QBM117 Business Statistics

Assignment 2

Due date: 6 January 2014

Value: 15%

Total 87 marks

Rationale

Assignment 2 is designed to:

demonstrate mastery of the subject material covered in the text, the lectures and the tutorials for the topics Probability, Probability Distributions, Sampling and Sampling Distributions;

detect difficulties with any concepts so the lecturer can provide feedback;

assess learning objectives 1, 3, 5, 6 and 7.

build a base for study in the later topics.

Question 1 (24 marks)

Marking Criteria

Marks will be awarded for:

the correct answers and appropriate detailed working, including the relevant formula;

appropriate well labelled diagrams;

a sentence which answers the questions asked.

a. If ()0.06PCD, P(C) = 0.4 and P(D) = 0.2,

i) find )|(DCP. (3 marks)

ii) find )(DCP. (3 marks)

b. In a drawer there are 30 ribbons, 16 are blue and the rest are red. Two ribbons are selected at random from the drawer, without replacement.

i) Draw a probability tree diagram to represent this problem. Label the end of each branch with the simple events and include the probabilities along each branch of the tree.

(3 marks)

ii) If two ribbons are drawn at random what is the probability of selecting a pair of matching ribbons (two ribbons the same colour)?

(2 marks)

c. Given that Z is the standard normal random variable,

i) find)36.1(ZP (2 marks)

ii) find P( -1.2< Z< 2) (3 marks)

An appropriate, well labelled diagram must be included for each part.

d. The life time of the Tuff brand of tyres is approximately normally distributed, with a mean of 63,000 km and a standard deviation of 3,000 km. The tyres carry a warranty of 55,000 km.

i) What proportion of tyres will fail before the warranty expires?

An appropriate, well labelled diagram must be included

(4 marks)

ii) The Tuff company claims that at least 10% of the tyres last longer than 68,000 km.

Provide evidence to support or disprove this statement.

An appropriate, well labelled diagram must be included.

(4 marks)

Question 2 (14 marks)

Marking Criteria

Marks will be awarded for:

the correct answers and appropriate detailed working, including the relevant formula;

a sentence which answers the questions asked.

a. On average 12 people every half hour check in at Counter A at the Qantas domestic terminal at Terminal 2 at Sydney airport. The people arrive at the counter randomly and independently.

i) Identify the type of probability distribution represented by this problem and write down the value(s) of the parameter(s). (2 marks)

ii) Calculate the probability of more than 13 people checking in at Counter A in the next half hour. (4 marks)

iii) Calculate the probability of exactly 5 people checking in at Counter A in the next 15 minutes. (5 marks)

b. Let X be the number of cars a mechanic repairs on a given day. The distribution of X follows.

No. cars X

2

3

4

5

6

Probability

0.10

0.15

0.35

0.23

0.17

Calculate the mean and standard deviation for this probability distribution .

(3 marks)

Question 3 (16 marks)

Marking Criteria

Marks will be awarded for:

the correct answers and appropriate detailed working including the relevant formula;

the correct use of Excel where appropriate;

a sentence which answers each question asked.

An office supply company conducted a survey before marketing a new paper shredder designed for home use. In the survey, 75% of the people who used the shredder were satisfied with it so the company decided to market it. Assume that 75% of all people who will use the new shredder will be satisfied.

a. Find the probability that for a random sample of 20 customers who purchased the new shredder, exactly 16 of these will be satisfied with it.

(4 marks)

b. Find the probability that for a random sample of 20 customers who purchased the new shredder, less than 14 of these will be satisfied with it.

(4 marks)

c. Find the expected number of dissatisfied customers in a sample of 50 customers who purchased the new shredder.

(2 marks)

d. Find the probability that for a random sample of 100 customers who purchased the new shredder, exactly 75 of these will be satisfied with it.

(3 marks)

e. Find the probability that for a random sample of 150 customers who purchased the new shredder, more than 95 will be satisfied with it.

(3 marks)

Hint: Use the appropriate statistical tables to determine the probabilities in parts a. and b. and use the appropriate Excel statistical function to determine the probabilities in parts d. and e. Include the Excel formula used (in d. and e.) when giving your answer.

Question 4 (11 marks)

Marking Criteria

Marks will be awarded for:

the correct answers and appropriate detailed working including the relevant formula;

appropriate well labelled diagrams;

a sentence which answers each question asked.

A Professor of statistics has noted from past experience that students who do all their assignments and tutorial questions have a 98% chance of passing the final exam, and if they don’t do any of the assignments and tutorial questions they have a 15% chance of passing the final exam. The Professor estimates that 40% of the students do not do all their assignments and tutorial questions.

a. Using letters of the alphabet and appropriate probability notation, define the two simple events described in this problem and their complements (4 definitions altogether)

(2 marks)

b. Draw a probability tree to represent the information given in the question using the letters that you used to define the simple events in part a. Label the end of each branch with the simple events and include the probabilities along each branch of the tree.

(3 marks)

c. What proportion of students fail the final exam?

(3 marks)

d. Given that a student failed the final exam, what is the probability this student did not do all their assignments and tutorial questions?

(3marks)

Question 5 (7 marks)

Marking Criteria

Marks will be awarded for:

the correct answers and appropriate detailed working including the relevant formula;

a sentence which answers each question asked.

a. A switch board operator receives an average of 25 calls every 15 minutes. Calls are received randomly and independently. The switch board operator used the following Excel command to determine a probability 1-POISSON(20, 25, TRUE)

Describe in words or write down using probability notation what probability he worked out in the context of this problem and then use Excel to evaluate it.

(3 marks)

b. An investment analyst collects data on shares. He notes whether dividends were paid on the shares, and whether the shares increased in price over a given period. The data collected is presented in the following table.

Price increase

No price increase

Total

Dividends paid

38

72

110

No dividends paid

85

55

140

Total

123

127

250

i) If a share is selected at random, what is the probability that it both increased in price and paid dividends? (2 marks)

ii) Given that a share has increased in price, what is the probability that it also paid dividends?

(2 marks)

Question 6 (15 marks)

Marking Criteria

Marks will be awarded for:

the correct answers and appropriate detailed working including the relevant formula and appropriate diagrams;

appropriate well labelled diagrams;

a sentence which answers each question asked.

a. The cost of rental for a two-bedroom apartment in a particular suburb is normally distributed with a mean of $2600 per month and a standard deviation of $450.

i) What is the probability that a randomly selected two-bedroom apartment in this suburb will cost less than $2300 per month?

(4 marks)

ii) If monthly rents for a random sample of 9 apartments in this suburb is selected, what is the probability that the mean rent for this sample is greater than $2400?

(5 marks)

iii) Only 5% of apartments will cost more than $x per month. Find x.

(3 marks)

An appropriate, well labelled diagram must be included for each part.

b. A local radio station wants to survey the population in their listening area on a particular issue. They ask people to call in and give their opinion regarding the issue.

From these calls the radio station draws a conclusion stating that the population in their listening area has this belief.

Explain the pitfalls in this type of sampling.

(3 marks