Assessment item 1

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Task

QUESTION 1 Probability 
Show all calculations/reasoning

19 marks - 3 for 1(a), 4 for 1(b), 4 for 1(c), 5 for 1(d), 3 for 1(e)

(a) Describe the classical method of determining probability. Give an example.

(b) What is the difference between a discrete probability distribution and a continuous probability distribution? Give your own example of each.

(c )Consider the following record of sales of lottery tickets for the last 50 days.

1.  What was the probability of selling 9 or 10 units on any one day?
2. What were the average daily sales?
3. What was the probability of selling 10 or more?
4. What was the probability of selling 11 or less? 

(d)   A class contains 40 students. 12 are female (F) and Australian citizens (A); 14 are male (M) and Australian citizens) (A); 10 are female and overseas citizens (OS); 4 are male and overseas citizens (OS).

1. Using Excel construct a table showing gender in the rows and citizenship in the columns with total of rows and columns. Then paste this Excel table into Word.
2. Now calculate the probability of a student being:

      (i)   Male
      (ii)  Overseas student
      (iii) Australian given a female
      (iv) Overseas student given a male

(e) The time to complete a construction project is normally distributed with a mean of 50 weeks and a standard deviation 10 weeks.

1. What is the probability that the project will be completed in 54 weeks or less?
2. What is the probability that the project will be completed in 62 weeks or less?
3. What is the probability that the project will take longer than 70 weeks?

QUESTION 2 Research Question, Constructing data table and calculating probabilities

14 marks - 5 for 1, 5 for 2, 4 for 3

The following question involves learning/employing research skills in searching out data on the Internet, presenting it in a well constructed and informative table, and calculating some probabilities showing calculation methods.

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 standard groups, eg, 0-14, 15-24, 25-54, 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.
3. Calculate from the table, showing your calculation methods:

• The probability that any person selected at random from the population is a male.
• The 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 15 and 24.
• The conditional probability that any person selected at random from the population is 25 or over given that the person is a male.

QUESTION 3 Statistical Decision Making and Quality Control 
Show all calculations/reasoning

 17 marks - 4 for a(1), 4 for a(2), 3 for a(3), 6 for (b)

(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 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-bar 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?

(b)
Hypothesis testing
ABC Company has retained a private consulting firm to determine methods that will reduce its inventory costs. The consulting firm assumes that the mean time between placement and receipt of an order is 120 days.

To test this assumption the production manager of ABC randomly selected 100 orders and found that the mean time until delivery was 118.5 days. Past experience indicates that the standard deviation of the lead time (sigma x) is 12 days.

Using an alpha level of 0.05 and a 2-tail test, determine whether there is sufficient evidence that the mean time is not 120 days? Explain your conclusion.

                     END OF ASSIGNMENT 1

Rationale

This assessment task covers topics 1 and 2: Probability concepts and distributions, and Statistical decisionmaking and quality control. It has been designed to ensure that you are engaging with the subject content. More specifically, it seeks to assess your ability to:

• apply probability concepts to decision making
• demonstrate problems solving skills in assessing, organising, summarising and interpreting relevant information for decision making
• demonstrate understanding of the application of statistical hypothesis testing to decisions, with particular emphais on quality control.