Probability: Statistical Decision Making

• QUESTION 1 Probability

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Guide to marks: 20 marks - (a) 3, (b) 3, (c1) 5, (c 2-6) 5 (1 each), (d) 4 (1 each)

(a) Define the term probability. How is it measured? What range can the measures take and what do they mean?

(b) What is meant by the term statistical independence? How can it be identified from a relationship between two variables in a given situation?

(c ) Consider the following record of daily sales of loaves of sourdough bread over the last 100 days. .

(1)Copy the above table into Excel and using formulas complete the missing column figures (note that the 5th and 6th columns refer to cumulative probability distributions) while the last 2 columns contain variance calculations. All cells (except for cols 1 and 2) are to contain formulas so no fudging. Answer the questions below by highlighting the answers in your table, and simply repeating these figures against answers 2 to 6. After answering the questions below paste your Excel model into Word twice, once showing the output and once showing formulas (with row and column headings). Insert the standard deviation below the variance.

(2)What were the average daily sales? Highlight your answer in the spreadsheet and repeat it here.
(3)What was the probability of selling 2 or more loaves on any one day? Highlight your answer in the spreadsheet and repeat it here.
(4)What was the probability of selling 2 or less? Highlight your answer in the spreadsheet and repeat it here.
(5)What is the variance of the distribution? Highlight your answer in the spreadsheet and repeat it here.
(6)What is the standard deviation? Highlight your answer in the spreadsheet and repeat it here.

(d)The average sales of oranges is 4,700 with a standard deviation of 500.

(1)What is the probability that sales will be greater than 5,500 oranges?
(2)What is the probability that sales will be greater than 4,500 oranges?
(3)What is the probability that sales will be less than 4,900 oranges?
(4)What is the probability that sales will be less than 4,300 oranges?

QUESTION 2 Research Question, Constructing data table and calculating probabilities

Guide to marks: 10 marks – (1) 3, (2) 3, (3) 4

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. 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 female.
•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 male and aged between 55 and 64.
•The conditional probability that any person selected at random from the population is aged between 25 and 64 given that the person is a female.

QUESTION 3 Statistical Decision Making and Quality Control
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Guide to marks: 20 marks – (a)10: 3 each for 1,2, and 3, 1 for conclusion, (b) 10: 2 for 1, 3 for 2, 2 for 3, 3 for 4

(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 30 hours with a standard deviation of 10 hours and is believed to be normally distributed. The company proposes to collect random samples of 64 observations to monitor labour time.

• If management wishes to establish x-bar 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.
• Management is considering three alternative procedures in order to maintain tighter control over labour time:
• Sampling more frequently using 16 observations and setting confidence intervals of 90%
• Maintaining 95% confidence intervals and increasing sample size to 64 observations
• Setting 95% confidence intervals and using sample sizes of 36 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
Active Insurance Company’s rates for fire insurance depend on the distance a home is from the nearest fire station. A progressive community claims that the average home in its town is within 5.5 km of the nearest fire station.

Active took a sample of 64 homes, which produced a mean of 5.8 km from the nearest fire station. Is there sufficient evidence to refute the town’s contention that the mean distance is not greater than the claimed 5.5 km if σ (sigma) = 2.4 km? Use α = 0.05.

1.Show the null and alternative hypotheses.
2.Calculate the critical value.
3.Should the town’s claim be accepted or rejected?
4.Sketch the situation.

END OF ASSIGNMENT 1

Rationale

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This assessment task will assess the following learning outcome/s:

• be able to demonstrate problem-solving skills in assessing, organising, summarising and interpreting relevant data for decision making purposes.
• be able to apply decision theory to business situations.
• be able to demonstrate understanding of the application of statistical hypothesis testing to decisions, with particular emphasis on quality control and interpreting the significance of regression coefficients in cost estimation.

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