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QUESTION 1 Probability
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(a) What are the two basic laws of probability?
(b) What is a random variable? What are the various types of random variables?
(c) Consider the following record of sales for a product for the last 100 days.
SALES UNITS NUMBER OF DAYS
0 10
1 25
2 30
3 25
4 10
100
1. What was the probability of selling 2 or 3 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?

(d) An urn contains 20 marbles. 6 are red, 10 are green and 4 are blue. Marbles are drawn and then replaced after each draw.
Calculate the probability of drawing:
(i) a blue marble on the first draw
(ii) a blue marble on the first draw and a red on the second draw
(iii) two green marbles in two draws
(iv) red marble on the second draw given a blue on the first draw
(e) The time to complete a construction project is normally distributed with a mean of 80 weeks and a standard deviation 8 weeks.
(i) What is the probability that the project will be completed in 84 weeks or less?
(ii) What is the probability that the project will be completed in 92 weeks or less?
(iii) What is the probability that the project will take longer than 90 weeks?
(f) 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, e.g. 0-14, 15-24, 15-54, 55-64, 65 and over. Insert total of each row and each column. Past 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 male.
• The marginal probability that any person selected at random from the population is aged between 25 and 54.
• The joint probability that any person selected at random from the population is a female and aged between 55 and 64.
• The conditional probability that any person selected at random from the population is 25 or over given that the person is a male.
QUESTION 2 Statistical Decision Making and Quality Control
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(a) 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 45 minutes with a standard deviation of 15 minutes 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 16 observations calculate the control limits covering the 95% confidence interval. (Round calculations to 2 decimal places)
(b) Quality control chart
Using the Excel Chart function prepare a quality control chart from the following sequential monthly cycle time data. The desired level of performance is 140, with an upper control limit of 160 and a lower control limit of 120.

Month Average Cycle Time
(Hours)
January 124
February 175
March 147
April 97
May 153
June 132
July 228
August 152
September 172
October 112
November 138
December 141
Explain whether the process is in control or whether corrective action is required.
Justify your conclusion.