QUESTION 1 - Probability
Show all calculations/reasoning
(a) What is a random variable? What are the various types of random variable? Describe the difference between them.
(b) What is meant by the term expected value? What does it measure? How is it computed for a discrete probability distribution? Give an example of its calculation.
(c) Consider the following record of daily sales of cars over the last 100 days.
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Sales Units (x)
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Number of days
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p(x)
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Exp Value
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More than
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Less than
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1
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20
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2
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40
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3
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20
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4
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10
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5
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10
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Total
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100
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(1) Copy the above table and complete the missing column figures (note that the last 2 columns refer to cumulative probability distributions). Then answer the following questions highlighting the answers to (3), (4) and (5).
(2) What was the probability of selling 1 or 2 units on any one day?
(3) What were the average daily sales?
(4) What was the probability of selling 3 or more?
(5) What was the probability of selling 4 or less?
(d) The average sales of apples is 5000 with a standard deviation of 500.
(1) What is the probability that sales will be greater than 5500 apples?
(2) What is the probability that sales will be less than 4900 apples?
(3) What is the probability that sales will be less than 4250 apples?
QUESTION 2 - Research Question, Constructing data table and calculating probabilities
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 15 and 24.
- 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
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 10 hours and is believed to be normally distributed. The company proposes to collect random samples of 64 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.
3. 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
Company A has invested a great deal of time and money in occupational safety training for its employees and claims its occupational sick days are now below the national average. The national average was found to be 1.5 occupational sick days. per 100 employees with a standard deviation of 0.3 days.
Company A randomly selected 100 employees for the last year and found the sample had a mean of 1.3 occupational sick days which Company A believed supported their claim.
Using hypothesis testing with an alpha level of 0.05 and a 1-tail test, show the null and alternative hypotheses, sketch the distribution showing mean and critical region, and determine whether Company A's belief is supported.