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)

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

1. What was the probability of selling 1 or 2 units on any one day?
2. What were the average daily sales units?
3. What was the probability of selling 3 units or more?
4. What was the probability of selling 2 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 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)

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.
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.

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

(e)

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

(e) 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?

Describe three types of heuristicsthat T&K suggest people use in judgments under uncertainty.

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

Attachment:- Data.xls