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Business Analytics — MGMT11169

PART A 15 MARKS

Answer three of the following four questions.

Each question is worth 5 marks (3 x 5 = 15 marks).

Question 1 5 Marks

(a) Categorise each of the following scenarios as a statistical problem in ‘Descriptive Statistics’, ‘Statistical Inference’ or ‘Probability’.

i. Scenario: 30% of households have electronic security systems implemented. What is the likelihood that a randomly selected residence does not have an electronic security system installed? (0.5 mark)

ii. Scenario: An article in health care journal described a study that contrasted the cardiovascular responses of 120 adult subjects during treadmill, trampoline, and carpeted jogging in place exercises. The average pulse rates of the entire adult population were substantially lower on the trampoline than on the treadmill and stationary jogging. (0.5 mark)

iii. Scenario: As an employee of an automobile dealership, you are tasked with analysing the sales volume for the past year, including calculating the average, the difference between the highest and lowest sales volumes, and creating a graph comparing sales volumes between 2002 and 2023. (0.5 mark)

(b) Your grade for “HRMT11011 Human Resource Management” subject will be either an HD, D, C, P or F. It will be kept in a CQU student database as the variable ‘grade’.

(i) Is the measurement scale of ‘grade’, an example of Nominal, Ordinal, Interval or Ratio? (0.5 mark)

(ii) Explain briefly why you decide this scale of measurement. (1 mark)

(c) The following are the population (in thousands) for three cities in United State of America. Present the data in a grouped bar graph on the graph paper below. (2 marks)

Question 2 5 Marks

The following data are the number of PCs produced on a production line at Australian Machine Pty Ltd during 20 randomly selected days.

(a) Complete the frequency table below by tallying the data and calculating the frequency f, the relative frequency rf, and rf (%), and the cumulative relative frequency crf (%) distributions using ‘2200 to less than 2500’ PCs as the first class: (1 mark)

(b) Construct a histogram for the frequency data on the graph paper below: (2 marks)

(d) Use the graph above to estimate the first (3 rd) quartile (i.e., 75%). An approximate answer is fine (show your working on the graph above.) (1 mark)

Question 3 5 Marks

During the summer season in Melbourne, the demand for bottled water rises. The number of 1-gallon bottles of water sold in one store over a 12-hour period during the summer season is:

60 84 65 67 75 72

80 85 63 82 70 75

(a) Calculate the value of the mean and median for this sample.

(Do not put the data into classes.) (2 marks)

(b) Which statistic (mean or median) best describes the centre of this distribution?

Explain the reason for your answer briefly. (1 mark)

(d) Calculate the 1 st and 3 rd quartiles

Question 4 5 Marks

(a) Consider the random experiment of rolling a pair of dice. Assume we are interested in the sum of the face values that are displayed on the dice.

i. How many outcomes are possible? (0.5 marks)

ii. List the outcomes. (0.5 marks)

iii. What is the probability of obtaining a value of 2? (0.5 marks)

iv. What is the probability of obtaining a value of 4? (0.5 marks)

(b) A data specialist obtains the following data at a community centre in Redfern, NSW. Of those visiting the community centre, 59% are male, 32% are alcoholics, and 21% are male and alcoholic.

i. Use this data to complete the table below. Assume 200 visitors in total as shown in the table. (2 marks)

What is the probability that a random visitor to the community centre is:

ii. both F and NAL, i.e., P (F ∩ NAL)? (0.5 mark)

iii. What is the probability that a random Male visitor is Alcoholic i.e., P(AL|M)? (0.5 mark)

PART B 15 MARKS

Answer three of the following four questions.

Each question is worth 5 marks (3 x 5 = 15 marks).

Question 1 5 Marks

(a) A process analysis conducted by Melbourne Production Company reveals 15% faulty screen smart phones. A superintendent of quality control is analysing ways to enhance the process. If he inspects a sample of 40 screen smart phone, compute the probability that he finds:

i. Exactly three faulty screen smart phones. (1 mark)

ii. One or more faulty screen smart phones. (1 mark)

iii. How many faulty screen smart phones would she expect in the sample of 40? (1 mark)

(b) The average rate of two complaint calls per minute at a Vodafone customer centre in Melbourne corresponds to a Poisson distribution.

What is the probability that the customer centre receives:

i. Exactly three complaint calls in the next minute? (1 mark)

ii. Two or more complaint calls in the next two minutes?

Question 2 5 Marks

Suppose that for a recent admissions class, a St Peters College in Brisbane received 2,851 applications for early admission. Of this group, it admitted 1,033 students early, rejected 854 outright and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 21% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2,175.

Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool.

i. Estimate P(E), P(R), and P(D). (1 mark)

ii. Are events E and D mutually exclusive? Find P (E ∩ D)? (1 mark)

iii. For the 2,175 students who were admitted, what is the probability that a randomly selected student was accepted during early admission? (1 mark)

iv. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process? (2 marks)

Question 3 5 Marks

The probability distribution of collision insurance damage claims paid by the Sydney Automobile Insurance Company is as follows.

i. Determine the collision insurance premium that would enable the company to breakeven based on the expected collision payment. (2 marks)

ii. The insurance company charges an annual rate of $600 for the collision coverage. What is the expected value of the collision policy for a policy holder? (Hint: it is expected payments from the company minus the cost of coverage.) Why does the policyholder purchase a collision policy with this expected value? (3 marks)

Question 4 5 Marks

The publication of a study on the earnings of entry-level workers in Australia is regularly conducted by the International Management Institute. In 2022, it was observed that male university graduates received the entry-level salary of $32.68 per hour, while their female counterparts earned $29.80 per hour. It is assumed that the standard deviation for male graduates is $3.30, and for female graduates it is $3.05.

i. What is the sampling distribution of x for a random sample of 60 male graduates? (2 marks)

ii. What is the sampling distribution of x for a random sample of 60 female graduates? (2 marks)

iii. In which of the preceding two cases, part (i) or part (ii), is the standard error of x smaller? Why? (1 mark)Sample Examination (Guide Only)

PART C 20 MARKS

Answer two of the following four questions.

Each question is worth 10 marks (2 x 10 = 20 marks).

Question 1 10 Marks

(a) The Brisbane Strawberries Company has raised concerns regarding the potential contamination of a significant number of recently harvested strawberry cases. The company opts to analyse a sample of randomly chosen instances.

The null hypothesis here is that the cases are contaminated.

i. What are the type I and type II errors for this problem? (2 marks)

ii. Would society want a small α or a small β if it had to choose one or the other to be small? Explain briefly. (2 marks)

(b) The Brisbane producer affirms that they have designed a clothesline with an average breaking strength of 8 kg. Historically, the population standard deviation, denoted as σ, has been noted at 0.5 kg, and no evidence suggests that σ has experienced any modifications. A sample consisting of 50 clotheslines was subjected to tensile testing. The average breaking strength of the sample is 7.8 kg, with a standard deviation of 0.7 kg.

i. Test the producer’s claim using a 5% level of significance i.e., use α = 0.05. (4 marks)

ii. Do you need to assume that breaking strength of a clothesline is normally distributed to test the producer’s claim? Explain your answer briefly. (2 marks)

Question 2 10 Marks

Consider the following gasoline time series data:

(a) Using the naïve method (most recent value) as the forecast for the next period, compute the following measures of forecast accuracy:

i. Mean absolute error (MAE) (1 mark)

ii. Mean squared error (MSE) (1 mark)

iii. Mean absolute percentage error (MAPE) (1 mark)

iv. What is the forecast for week 7? (1 mark)

(b) Using the average of all the historical data as a forecast for the next period, compute the following measures of forecast accuracy:

i. Mean absolute error (MAE) (1 mark)

ii. Mean squared error (MSE) (1 mark)

iii. Mean absolute percentage error (MAPE) (1 mark)

iv. What is the forecast for week 7? (1 mark)

Question 3 10 Marks

A Sydney-based winemaker is tasked with calculating the ideal quantity of two wine varietals to yield from specific grapes. Each litre of “T” wine yields $8 profit, while each litre of “D” wine produces $3 profit. The labour hours and bottling process time used for type of wine are given in the table below. Resources available include 200 labour hours and 110 hours of bottling process time. Assume the winemaker has more than enough grapes available to supply any feasible production plan.

i. What is the objective function? (2 marks)

ii. List all the constraints. (3 marks)

iii. Use graphical approach to determine an optimal solution. (5 marks)

Question 4 10 Marks

The following pairs of values give x, umbrella marketing expense (in hundreds of dollars) during a certain period and y, the amount of company's sales (in thousands of dollars), for the same period.