Probability and Statistical Quality Control
(a)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?
(b)The lifetime of a certain type of colour television picture tube is known to follow a normal distribution with a mean of 5000 hours and a standard deviation of 500 hours.
Calculate the probability that a single randomly chosen tube will last
- more than 6000 hours
- less than 4500 hours
- between 4800 and 5250 hours
(c)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 30 hours with a standard deviation of 12 hours and is believed to be normally distributed. The company proposes to collect random samples of 36 observations to monitor labour time.
- If management wishes to establish x ̅ control limits covering the 95% confidence interval, calculate the appropriate UCL and LCL.
- If management wishes to use smaller samples of 16 observations calculate the control limits covering the 95% confidence interval.
(d)Below are 15 weeks’ observations of average service time for cleaning motel rooms at the Sleep-In Motel:
1.Enter this data in an Excel Worksheet in columns A and B. Use this historical data to calculate the mean and standard deviation of the average cleaning times over the 15 week period. You can use Excel to do this, Data/Data Analysis/Descriptive Statistics. You must have the Add-in Analysis ToolPak to access Data Analysis – if Data Analysis is not available, add it in (File/Options/Add-ins/Go/Analysis ToolPak)
2.Add three more columns to your data: Col C Mean (from output in (1) and upper and lower control limits in Col D UCL and Col E LCL. Calculate 2-sigma limits for the UCL and LCL using the standard deviation (from output in (1). The figures in each of these 3 columns should be the same for every week.
3.Prepare a statistical control chart by plotting the weekly average cleaning times over the 15 weeks, and showing the mean and upper and lower control limits of 2 standard deviations. You can use the Excel Chart Function (Insert Line Chart) to do this. Make sure you highlight all the data including the headings before the chart command.
4.Explain whether the process is in control or whether corrective action is required. Justify your conclusion.
(e)1.Search the Internet for the latest figures you can find on the age and sex of the Australian population.
2.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 column. Paste the table into Word as a picture.
Give the table a title, and below the table quote the source of the figures.
3.Calculate from the table, showing your calculation methods:
- The marginal probability that any person selected at random from the population is a female.
- The marginal probability that any person selected at random from the population is aged between 15 and 24 (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 male 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 female.
A vendor of ice cream buckets at the Sydney Cricket Ground has determined that sales at one-day cricket matches are a function of the maximum daily temperature:
The ice creams sell for $8.00 each and cost $4.00. Unsold ice creams will be worthless because the vendor has no way of storing them until the next match.
(a) Prepare a 5×5 payoff matrix showing conditional profits
(b) If the vendor were a pessimist how many ice creams would he order?
(c) If the vendor were an optimist how many ice creams would he order?
(d) If the vendor followed the criterion of regret how many ice creams would he order?
(e) Using the Laplace criterion how many ice creams would he order?
(f) Based on meteorological information the vendor estimates the following probability distribution for temperature at a coming match:
If the vendor based his decision on maximum expected value how many ice creams would he order?
The tasks in this question are similar to those tasks that would be carried out in the workplace and use similar data.
The State Opera Theatre gains significant revenue from ticket sales at each opera performed during the season. The sale of souvenir programs for all performances of each opera also adds to profitability. Each program costs $1.60 to produce and sells for $4.00. Any programs unsold at the end of any opera are donated to a recycling centre and do not produce any revenue.
Records of the programs sold for each opera show the following:
(a) Your manager has asked you as the management accountant to determine the profitability of the souvenir program production. In particular you have been asked to investigate the strategy where the number of programs to be printed should equal the number demanded at the previous opera. You decide to use Excel to simulate the sale of programs at 10 operas in a season together with the profit or loss on programs for each opera. You will have to generate a dummy sale for a “previous” opera (Opera 0) to begin with so that you have a starting point for Opera 1. Include a calculation of the total profit/loss for the season and the average profit/loss per opera.
Hints: Your model should have 8 columns: Opera #, RN Demand, Demand, Production #, Sales Units, Sales Revenue, Production Costs, Profit/Loss. The model must be completely formula driven – there must be no data in the model or the model formulas – all data should be in a data input section above the model. An IF or MIN function is required in the formulas in the Sales Units column. After completing the model you can vary the results by pressing F9 (recalculate) a number of times to view such variations resulting from changes in the random numbers generated.
Show the data and the model in two printouts: (1) the results, and (2) the formulas. Both printouts must show row and column numbers and be copied from Excel into Word. See Spreadsheet Advice in Interact Resources for guidance.
(b)The manager has also asked you to compare the strategy of printing the number of programs demanded at the previous opera [results already calculated in (a)] with an alternative strategy of printing 3000 programs for each opera. Calculate the profit or loss in each of the 10 operas and the average profits for the whole season when adopting this second strategy. You can do this by copying your simulation model in (a) into a second worksheet and adjusting the model where necessary to incorporate any additional or different calculations. You may generate different random numbers for (b) but press F9 a number of times to see the comparisons between (a) and (b).
When you are satisfied with the results show the data and the model in two printouts: (1) the results, and (2) the formulas. Both printouts must show row and column numbers and be copied from Excel into Word.
(c)Write a report to your manager explaining which of the two production strategies [the one in (a) or the one in (b)] you would recommend and why. You may support your report by including reference to the profits under each strategy, noting any limitations to the methods you have adopted to analyse the situations.
The report must be dated, addressed to the Manager and signed off by you.
This assessment task covers topics 1 to 5: Probability concepts and distributions, statistical decision making and quality control, decision analysis under uncertainty and risk, value of information, and simulation. 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:
- apply probability concepts to decision-making
- demonstrate an understanding of statistical hypothesis testing in quality control
- 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