# Probability Assignment

QUESTION 1 Probability Show all calculations/reasoning

Guide to marks: 18 marks - 4 for (a), 5 for (b), 6 for (c), 3 for (d)

(a)  Define what is meant by a probability distribution.

What is the difference between a discrete probability distribution and a continuous probability distribution?

Give an example of your own for each type of distribution.

(b)  Consider the following record of daily sales of a baker’s top selling loaf of bread over the last 100 days:

 NUMBER SOLD NUMBER OF DAYS 0 5 1 15 2 20 3 25 4 20 5 15 Total 100
• What was the probability of selling 3 or 4 loaves on any one day?
• What were the average daily sales over the period?
• What was the probability of selling 2 or more loaves on any one day?
• What was the probability of selling 4 loaves or less on any one day?

(c)  A coin is tossed twice. Calculate the probability of each of the following:

• A head on the first toss.
• A tail on the second toss given a head on the first toss
• Two tails
• A tail on the first toss and a head on the second toss
• A tail on the first and a head on the second or a head on the first and a tail on the second
• At least one head on the first two tosses

(d) The average sales of apples is 5000 with a standard deviation of 600.

• What is the probability that sales will be greater than 5600 apples?
• What is the probability that sales will be less than 5240 apples?
• What is the probability that sales will be less than 4400 apples?

QUESTION 2 Research Question: Constructing data table and calculating probabilities

Guide to marks: 14 - 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.

• Search the Internet for the latest figures you can find on the age and sex of the Australian population.
• 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.
• 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 55 and 64.
• The joint probability that any person selected at random from the population is a female and aged between 15 and 24.
• The probability that any person selected at random from the population is 55 or over.

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

Guide to marks: 18 marks - 4 for a(1), 4 for a(2), 2 for b(1), 3 for b(2), 2 for b(3), 3 for b(4)

(a)  A company wishes to set control limits for monitoring the direct labour time to produce an important product. In the past the mean time has been 20 hours with a standard deviation of 5 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.

(b) Hypothesis testing

Argon Advertising Agency has suggested to Specialty Store that it use mail circulars for the area within a 15 km radius of its store as its major source of advertising. Asked, “Why not extend the area beyond 15 km?” Argon’s representative answered: “Your average customer lives no more than 9 km from the store, so that 15 km will cover almost all your potential customers.”

The owner of Specialty Store was not convinced of the truth of this answer and decided to test the statement that the average customer lives no more than 9 km from the store. A sample of 50 customers was taken, and a mean travelling distance of 10.22 km was found.

Based on the sample results, test the hypothesis that the average customer lives no more than 9 km from the store at a significance level of  = 0.05. Experience shows that  = 5 km. Your answer should respond to the following four requirements:

• Show the null and alternative hypotheses.
• Calculate the critical value.
• Sketch the situation.
• Assess the appropriate conclusion.

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