 510 1,900

4,300 70 380 1,400

5,300 83 350 1,600

7,500 74 490 1,650

8,000 100 560 1,850

10,000 105 770 1,250

(a) The CEO suggests that he has heard that the high-low method of estimating costs works fairly well and should be inexpensive to use. Write a response to this suggestion for the CEO indicating the advantages and disadvantages of this method. Include the calculation of a cost equation for this data using Machine Hours as the cost driver.

(b)Using Excel develop three scatter diagrams showing overhead costs against each of the three proposed independent variables. Comment on whether these scatter diagrams appear to indicate that linearity is a reasonable assumption for each.

(c) Using the regression module of Excel's Add-in Data Analysis, perform 3 simple regressions by regressing overhead costs against each of the proposed independent variables. Show the output for each regression and evaluate each of the regression results, indicating which of the three is best and why.

Provide the cost equations for those regression results which are satisfactory and from them calculate the predicted overhead in a month where there were 100 Indirect Workers and 500 Machine Hours and 1,000 Jobs worked.

(d)Selecting the two best regressions from part (c) conduct a multiple regression of overhead against these two independent variables. Evaluate the regression results.

Draw conclusions about the best of the four regression results to use.

QUESTION 5 Forecasting

All forecasts are never 100% accurate but subject to error.

11. How is forecast error calculated? (1 mark)

12. Identify and describe three common measures of forecast error. Then illustrate how each is calculated by constructing a 4-period example. (4 marks)

(b) Consider the following table of monthly sales of car tyres by a local company:

Month Unit Sales

January 400

February 500

March 540

April 560

May 600

June ?

(i) Using a 2-month moving average develop forecasts sales for March to June inclusive.

(ii) Using a 2-month weighted moving average, with weights of 2 for the most recent month and 1 for the previous month develop forecasts sales for March to June inclusive.

(iii) The sales manager had predicted sales for January of 400 units. Using exponential smoothing with a weight of 0.3 develop forecasts sales for March to June inclusive.

(iv) Which of the three techniques gives the most accurate forecasts? How do you know?

(c)Describe the four patterns typically found in time series data. What is meant by the expression "decomposition" with regard to forecasting? Briefly describe the process.

Rationale

This assessment task covers topics 1 to 7: Probability concepts and distributions, statistical decision making and quality control, decision analysis under uncertainty and risk, value of information, simulation, correlation and regression analysis, and forecasting techniques. It has been designed to ensure that you are engaging with the subject content on a regular basis. More specifically, it seeks to assess your ability to:

o demonstrate problems solving skills in assessing, organising, summarising and interpreting relevant data for decision making purposes

o apply decision theory to business situations

o use simulation in complex decisions

o demonstrate understanding of statistical hypothesis testing

o use accepted time series forecasting methods

Marking criteria

Assessment Item 1: Marking Guidelines

GRADE REQUIREMENTS

In each of the five questions students must score the marks as shown below to gain the appropriate grade:

PS: At least 10 and less than 13 out of 20 marks.

CR: At least 13 and less than 15 out of 20 marks.

DI: At least 15 and less than 17 out of 20 marks.

HD: At least 17 out of 20 marks.

CRITERION GRADE REQUIREMENTS

Question 1

Parts (a) to (d)

1. Apply probability concepts to decision making To pass students must score at least 7 out of 14 marks.

Part (e)

2. Demonstrate problem solving skills in assessing, organising, summarising and interpreting relevant data for decision making. To pass students must score at least 3 out of 6 marks.

Question 2

Apply decisions theory to business situations. To pass students must score at least 10 out of 20 marks.

Question 3

Use simulation in complex decisions. To pass students must score at least 10 out of 20 marks.

Question 4

1 Apply decisions support tools to management decision making.

2 Apply statistical hypothesis testing to determine significance of regression coefficients in cost estimation. To pass students must score at least 10 out of 20 marks.

Question 5

Use accepted time series forecasting methods. To pass students must score at least 10 out of 20 marks.

Assessment item 2

Final Exam

Value: 70%

Date: To be advised

Duration: 2 hours

Submission method options

N/A - submission not required/applicable

Rationale

This assessment task covers all topics in the subject and has been designed for you to demonstrate your mastery of the subject material and your ability to employ decision support tools in management decision making.

Requirements

The examination consists of:

18. 20 multiple choice questions covering all topics

19. 3 long answer questions covering topics 8, 9, 10 and 11.

All questions must be answered

Marking criteria

In order to pass the exam you must obtain between 50% and 64%.

In part A students are required to answer 20 multiple choice questions designed to assess your knowledge of topics 1-7.

In part B students are required to answer 3 longer questions involving the ability to apply the decision support tools covered in topics 8 to 11.

The exam is designed to assess your ability to apply decision support tools to a variety of business decision situations.

Material provided by the University

1 general purpose answer sheet GPAS-200R

1 answer booklet 12 page.

Material required by the student

Writing implements including a 2B pencil and an eraser

Any calculator allowed, including programmable calculators (hand held, no printer). I-pads and other hand-held computers are not accepted as calculators.

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