statistics-probability-and-statistical-quality

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Assessment item 1

Assignment questions

Value:30%

Due date:06-May-2014

Return date:27-May-2014

Submission method
options

Hand delivery (option applies to Internal only)
Alternative submission method

Task

QUESTION
1 Probability and Statistical Quality Control 20 marks
Show all calculations/reasoning

(a)

5 marks, one for each part

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:
  • Sampling more frequently using
    9 observations and setting confidence intervals of 80%
  • Maintaining 95% confidence
    intervals and increasing sample size to 64 observations
  • 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:

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

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 20 marks

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)
10 marks
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) 3 marks
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)
3 marks
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)
4 marks

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

QUESTION
4 Regression Analysis and Cost Estimation 20 marks

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)
5 marks
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)
5 marks
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)
5 marks
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)
5 marks
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 20 marks

(a)
5 marks

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

  1. How is forecast error
    calculated? (1 mark)
  2. Identify and describe three
    common measures of forecast error. Then illustrate how each is calculated
    by constructing a 4-period example. (4 marks)

(b) 10 marks as indicated below

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)
3 marks

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

(ii)
3 marks

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)
3 marks

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)
1 mark

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

(c)
5 marks

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:

  • demonstrate problems solving
    skills in assessing, organising, summarising and interpreting relevant
    data for decision making purposes
  • apply decision theory to
    business situations
  • use simulation in complex
    decisions
  • demonstrate understanding of
    statistical hypothesis testing
  • 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.

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