A market survey found that 90% of the individuals questioned owned a colour televisions set. The survey also found that 70% owned a stereo of some kind. Of those who own a colour TV set 68% own a stereo.

1. What is the probability that a randomly selected individual will own both a colour television and a stereo set?

2. What is the probability that a randomly selected individual will own a colour television or a stereo set?

(b) Consider the following record of sales for a product for the last 100 days.
Sales Units Number of Days
0 4
1 20
2 30
3 40
4 6
100

1. What was the probability of selling 1 unit on any one day?

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

3. What were the average daily sales units?

4. What was the probability of selling 1 unit or more?

5. What was the probability of selling 3 units or less?

(c) The lifetime of a certain type of colour television picture tube is known to follow a normal distribution with a mean of 2400 hours and a standard deviation of 200 hours.

Calculate the probability that a single randomly chosen tube will last
1. more than 2500 hours
2. less than 2250 hours
3. between 2350 and 2450 hours
4. less than 2600 hours
5. more than 2225 hours

(d) 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 ¯ control limits covering the 95% confidence interval, calculate the appropriate UCL and LCL.

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

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.

Calculate the control limits for each of the 3 alternatives.

Which procedure will provide the narrowest control limits? What are they?

QUESTION 2 Decision Analysis

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

A member of a group of medical practitioners is considering opening his own private medical practice. He estimates that if demand for his services is high he could realise a profit of $500,000. If demand is low he could lose $200,000.

(a) If the medical practitioner follows the criterion of regret what should he do? Construct a regret matrix to derive your answer.

(b) His best guess is that there is a 50-50 chance that the practice would be successful.

What should he do if he follows the EMV criterion? Show calculation.

(c) Calculate the expected value of perfect information?

(d) A market research firm offers to perform a study of the market for a fee of $25,000. Their past experience enables them to make the following claims:

There is a 90% chance that they would successfully predict a favourable market (ie demand would be high) and an 80% chance they would successfully predict an unfavourable market.

Using the market research experience, calculate the revised probabilities of demand given predictions of a favourable market and an unfavourable market.

(e) Based on these revised probabilities what should the medical practitioner do? Support your answer with EVSI and ENGSI calculations.

(f) Tversky and Kahneman describe three types of heuristics that people use in judgments under uncertainty. What do they mean by the term heuristics? Briefly describe the ones that they mention. Give one example from your own experience of a bias that might result from each of these heuristics.

QUESTION 3 Regression Analysis and Cost Estimation

The CEO of Milton Manufacturing Company has asked you to develop a cost equation to predict monthly overhead costs in its production department. You have collected the following data for the last 10 months: Overhead costs (OH$) and the proposed independent variables: Number of machine hours worked (MH), number of direct labour hours (DLH) and number of indirect labour workers (IL Workers).
OH ($) MH DLH IL Workers
2,000 9,500 1,800 3
4,500 20,000 4,200 8
3,000 14,000 2,500 15
2,700 13,000 2,400 10
6,000 28,000 5,000 16
5,100 25,000 4,800 12
8,000 42,000 8,100 6
4,800 25,000 4,500 8
7,500 35,000 6,900 14
6,500 32,000 6,000 11

(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, including the calculation of a cost equation for this data using MH as the cost driver.

(b) Using Excel develop three scatter diagrams showing overhead costs against each of the proposed independent variables. Comment on whether these scatter diagrams 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 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 10,000 MH and 3,000 DLH 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.

If there should be a problem identify the potential cause and test for confirmation.

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

QUESTION 4 Forecasting

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

1 How is forecast error calculated?

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

(b) Consider the following table of monthly sales of car tyres by a local company:
Month Unit Sales
Jan 400
Feb 500
Mar 540
Apr 560
May 600
Jun ?

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.