Get questions and answers for Business Statistics

The Sally Coffee Shop is considering closing one hour earlier in the evening through the week. Sally obtained the following data from a study taken last week of the number of customers between the hours of 10 and 11 p.m., and the dollar value spent.

a. Draw a scatter diagram.
b. Determine the regression equation.
c. Determine the value of y′ when x is 7

The Simone Company is considering a new computer program to speed up the processing of its inventory. Two programs are being considered: Program A and Program B. To test the programs, 23 employees were divided into two groups of 12 and 11. The mean processing time of the 12 employees for Program A was 35 seconds, with a standard deviation of 8 seconds. The mean processing time of the 11 employees for Program B was 34 seconds, with a standard deviation of 12 seconds. Is there a difference in the variation of the two programs? Test at the 0.10 significance level.

There are four auto body shops in Shell River, Manitoba, all of which claim to serve customers promptly. To check if there is any difference in service, customers are randomly selected from each repair shop and their waiting times in days are recorded. The output from a statistical software package is as follows:

Is there evidence to suggest a difference in the mean waiting times at the four body shops? Use the 0.05 significance level.

Jack Wolicki is a production supervisor in an industrial manufacturing company. He would like to determine whether there is more variation in the number of units produced on the afternoon shift than on the day shift. A sample of 15 day-shift workers showed that the mean number of units produced was 200 with a standard deviation of 19.6. A sample of 10 afternoon-shift workers showed that the mean number of units produced was 205 with a standard deviation of 23.8. At the 0.05 significance level, is there more variation in the number of units produced on the afternoon shift?

A physician who specializes in weight control has three different diets she recommends. As an experiment, she randomly selected 15 patients and then assigned 5 to each diet. After three months the following weight losses, in kilograms, were noted. At the 0.05 significance level, can she conclude that there is a difference in the mean amount of weight loss among the three diets?

A real estate agent in Northern Ontario wants to compare the variation in the selling price of homes on waterfronts with those one to three blocks from the water. A sample of 21 waterfront homes sold within the last year revealed the standard deviation of the selling prices was $45 600. A sample of 18 homes, also sold within the last year, that were one to three blocks from the water, revealed that the standard deviation was $21 330. At the 0.01 significance level, can we conclude that there is more variation in the selling prices of the waterfront homes?

What is the critical F-value for a sample of six observations in the numerator and four in the denominator? Use a two-tailed test and the 0.10 significance level.

A company is researching the effectiveness of a new design to decrease the time to access a website. Ten website users were randomly selected—five accessed websites using the old web design, and the other five accessed websites using the new web design, and their times (in seconds) to access the website with the old and new designs were recorded. The results are as follows:

Can the company conclude that the new web design allows users to access websites faster than with the old design? Test at the 0.05 level.

After a losing season, there is a great uproar to fire the head basketball coach. In a random sample of 200 college students, 80 favour keeping the coach. Test at the 0.05 level of significance whether the proportion of students who support the coach is less than 50%.

Differentiate between data and information.

For each of the following companies, give examples of data that could be gathered and what purpose these data would serve: Bluescope Steel, AAMI, Jetstar, IKEA, Telstra, ANZ Bank, Sydney City Council, and Black and White Cabs.

Classify each of the following as nominal, ordinal, discrete or continuous data.

(a) The RBA interest rate

(b) The return from government bonds

(c) The customer satisfaction ranking in a survey of a telecommunications company

(d) The ASX 200 index

(e) The number of tourists arriving in Australia each month

(f) The airline a tourist flies with into Australia

(g) The time to serve a customer in a caf´e queue

Cricket Australia wants to run a marketing campaign to increase attendance at test matches. You have been hired as a consultant to conduct a survey and prepare a report on your findings.

(a) What variables do you consider affect a person’s interest in cricket test matches?

(b) Design a questionnaire of 10 to 15 questions that will enable you to decide which section of the population the marketing campaign should target.

Compile a list of every piece of data you have provided to business organisations over the past 24 hours (choose a shorter or longer time if necessary). Do you think these data are valuable to those organisations? Why/why not?

Define machine learning in the context of data mining.

For the following data, construct a frequency distribution table with six classes.

The following data are the average monthly after tax salary in 2020 for 162 countries.

a. Construct a frequency distribution for the data using five class     intervals.

b. Construct a frequency distribution for the data using 10 class     intervals.

c. Examine the results of (a) and (b) and comment on the usefulness of the frequency distributions in summarising these data.

Data were collected on the number of passengers at each train station in Melbourne. The numbers for the weekday peak time, 7 am to 9:29 am, is given below. Construct a frequency distribution for these data. What does the frequency distribution reveal about train usage in Melbourne?

A company manufactures a metal ring, which usually weighs about 1.4 kg, for industrial engines. A random sample of 50 of these metal rings produced the following weights (in kg).

Construct a frequency distribution for these data using eight classes. What can you observe about the data from the frequency distribution?

In a medium-sized New Zealand city, 90 houses are for sale, each with about 180 m² of floor space. The asking prices vary. The frequency distribution shown contains the price categories for the 90 houses. Construct a histogram, a frequency polygon and an ogive from these data.

Asking price ($000) ............... Frequency
(120, 130] .................................... 21
(130, 140] .................................... 27
(140, 150] .................................... 18
(150, 160] .................................... 11
(160, 170] ...................................... 6
(170, 180] ...................................... 3
(180, 190] ...................................... 4

The following figures are type and corresponding proportion of expense required to create a new processed food product, ready to introduce to the market. Produce a pie chart and a Pareto chart for these data and comment on your findings.

Expense Proportion .......................... (%)
Technical support ................................ 8
Project management .......................... 5
Administrative support ...................... 4
Other overhead ................................... 2
Development ..................................... 56
Research ............................................ 25

Produce a pie chart and a Pareto chart for these data and comment on your findings.

The table below shows Department of Foreign Affairs and Trade data for the value (in A$ million) of Australian exports to the top 10 buyers of Australian goods. Construct a pie chart to represent these data. Label the slices with the corresponding percentages and give the chart an appropriate title. Comment on the effectiveness of using a pie chart to display these data.

Export market ......................... A$ million
China 94 ........................................... 655
Japan 47 ........................................... 501
South Korea 19 ............................... 610
USA 9 ............................................... 580
India 9 ............................................. 517
New Zealand 7 ............................... 399
Singapore 5 .................................... 659
Taiwan 7 ......................................... 356
UK 3 ................................................ 859
Malaysia 5 ...................................... 561

The table below shows the weekly expenses for a family of four. Construct a pie chart displaying his information.

Item Expense ..................................................................... ($)
Mortgage repayments .................................................... 450
Other housing costs ......................................................... 15
Education ......................................................................... 100
Cars .................................................................................. 125
Groceries ......................................................................... 250
Dining out ........................................................................ 100
Sport, recreation and other entertainment .................. 55

The table below shows Department of Foreign Affairs and Trade data for the value (in A$ million) for the top 10 import sources of goods into Australia. Produce a bar chart for these data. On the basis of the charts for this and the previous problem, comment on Australia’s major business partners.

Import source ............................................... A$ million
China 49 .............................................................. 329
USA 39 ................................................................. 181
Japan 21 .............................................................. 221

Singapore 17 ...................................................... 878
Thailand 13 ......................................................... 832
Germany 13 ....................................................... 099
UK 12 .................................................................. 044
Malaysia 10 ........................................................ 944
South Korea 10 .................................................. 813
New Zealand 10 ................................................. 532

The Bureau of Infrastructure, Transport and Regional Economics publishes transport statistics. The table below shows the number of sectors flown by each domestic airline. Construct a pie chart for these data and comment on your findings.

Airline .............................. Number of sectors flown
Jetstar 7 .................................................... 315
Qantas 9 ................................................... 520
QantasLink 10 .......................................... 243
Regional Express 5 .................................. 715
Tiger Air 1 ................................................. 872
Virgin Australia 10 ................................... 850
Virgin Australia Regional Airlines 2 ....... 533

Sometimes when we create a stem-and-leaf plot, we get too many leaves per stem to give a good representation of the data. In such a case, we can further split the data space for each stem equally. Suppose 100 CPA firms are surveyed to determine how many audits they perform over a certain time. In this case, the best representation of the data is given by splitting each stem into five ranges. This results in the stem-and-leaf plot shown. The first stem, 1, has been split into five, the first of which contains data in the range 10 to 11 inclusive (and there are no data points in this range), the second in the range 12 to 13, the third in the range 14 to 15, the fourth in the range 16 to 17 and the fifth in the range 18 to 19. Similarly, the other stems have been divided into five ranges. This method is similar to how we define intervals for histograms. What can you learn from this plot about the number of audits being performed by these firms?

A researcher wants to determine if the number of international students studying in New Zealand and the number of visitors from their corresponding countries are related. Data on these variables for eight Asian countries are shown in the following table.

Construct a scatterplot of the data. Examine the plot and discuss the strength of the relationship between the number of international students and the number of visitors from their countries.

The following data are monthly downloads (megabytes) on handheld devices. Construct a stemand- leaf plot of the data using the whole part for the stem and the decimal part for the leaf. What does this plot tell you about monthly downloads on handheld devices?

Describe the advantages of multidimensional visualisation over univariate visualisation.

What is ‘chart junk’? Why do you think it arises?

What visualisation features can be used to represent:
(a) categorical information
(b) numerical information
(c) temporal information?

What is the advantage of using multiple panels in a visualisation?

How does a scatterplot matrix differ from a basic scatterplot?

What is rescaling?

What distinguishes an interactive visualisation from a static visualisation?

Do you agree with the quote ‘A picture is worth a thousand words. An interface is worth a thousand pictures.’ Why? Why not?

How useful do you think the ability to zoom in on multiple linked graphics is for data analysis? Is it a powerful tool for exploring data and understanding relationships, or is it overcomplicated?

An NRMA report states that the average age of a car in Australia is 10.5 years. Suppose the distribution of ages of cars on Australian roads is approximately bell-shaped. If the standard deviation is 2.4 years, between what two values would 95% of the car ages fall?

According to the Australian Taxation Office, the average taxable income in an affluent suburb of Sydney is $94720. Suppose the median taxable income in this suburb is $90050 and the mode is $89200. Is the distribution in this area skewed? If so, how? Which of these measures of central tendency would you use to describe these data? Why?

A hire car company is interested in summary statistics that are useful in describing travel times between the CBD and the domestic terminal at Sydney Airport. The company locates a report that indicates that the average total time for travel by car is 14 minutes. The shape of the distribution of travel times is unknown, but in addition it is reported that 35% of travel times are between 10.5 and 17.5 minutes. Use Chebyshev’s theorem to determine the value of the standard deviation associated with travel times.

A survey of drivers asked respondents to list the age in years of the vehicle that they usually drive. The following data represent a sample of 18 responses provided. Use these data to construct a box-and-whisker plot. List the median, Q₁, Q₃, the endpoints of the inner fences and the endpoints of the outer fences. Are any outliers present in the data?

An online retailer that sells board games and puzzles has produced summary measures, shown in the right-hand column, describing the cost charged to consumers for shipping. Write a short description of the data incorporating a discussion of symmetry and skewness to inform the retail owners whether the data related to shipping charges are bell-shaped and how this is reflected in the measures of central tendency, particularly the median and mean.

Shipping charges
Mean .......................................................... 10.5331
Standard error ....................................... 0.171893
Median .............................................................. 9.8
Mode .................................................................... 8
Standard deviation ............................... 2.881457
Sample variance .................................... 8.302794
Kurtosis .................................................. 3.002015
Skewness ............................................... 1.855815
Range ................................................................. 13
Minimum .......................................................... 7.7
Maximum ....................................................... 20.7
Sum ............................................................ 2959.8
Count ............................................................... 281

Suppose Event A occurs 1050 times, Event B occurs 720 times, Event C occurs 120 times and Event D occurs 1110 times. Calculate the relative frequency of each event and report the probabilities of each event.

(a) P (A) = __________

(b) P (B) = __________

(c) P (C) = __________

(d) P (D) = __________

Use the values in the contingency table to solve the equations given.

(a) P(F) = ___________
(b) P(B ∪ F) = ___________
(c) P(D ∩ F) = ___________
(d) P(B | F) = ___________
(e) P(A ∪ B) = ___________
(f) P(B ∩ C) = ___________
(g) P(F | B) = ___________
(h) P(A | B) = ___________
(i) P(B) = ___________

(j) Based on your answers to these calculations in parts (a), (d), (g) and (i) are variables 1 and 2 independent? Why or why not?

A car fleet manager working for a local council is thinking of gradually replacing the current fleet of vehicles used by the council with vehicles that use LPG (gas) rather than petrol. The concern is not so much the average price of petrol but rather the variability in price that occurs, as this becomes problematic for budgeting and managing reimbursements to employees. The fleet manager will upgrade the fleet to LPG-powered vehicles so long as the variability in the LPG price is lower than that of petrol. The fleet manager uses a website that collates data on fuel prices in the council region to produce the following summary statistics based on a random sample of prices drawn from the past year.

Interpret the output and comment on the variability observed for the price of each fuel. If you make this comparison using the standard deviation for each fuel type, what conclusion can you reach?

Suppose the fleet manager tells you that he prefers to compare variability relative to the size of each fuel’s mean price. Make a recommendation to the fleet manager about which vehicles should be used in the upgrade.

A wine industry association reports in its e-newsletter that a particular fine wine is being marketed by online wine distributors with an average market price of $125 per bottle and standard deviation of $12, with the distribution of prices being approximately bell-shaped. One boutique wine distributor is concerned by this report as it is charging $50 per bottle for this particular wine.

Between what two price points would approximately 68% of prices fall? Between what two numbers would 95% of the prices fall? Between what two values would 99.7% of the prices fall?Write a short report informing the distributor whether the current price being charged is comparable to others.

The size in square metres of properties sold by a major real estate firm in Australia in the last year was analysed using descriptive summary measures. The mean size of a one-bedroom residential apartment sold by this firm was 60 square metres, the median was 55 square metres and the standard deviation was 12 square metres. Compute the value of the Pearsonian coefficient of skewness and interpret the result.

The following scatterplot examines the potential association between years of education and weekly salary. The data was obtained from the Combaro dataset on the student website (Combaro.xls). Write short descriptions about what you would expect to see in the graph and what you actually see. In examining both descriptions, estimate the correlation coefficient in each case. Using the Combaro dataset, calculate the correlation coefficient. What does your analysis suggest about the years of education an employee has completed and the amount they are paid?

The CEO of Combaro would like to know whether employees are satisfied in their positions. In particular, the CEO would like to know about the central tendency of the data and whether they are skewed in some way. For instance, the CEO suspects that there may be many employees who are quite satisfied but average satisfaction levels are being distorted by a few individuals who are extremely unhappy. Using the data provided on the student website (Combaro.xls), create a boxplot to investigate whether this is the case. Write a report to the CEO on your findings, including supporting numerical measures such as skewness.

A dataset contains the following seven values.

(a) Calculate the range.
(b) Calculate the population variance.
(c) Calculate the population standard deviation.
(d) Calculate the interquartile range.
(e) Calculate the z-score for each value.
(f) Calculate the coefficient of variation.

A hairdresser franchisor is concerned about the time taken by staff to complete a standard haircut for male customers at one of its newly opened stores. She decides to visit the store and record the time taken for such a category of haircut on 30 random occasions. A benchmark of 20 minutes has been set as a reasonable objective based on the franchisor’s experience at her other stores. Interpret the following output to help the franchisor understand the time taken to provide male customers with a standard haircut in relation to this benchmark.

Summary measure ............................. Time taken (minutes)
10th percentile ............................................. 8.2
Quartile 1 .................................................... 10.5
Median ........................................................ 12.5
Quartile 3 ................................................... 19.7
90th percentile ............................................. 32

Compute P₃₅, P₆₅, P₉₀, Q₁, Q₂ and Q₃ for the following data.

Compute the 20th percentile, the 60th percentile, Q₁, Q₂ and Q₃ for the following data.

Financial analysts like to use the standard deviation as a measure of risk for a stock. The greater the deviation in a stock price over time, the more risky it is to invest in the stock. However, the average prices of some stocks are considerably higher than the average prices of others, allowing for the potential of a greater standard deviation of price. For example, a standard deviation of $5.00 on a $10.00 stock is considerably different from a $5.00 standard deviation on a $40.00 stock. In this situation, a coefficient of variation might provide insight into risk. Suppose Stock X costs an average of $13.21 per share and has shown a standard deviation of $5.28 for the past 30 days. Suppose Stock Y costs an average of $2.52 per share and has shown a standard deviation of $0.50 for the past 30 days. Use the coefficient of variation to determine the variability for each stock. Based on the coefficient of variation, which is the riskier stock?

A fitness consultant working with a leading rugby league team measures the height of a sample of 100 male players. The output is broken down by position in terms of summary measures associated with a sample of 50 players who predominantly play in the forwards and a sample of 50 players who predominantly play in the backs. Explain in plain language what these figures imply.

The owner of a new Indian restaurant is wondering how its prices compare with others in the local area. Use the following price information on a sample of 16 rogan josh main dishes to write a short report to the restaurant owner about the central tendency of the data.

The Australian Census of Population and Housing asks for each resident’s age. Suppose that a sample of 40 households taken from the census data shows the age of the first person recorded on the census form as follows.

Compute P₁₀, P₈₀, Q₁, Q₃, the interquartile range and the range of these data.

Determine the mode, median and mean of the following data.

The Australian Census of Population and Housing asks every household to report information on each person living in the house. Suppose that, for a sample of 30 households, the number of persons living in each is reported as follows.

Compute the mean, median, mode, range, lower and upper quartiles, and interquartile range for these data and interpret them in a brief plain-language report.

A hardware store determines that 70% of its customers do not use the self-checkout system to make their purchases. It also determines that 80% of its customers pay by credit card. Among those using the self-checkout system, however, only 60% pay by credit card.

(a) Use this information to determine the probability that a customer uses the self-checkout system and pays by credit card.

(b) If use of the self-checkout system and payment by credit card are independent, what would the probability in part (a) be?

According to a consumer report, approximately 3% of New Zealanders bought a new vehicle in the past 12 months. The report also indicates that 10% commenced a new job in the same period. The report further reveals that in the past 12 months, 2% bought a new vehicle and commenced a new job. A New Zealander is randomly selected.

(a) What is the probability that in the past 12 months the individual has purchased a new vehicle or commenced a new job?

(b) What is the probability that in the past 12 months the individual has purchased a new vehicle or commenced a new job but not both?

(c) What is the probability that in the past 12 months the individual has neither bought a new vehicle nor commenced a new job?

(d) Why does the special law of addition not apply to this problem?

According to a recent survey, 40% of millennials (those born in the 1980s or 1990s) view themselves more as ‘spenders’ than ‘savers’. The survey also reveals that 75% of millennials view social networking as important to find out about products and services. A social media expert wants to determine the probability that a randomly selected millennial either views themselves as a ‘spender’ or views social networking as important to find out about products and services. Can this question be answered? Under what conditions can it be solved? If the problem cannot be solved, what information is needed to make it solvable?

Use the values in the following matrix to solve the equations given.

(a) P(A ∪ D) =  __________
(b) P(E ∪ B) =  __________
(c) P(D ∪ E) =  __________
(d) P(C ∪ F) =  __________

Given P(A) = 0.10, P(B) = 0.12, P(C) = 0.21, P(A ∩ C) = 0.05 and P(B ∩ C) = 0.03, solve the following equations.

(a) P(A ∪ C) = __________

(b) P(B ∪ C) = __________

(c) If A and B are mutually exclusive, P(A ∪ B) = __________

The operations manager of a cinema complex is interested in how patrons using the candy bar purchase food and drinks in various combinations. In particular, the manager wants to know the probability that a patron will purchase water and popcorn, juice and popcorn, soft drink and popcorn, and popcorn with no drink. The table lists a random sample of 730 purchase outcomes recently made by patrons. To simplify the experiment, only outcomes in which no beverage or one beverage was purchased are examined. Hence, any event in any row is mutually exclusive of any event in any other row (e.g. a single purchase could be listed as purchasing water only, but outcomes relating to a single purchase involving water and juice are not recorded). Transform this table to list probability information of purchases and use it to respond to the manager’s questions. Overall, what is the probability that popcorn will be purchased?

A local GP is concerned that the other doctors in the general practice where she works provide a proportionally higher number of short consultations. A random sample of 200 consultations is extracted from the records and the results are detailed in the following contingency table. Calculate the conditional probability that a short consultation is provided given that the patient sees Dr Howell. Repeat the calculations for each other doctor. Overall, what is the probability that a consultation is short? Does there appear to be a relationship between the doctor a person sees and whether the consultation is short or long? Write a short interpretation of your findings for the local GP.

Given P(A) = 0.40, P(B) = 0.25, P(C) = 0.35, P(B | A) = 0.25 and P(A | C) = 0.80, solve the following.

(a) P(A ∩ B) = __________

(b) P(A ∩ C) = __________

(c) If A and B are independent, P(A ∩ B) = __________

(d) If A and C are independent, P(A ∩ C) = __________

According to the ABS, 73% of women aged 25 to 54 participate in the labour force. Suppose that 78% of women aged 25 to 54 are married or partnered. Suppose also that 61% of all women aged 25 to 54 are married/partnered and participating in the labour force. What is the probability that a randomly selected woman aged 25 to 54:

(a) is married/partnered or participating in the labour force

(b) is married/partnered or participating in the labour force but not both

(c) is neither married/partnered nor participating in the labour force?

A company is wondering whether a consumer’s preferred brand, among its own and competing brands, is related to the age of the consumer. The company conducts a survey of 2000 consumers, asking participants to indicate which brand they prefer and which age group they belong to. The responses to these two questions are summarised using a contingency table.

For each set of respondents in a particular age group, use the information in the table to determine the probability that a consumer prefers Brand A. Do the same for Brands B, C and D. Arrange the probabilities that you have calculated into a suitable table. Write a short report to the company discussing these results and whether you think the preference towards a particular brand is related to a consumer’s age.

Convert the following contingency table to a probability matrix to solve the equations given.

(a) P(A) = __________
(b) P(E) = __________
(c) P(A ∩ E) = __________
(d) P(A = E) = __________

Use the values in the following contingency table to solve the equations given.

(a) P(A) = __________

(b) P(Z) = __________

(c) P(A ∩ X) = __________

(d) P(B ∩ Z) = __________

(e) P(A ∪ C) = __________

(f) P(A ∩ B) __________

A government department investigating the issues of tax avoidance and evasion determines that the following outcomes have occurred in a random sample of 400 cases.

(a) What is the probability that a tax return will be filed? Show how considering the complement of this event provides at least two methods of answering this question.

(b) Using the information provided, explain why the event of some form of evasion requiring further investigation occurring among those who do file an assessment is not an elementary event.

A study of tweeting behaviour revealed that, among 36- to 45-year-olds, the number one tweeted topic relates to family, with 28% making some reference to it among those tweets that could be categorised in a mutually exclusive manner. Other tweets were categorised as referring to the arts (22%), entertainment (18%), education (17%) and sports (11%), while 4% referred to some other topic. If a tweet by a 36- to 45-year-old is randomly selected and able to be categorised in the same way, determine the probabilities of the following.

(a) The tweet is about family.

(b) The tweet is not about sports.

(c) The tweet is about the arts or entertainment.

(d) The tweet is neither about family nor about education.

An external agency measures the number of listeners of radio stations in a number of different regions. For one particular region in which there are 20 radio stations, it claims that listeners are fairly random in choosing which station to listen to. Station ABC and Station XYZ are just two of the stations in this particular region. A person driving in their car switches on the radio and chooses one station to listen to.

(a) If listening behaviour was as random as the agency claims, what would be the probability of the driver tuning into Station ABC when choosing one station to listen to?

(b) What would be the probability of the driver choosing Station XYZ?

(c) What is the probability of the driver listening to either ABC or XYZ?

(d) What is the probability of the driver listening to both ABC and XYZ at the same time? Use your answer to explain how the concept of a mutually exclusive event is relevant in this particular context.

Octopus Travel studied the types of work-related activity that Australians do while on holiday. Among other things, 20% take their work with them on holiday. Fifteen per cent negotiate new deals or contracts on holiday. Respondents were allowed to select more than one activity. Suppose that, of those who take their work on holiday, 68% negotiate new deals or contracts. One of these survey respondents is selected randomly.

(a) What is the probability that, while on holiday, this respondent negotiated a new deal or contract and took their work?

(b) What is the probability that, while on holiday, this respondent did not take their work and did not negotiate a new deal or contract?

(c) What is the probability that, while on holiday, this respondent took their work given that the respondent negotiated a new deal or contract?

(d) What is the probability that, while on holiday, this respondent did not negotiate a new deal or contract given that the respondent took their work?

(e) What is the probability that, while on holiday, this respondent did not negotiate a new deal or contract given that the respondent did not take their work?

(f) Construct a probability matrix for this problem.

A white goods manufacturer is sourcing parts for its air-conditioning units. Management is currently putting together a timeline for the next stages of manufacturing. To do so, management considers that one set of parts relating to the motor are coming from Germany and one set of parts relating to the casing will come from the USA. Past experience suggests there is an 8% probability that the German parts will arrive late due to disruptions in shipping. The probability of parts arriving late from the USA is deemed to be 5%.

(a) What is the probability that parts from Germany will not be delayed?

(b) What is the probability that parts from the USA will not be delayed?

(c) Assume that the events in the two previous questions are independent. What is the probability that the parts from Germany will arrive late given that parts from the USA arrived late?

A poll is conducted by a television network to evaluate public opinion on the state of the economy in New Zealand using a countrywide representative sample. It is found that 35% of respondents believe the economy is in an acceptable position. Of the respondents living in rural areas, 75% do not believe the economy to be in an acceptable position. Assume that, of the people surveyed, 85% are not living in a rural area. One New Zealander is selected randomly.

(a) What is the probability that the person is living in a rural area?

(b) What is the probability that the person is not living in a rural area and does not believe the economy is in an acceptable position?

(c) If the person selected does not believe the economy is in an acceptable position, what is the probability that the person is living in a rural area?

(d) What is the probability that the person is living in a rural area or believes the economy is in an acceptable position?

A survey company asked purchasing managers what traits in a sales representative impressed them most. Seventy-eight per cent selected ‘thoroughness’. Forty per cent responded ‘knowledge of your own product’. The purchasing managers were allowed to list more than one trait. Suppose 27% of the purchasing managers listed both ‘thoroughness’ and ‘knowledge of your own product’ as the traits that impressed them most. A purchasing manager is sampled randomly.

(a) What is the probability that the manager selected ‘thoroughness’ or ‘knowledge of your own product’?

(b) What is the probability that the manager did not select ‘thoroughness’ and did not select ‘knowledge of your own product’?

(c) If it is known that the manager selected ‘thoroughness’, what is the probability that the manager also selected ‘knowledge of your own product’?

(d) What is the probability that the manager did not select ‘thoroughness’ but did select ‘knowledge of your own product’?

A superannuation company offers seven different types of funds that members can choose to invest in. Two of these funds involve a portfolio that exposes investors to a high level of risk. Using only this information, what method of assigning probabilities can be used to determine the probability that a member will invest in a fund that is deemed to be of a high risk? Determine this probability. Write a short paragraph that describes the advantages and disadvantages of the method used to determine the probability.

The following probability matrix contains a breakdown of the age and gender of general practitioners working in Australia.

What is the probability that one randomly selected general practitioner:
(a) is 35–44 years old
(b) is both female and 45–54 years old
(c) is male or is 35–44 years old
(d) is less than 35 years old or more than 54 years old
(e) is female if they are 45–54 years old?

(f) What is the probability that a randomly selected physician is neither a woman nor 55–64 years old?

A company advertising used cars for sale classifies vehicles, based on their body type, into one of eight mutually exclusive categories. The following table summarises the company’s current range of advertised vehicles.

Body type .............................. Number of vehicles for sale
4WD/SUV 8 ..................................... 330
Coupe 1 .......................................... 190
Hatchback 17 ................................. 850
People mover 9 ............................. 520
Sedan 47 ........................................ 600
Ute 5 ............................................... 950
Van 4 .............................................. 760
Wagon 23 ...................................... 800

What is the probability that a randomly selected vehicle will be:

(a) a ute

(b) a sedan

(c) a sedan or a hatchback?

Use the values in the contingency table to solve the equations given.

(a) P(A ∩ G) = ___________
(b) P(A | G) = ___________
(c) P(A) = ___________
(d) P(A ∪ G) = ___________
(e) P(F | B) = ___________
(f) P(B | F) = ___________
(g) P(A ∪ B) = ___________
(h) P(F) = ___________

If X is the set of events that lists the seven days of a week, what is the probability that any single day chosen from X will be:

(a) A Monday

(b) Not a Monday

(c) A weekday

(d) A weekend day?

In any given month, a busy port can have many cargo ships arriving. The following table displays the data collected over a month of ship arrivals at a particular port, with a total of 76. This data will allow the manager of the port to formulate staffing plans for crews to load and unload the vessels.

Assume that the number of arrivals per day has a Poisson distribution.

(a) What is the probability of zero arrivals on any given day?

(b) What is the probability of two arrivals on any given day?

(c) The manager’s standard plan should provide a 90% service rate — it should include adequate labour and other resources to service 90% of the vessels on their arrival date. How many arrivals per day should the standard plan anticipate?

Tropical cyclones are quite common in northern Australia during the summer months. Suppose the number of cyclones is Poisson distributed with a mean of 4.5 cyclones per season.

(a) What is the probability of having no cyclones over a season?

(b) What is the probability of having between 3 and 4 cyclones in a given season?

(c) What is the probability of having over 6 cyclones in a given season? If this actually occurred, could you make any conclusions to support the arguments for climate change? What might you conclude about lambda?

Many organisations, including the Cancer Society of New Zealand, endorse the recommendation that people consume three servings of vegetables and two servings of fruit per day as this provides some protection from heart disease and cancer. A report states that 49% of New Zealanders regularly eat fresh fruit and vegetables and that, among this same group, 80% agree it is good for their health. Interestingly, in addition the report states that 94% of those who do not eat fresh fruit and vegetables regularly also believe eating fresh fruit and vegetables regularly is good for health. If a New Zealander is selected randomly, determine the probability that this person:

(a) regularly eats fresh fruit and vegetables, and believes this to be a dietary practice that is good for their health.

(b) does not regularly eat fresh fruit and vegetables.

(c) does not regularly eat fresh fruit and vegetables, yet believes this to be a dietary practice that is good for health.

(d) believes regularly eating fresh fruit and vegetables is a dietary practice that is good for health.

(e) regularly eats fresh fruit and vegetables, given they believe this to be a dietary practice that is good for their health.

A company has recently undertaken a program to replace the desktop computers of its staff. The new computers have a different operating system and newer software. Initially, employees appeared disgruntled about having to learn how to use the computers. The company offered, at great cost, a series of training programs; unfortunately, some employees still appear to be unsatisfied, although management suspects this stems from part-time employees who have had fewer opportunities to use the new computers.

Assess a survey that was conducted on a random sample of 200 employees to report on the interest in training overall, but also determine whether the additional training sought by employees is conditional on their employment status being full-time or part-time. Write a short summary to help management decide whether to offer more training and whether this arises from the part-time cohort.

Consider the following results of a survey asking, ‘Have you visited a museum in the past 12 months?’ and ‘Do you have a child less than 10 years of age?’

Is the variable ‘museum visitor’ independent of the variable ‘child under 10’? Why or why not?

The ABS energy survey reports that 45% of all Australian households have an air conditioner and that 30% of all Australian households have a dishwasher. An Australian household is randomly selected.

(a) Assume that whether a household has a dishwasher is unrelated to whether the same household has an air conditioner. Use the special law of multiplication to determine the probability that the household has both an air conditioner and a dishwasher.

(b) Suppose another report states that if an Australian household has a dishwasher, the probability of this household having an air conditioner is 80%. Use the general law of multiplication to determine the probability that the household has both an air conditioner and a dishwasher. Does it appear that whether a household has a dishwasher is related to whether the same household has an air conditioner?

A data-management company records a large amount of data. Historically, 0.8% of the pages of data recorded by the company contain errors. If 150 pages of data are randomly selected, what is the probability that:

(a) 5 or more pages contain errors

(b) more than 15 pages contain errors

(c) none of the pages contain errors

(d) fewer than 4 pages contain errors?

A university conducted a study into students’ perceived usability of a proposed upgrade to the existing learning management system (LMS). Students were asked to visit a test site and evaluate the proposed LMS relative to their perceptions of the existing LMS. To further evaluate the results, participants were asked to nominate one field that represented their primary area of study. Some 84% of students perceived the proposed LMS to be superior in usability to the existing system. Of the students sampled, 45% were primarily studying business and rated the proposed system to be superior in usability. Of those students primarily studying science, 95% found the proposed LMS easier to use; these students represented 30% of the surveyed participants. Together, those students primarily studying science and those primarily studying business made up 85% of the sample. If a student is selected randomly, determine the probabilities of the following.

(a) The student rates the proposed system as superior in usability given that the student primarily studies business.

(b) The student rates the proposed system as superior in usability given that the student primarily studies in an area other than business.

(c) The student is studying science, given they do not believe the proposed system to be superior in usability.

(d) The student is primarily studying science and believes the proposed system to be superior in usability.

Solve the following binomial distribution problems by using the binomial formula.

(a) If n = 12 and p = 0.33, what is the probability that x = 5?

(b) If n = 7 and p = 0.60, what is the probability that x ≥ 1?

(c) If n = 8 and p = 0.80, what is the probability that x > 9?

(d) If n = 12 and p = 0.75, what is the probability that x ≤ 5?

Find the following values by using the Poisson formula.
(a) P(x = 4|???? = 3.2)
(b) P(x = 3|???? = 2.3)
(c) P(x ≤ 4|???? = 5.1)
(d) P(x = 0|???? = 3.5)
(e) P(x = 2|???? = 3.7)
(f) P(2 < x < 5|???? = 3.6)

According to ABS data, 40% of Australians above the age of 65 have chronic heart disease. Suppose you live in a state where the environment is conducive to good health and low stress, and you believe the conditions in your state promote healthy hearts. To investigate this theory, you conduct a random telephone survey of 20 persons over 65 years of age in your state.

(a) On the basis of the figure from the ABS, what is the expected number of persons in your survey who have chronic heart disease?

(b) Suppose only 1 person in your survey has chronic heart disease. What is the probability of finding 1 or fewer people with chronic heart disease in a sample of 20 if 40% of the population in this age bracket have this health problem? What do you conclude about your state from the sample data?

The two most popular majors in a business degree are marketing and international business, with 30% of students enrolled in marketing and 18% enrolled in international business. Suppose 20 students are selected at random.

(a) What is the probability that:

(i) at least half of them are studying a marketing major

(ii) no more than a quarter are studying an international business major

(iii) between 10 and 15 are studying a marketing major?

(b) Find the mean and variance of the number of students studying a marketing major. Find the mean and variance of the total number studying a marketing or international business major.

One of the most useful applications of the Poisson distribution is in analysing visitor numbers on websites. Analysts generally believe visitor numbers (‘hits’) are Poisson distributed. Suppose a large e-commerce website averages 12600 visits per minute.

(a) What is the probability that there will be 13000 hits in any given minute?

(b) If a server can handle a maximum of 15000 hits per minute, what is the probability that it will be unable to handle the hits in any 1-minute period?

(c) What is the probability that between 12000 and 13000 hits will occur in any given minute?

(d) What is the probability that 5000 or fewer hits will arrive in a 15-second interval?

An importer of shirts made in China buys the shirts in lots of 100 from a supplier who guarantees that no more than 2% of the shirts are faulty. On a given day, 15 lots are delivered to the importer.

(a) What is the probability that, in a given lot of 100 shirts, more than 5 shirts are faulty?

(b) What is the expected number of faulty shirts in a lot selected at random?

(c) A lot is rejected if it has more than 4 faulty shirts. What is the probability that a lot is rejected?

(d) What is the probability that more than 3 lots out of the 15 are rejected?

(e) On average, how many lots out of the 15 will be rejected?

Find the following values by using table A.2 in the appendix.
(a) P(x = 6|λ  = 3.8)
(b) P(x > 7|λ  = 2.9)
(c) P(3 ≤ x ≤ 9|λ  = 4.2)
(d) P(x = 0|λ  = 1.9)
(e) P(x ≤ 6|λ  = 2.9)
(f) P(5 < x ≤ 8|λ  = 5.7)

Use the Poisson formula to solve the following Poisson distribution problems.

(a) If λ = 1.10, what is the probability that x = 3?

(b) If λ = 7.57, what is the probability that x ≤ 1?

(c) If λ = 3.6, what is the probability that x > 6?

Suppose that 20% of all sharemarket investors are retirees. Suppose a random sample of 25 sharemarket investors is taken.

(a) What is the probability that exactly 7 are retirees?

(b) What is the probability that 10 or more are retirees?

(c) How many retirees would you expect to find in a random sample of 25 sharemarket investors?