Business Statistics

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 ( ) 0.06 P C D ? ? , P(C) = 0.4 and P(D) = 0.2,

i) find ) | ( D C P .

(3 marks)

ii) find ) ( D C P ? .

(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 ( ? Z P (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)

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 ( ) 0.06 P C D ? ? , P(C) = 0.4 and P(D) = 0.2,

i) find ) | ( D C P .

(3 marks)

ii) find ) ( D C P ? .

(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 ( ? Z P (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)