Faculty of Business, Economics and Law

Bachelor of Business

Statistics for Business

ECON862

Trimester 2 2020

ASSIGNMENT 2

45 Marks

Section 1 – Basic Probability Theory

Question 1 (9 Marks)

A disease has a 6% prevalence in the population. You need to choose between two tests.

1) If the disease is present, the first test will give a positive result with probability 0.90. If the disease is not present, it will give a positive result with probability 0.07.

2) If the disease is present, the second test will give a positive result with probability 0.96. If the disease is not present, it will give a positive result with probability 0.10.

a) For each of the tests, what is the probability that someone has the disease given a positive test result?

b) The disease is highly contagious, and so those with the disease need to be quarantined. Which test would you recommend? Explain your reasoning.

Question 2 (9 Marks)

A village fete has a children’s running race each year, run in heats of up to ten children. For each heat the first three contestants past the finishing line qualify for the final. There are three prizes in the final for 1st, 2nd and 3rd places. One year 29 children enter the race so there are three heats, of ten, ten and nine children.

1) What is the probability that three randomly chosen competitors win prizes?

2) What is the probability that two randomly chosen competitors win prizes?

3) How many ways are there to select ten competitors for the first heat?

4) Once the competitors have been selected for the first heat, how many different groups of three qualifiers are possible from this heat?

5) In the final, of nine qualifiers, how many different ways can the three prizes be awarded?

6) What is the probability that of the final nine qualifiers, two of the three prize winners are from the first heat?

Section 2 – Discrete Probability Distributions

Question 3 (7 Marks)

You wish to form an investment portfolio, and have three assets to choose from, a high-risk asset A, and medium-risk asset B, and a zero-risk asset C.

The returns per $1,000 invested for each of these assets in each of three possible economic conditions is give in the table below:

Asset A Asset B Asset C

Recession (P=0.2) -$170 -$40 $30

Normal (P=0.5) $10 $90 $30

Expansion (P=0.3) $210 $140 $30

1) What is the expected return and variance of returns for each asset?

2) What is the expected return and variance of returns for a portfolio that invests $600 in A, $300 in B, and $100 in C?

3) If the economic forecast changes such that there is now a 35% chance of recession, a 50% chance of a normal economy and only a 15% chance of expansion, determine the new expected return for the same investments as in Part 2.

Question 4 (5 Marks)

A telephone receptionist takes an average of six calls per hour. Calculate the probability that the receptionist takes:

a) exactly five calls in the next 45 minutes.

b) two or more calls in the next 30 minutes.

c) from eight to twelve calls inclusive in the next two hours.

Question 5 (5 Marks)

An insurance company has evidence to suggest that 24% of cars are not insured. Calculate the following probabilities:

a) That of the next seven cars involved in accidents, exactly two are not insured.

b) That of the next twelve cars involved in accidents, fewer than three are not insured.

c) That of the fifteen cars involved in accidents, more than twelve are insured.

Section 3 – Continuous Probability Distributions

Question 6 (5 Marks)

A trucking company has worked out that on average its trucks drive 115,000 kilometres a year, with a standard deviation of 22,000 kilometres. The distances driven are normally distributed.

a) What percentage of the trucks will drive between 100,000 and 120,000 kilometres a year?

b) What percentage of trucks will drive less than 60,000 or more than 140,000 kilometres per year?

c) What minimum distance will be driven by at least 75% of the trucks?

Section 4 – Sampling Distributions

Question 7 (5 Marks)

In the first quarter of 2014, the rental cost of a three-bedroom house in a regional town was $300 with a standard deviation of $30. Assume that the rental costs are normally distributed. If you select a random sample of ten rental properties, what is the probability that the sample will have a mean rental cost of:

a) Less than $275?

b) Between $280 and $300?

c) Greater than $310?

~end~

Bachelor of Business

Statistics for Business

ECON862

Trimester 2 2020

ASSIGNMENT 2

45 Marks

Section 1 – Basic Probability Theory

Question 1 (9 Marks)

A disease has a 6% prevalence in the population. You need to choose between two tests.

1) If the disease is present, the first test will give a positive result with probability 0.90. If the disease is not present, it will give a positive result with probability 0.07.

2) If the disease is present, the second test will give a positive result with probability 0.96. If the disease is not present, it will give a positive result with probability 0.10.

a) For each of the tests, what is the probability that someone has the disease given a positive test result?

b) The disease is highly contagious, and so those with the disease need to be quarantined. Which test would you recommend? Explain your reasoning.

Question 2 (9 Marks)

A village fete has a children’s running race each year, run in heats of up to ten children. For each heat the first three contestants past the finishing line qualify for the final. There are three prizes in the final for 1st, 2nd and 3rd places. One year 29 children enter the race so there are three heats, of ten, ten and nine children.

1) What is the probability that three randomly chosen competitors win prizes?

2) What is the probability that two randomly chosen competitors win prizes?

3) How many ways are there to select ten competitors for the first heat?

4) Once the competitors have been selected for the first heat, how many different groups of three qualifiers are possible from this heat?

5) In the final, of nine qualifiers, how many different ways can the three prizes be awarded?

6) What is the probability that of the final nine qualifiers, two of the three prize winners are from the first heat?

Section 2 – Discrete Probability Distributions

Question 3 (7 Marks)

You wish to form an investment portfolio, and have three assets to choose from, a high-risk asset A, and medium-risk asset B, and a zero-risk asset C.

The returns per $1,000 invested for each of these assets in each of three possible economic conditions is give in the table below:

Asset A Asset B Asset C

Recession (P=0.2) -$170 -$40 $30

Normal (P=0.5) $10 $90 $30

Expansion (P=0.3) $210 $140 $30

1) What is the expected return and variance of returns for each asset?

2) What is the expected return and variance of returns for a portfolio that invests $600 in A, $300 in B, and $100 in C?

3) If the economic forecast changes such that there is now a 35% chance of recession, a 50% chance of a normal economy and only a 15% chance of expansion, determine the new expected return for the same investments as in Part 2.

Question 4 (5 Marks)

A telephone receptionist takes an average of six calls per hour. Calculate the probability that the receptionist takes:

a) exactly five calls in the next 45 minutes.

b) two or more calls in the next 30 minutes.

c) from eight to twelve calls inclusive in the next two hours.

Question 5 (5 Marks)

An insurance company has evidence to suggest that 24% of cars are not insured. Calculate the following probabilities:

a) That of the next seven cars involved in accidents, exactly two are not insured.

b) That of the next twelve cars involved in accidents, fewer than three are not insured.

c) That of the fifteen cars involved in accidents, more than twelve are insured.

Section 3 – Continuous Probability Distributions

Question 6 (5 Marks)

A trucking company has worked out that on average its trucks drive 115,000 kilometres a year, with a standard deviation of 22,000 kilometres. The distances driven are normally distributed.

a) What percentage of the trucks will drive between 100,000 and 120,000 kilometres a year?

b) What percentage of trucks will drive less than 60,000 or more than 140,000 kilometres per year?

c) What minimum distance will be driven by at least 75% of the trucks?

Section 4 – Sampling Distributions

Question 7 (5 Marks)

In the first quarter of 2014, the rental cost of a three-bedroom house in a regional town was $300 with a standard deviation of $30. Assume that the rental costs are normally distributed. If you select a random sample of ten rental properties, what is the probability that the sample will have a mean rental cost of:

a) Less than $275?

b) Between $280 and $300?

c) Greater than $310?

~end~

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