BEA140 Quantitative Methods Assignment

Question 1

Homer and Marg wish to buy a block of land, and pay it off in six years. For the first four years they believe that they can make payments of $1500 per month. After that, once their youngest child Maggie starts school, they believe that they afford to pay $2000 a month. They have access to a loan from the Captial City Bank with interest being charged at j12 = 5.4% p.a. They wish to borrow the maximum amount that they can afford to pay back in six years.

a) Assuming they make the maximum payments that they have budgeted for, illustrate the cash flows associated with the loan as a fully labelled time line diagram. (Assume the first repayment occurs one month after they take out the loan.).

b) Determine the maximum Homer and Marg can afford to borrow.

c) Describe and perform a sanity check on your answer from part (b).

d) Assuming they borrow the maximum possible, construct an amortization table for the last two payments. Ensure that you show how you obtained your starting value for the table, and a set of example calculations for one line of the table. Describe and apply a sanity check to your table.

e) If Maggie is able to start school a year earlier than expected, Marg and Homer will be able to switch from monthly payments of $1500 to monthly payments of $2000 a year earlier than expected. Determine how much this will increase the maximum amount they can borrow.

Question 2

Today, your investment company has the opportunity to buy an Office Block for $6,300,000. The current tennants have a twelve year lease, and after costs the building owners receive $80,000 at the end of each month. At the end of twelve years the building is due to be demolished to make way for a new freeway, at which time the government will pay the owners $3,000,000 compensation. (That is, at the end of the last month of ownership, the owner will receive three million dollars from the government as well as the last regular rent payment.)

a) Illustrate the cashflows associated with this office block using a fully labelled time lime diagram.

b) The corporate discount rate associated with investments of similar risk is r = j1 = 15.4% p.a. Determine the present value of the entire cashflow, and thus make a recommendation as to whether your company should buy this office block. [Hint: Be careful to find the appropriate equivalent discount rate to use in your calculations.]

You have just identified a better opportunity which will become available in 15 months, but will require you to have 10 million dollars. You decide to establish a sinking fund into which you will immediately deposit the $6,300,000 that you were thinking about spending on the office block. You also plan to make monthly deposits of size R into the sinking fund, to help you achieve your target of 10 million dollars. The first deposit will be in one month’s time. The fund will earn interest at a rate of j12 = 9% p.a.

c) Illustrate this deposit sequence as a fully labelled time line diagram.

d) Find the size of the required regular deposit, R. (Hint: Do not forget that the initial deposit of $6.3 million will attract interest.)

e) Construct a sinking fund table showing the last deposit. Ensure that you show how you arrived at your starting value.

Question 3

a) A university lecturer is considering three different approaches to getting feedback on her lecturing style:

1. An email to all the students in the class inviting them to take part in an anonymous survey.
2. Having an assistant come into the last face-to-face lecture of the year and hand out and collect an anonymous paper form.
3. Picking a random sample of 20 students from the class and doing an in-depth interview with them herself.

For each possible approach explain why it might produce a biased view and identify which form of bias it will be most susceptible to.

b) Research has revealed that of the 100 most successful products in the health foods market, 63% have predominantly green coloured packaging. Explain whether you should be confident that using green packaging for your new health food, “Kale and Tofu Surprise”, will contribute to its success.

c) Consider the the following outcomes for 9 students who undertook a first aid course.

For each outcome variable (Mark, Grade, Competent) identify which measures of central tendency (mean, median, mode) are feasible, and then determine these.

d) In a wheat farming district, 25 fields (each one hectare in size) were randomly chosen to trial a new variety of wheat. At the end of the growing season, the wheat was harvested and the yields are summarised in the table below;

i. Determine the mean yield, median yield, and the standard deviation of the yield for this sample. Describe and conduct a sanity check for each of these three values.
ii. A researcher with access to the raw data found that the values for mean and standard deviation were 5.08 tonnes/hectare and 1.01 tonnes/hectare respectively. Explain why they are different to your answers, and identify which set of results is more accurate.

e) Simpson Enterprises owns two water parks, one in Springfield and one in Ogdenville. A box and whisker plot of their daily attendances for 2018 appears in the diagram to the right.

i. Estimate the five number summary for Springfield daily attendance.
ii. Identify the shape of the distribution of Ogdenville’s daily attendance. Repeat for Springfield. Be careful to explain how you identified these shapes.
iii. Assuming both parks were open for 365 days in 2018, estimate the total attendance at Simpson Enterprise water parks for 2018.

Question 4

a) Mary runs a flower stall at the local Sunday market. She can buy bunches of roses from her supplier for $10 each. She can sell bunches of roses for $20 each. On Sundays the average demand for bunches of roses at her stall is 2.5 bunches per Sunday. Any bunches of roses that are not sold on Sunday go bad before the next weekend. Mary’s current policy is to stock 3 bunches of roses.

i. Suggest a distribution that may describe the demand for bunches of roses on Sundays. List the assumptions would need to be met in order for this distribution to be a good fit to this scenario?

ii. Complete the table below in your answer booklet (some entries have been completed for you).

iii. Determine the probability that Mary will make a loss?

iv. Determine Mary’s expected daily profit. What is the variance of the daily profit?

b) When a pill making machine in a pharmaceutical factory is running correctly it produces paracetamol tablets with a mean weight of 500.00 mg, and a standard deviation of 0.35 mg.

i. Assuming that tablet weights follow a normal distribution, determine which of the following events is more unlikely if the machine is running correctly:

• A randomly chosen tablet weighs more than 501.00 mg
• Three randomly chosen tablets each weigh more than 500.50 mg

ii. Thus explain which event gives a stronger signal to the operator that the machine is not running properly.

iii. Describe two ways that you could easily check whether the assumption of normality seems reasonable.

Question 5

a) It has been estimated that at any point in time 2% of the computers have a virus. If a computer has a virus, tests reveal that SuperGuard virus detection software will: 

• Report that a virus has been detected 94% of the time.
• Report the computer as clean the rest of the time. (a false negative)

However, if a computer is not infected with a virus, SuperGuard will:

• Report that a virus has been detected 9% of the time (a false postive)
• Report the computer as clean the rest of the time.

i. Present the various combinations of events (has virus / doesn’t have virus, virus reported / virus not reported) in a fully labelled tree diagram, showing marginal, conditional and joint probabilities.
ii. Determine the probability that SuperGuard will report a virus on a randomly selected computer.
iii. Thus determine the probability that a computer will have a virus if SuperGuard reports that it has one.
iv. Determine the probability that a computer will NOT have a virus, if SuperGuard does NOT report one. Compare this with your answer from (iii) and comment on whether SuperGuard is a good piece of virus detection software. (One or two sentences will do.)

b) If 2% of computers are infected with a virus, determine the probability that a randomly selected sample of 100 computers have no more than 1% infected (i.e. no more than 1 computer infected).

c) Determine the approximate probability that a randomly selected sample of 1000 computers have no more than 1% infected (i.e. no more than 10 computers infected). 

Question 6

In the state of Kingsland, a health economist is working on a project to target towns that have a high average cholesterol level with an educational program about exercise and healthy eating. From previous studies it is known that the cholesterol level of adults in Kingsland follows a normal distribution with a mean of 180 millgrams per decilitre. In the town of Shelbyville, a random sample of 40 adults are given a cholesterol test. The sample mean cholesterol level is 200 and the sample standard deviation is 65. The health economist uses the BEA140 Simple Single Sample Test, with ? = 5%, in order to determine whether Shelbyville’s average cholesterol level is different to Kingsland’s. The printout appears below. (NB Calculated values have been replicated in a larger font to make them easier to read.)

Using information from the printout above:

a) Explain why they have used a t test rather than a z test.

b) Explain whether the null hypothesis would have been retained or rejected if the Test Significance was chosen to be ? = 10%. [Note: You do not need to perform an entire hypothesis test, but rather simply focus on how the change in ? impacts the decision.] 

c) Explain whether the null hypothesis would have been retained or rejected if the Test Significance remained at ? = 5%, but the null hypothesis was H0: ? = 175. [Note: You do not need to perform an entire hypothesis test, but rather simply focus on how the change in H0 impacts the decision.]

d) The health economist forgot that sampling was done without replacement for Shelbyville’s finite population of 400. Explain whether this piece of information would change the decision that appears in the printout. [Note: You do not need to perform an entire hypothesis test, but rather simply focus on how knowing the population size, N, impacts the test statistic.]

e) The health economist then realises that what they actually wanted to know was whether average cholesterol levels in Shelbyville are significantly HIGHER than Kingsland, NOT whether they are significantly DIFFERENT. Identify what the null and alternate hypotheses should have been if a one tail test was used, and then go on to explain whether this would have changed our decision in part (d). [Note: No calculations required, an explanation is sufficient.]

f) If we use the sample standard deviation of cholesterol for Shelbyville as an estimate for the standard deviation of cholesterol for Capital City, determine the minimum sample size required if we need to be 99% confident that the sample mean for Capital City is within ± 20 milligrams per decilitre of its true population mean.
continued....

Question 7

a) You are the manager of a supermarket that has just received a shipment of 1024 pumpkins. If a pumpkin has any imperfections it is classified as B-grade, otherwise it is classified as A grade. As A-grade pumpkins sell for a higher price you wish know what percentage of the
shipment are A-grade pumpkins. You don’t have time to inspect the whole shipment, so you inspect a random sample of 144 pumpkins, and find that 108 are A-grade. Determine a 95% confidence interval for the proportion of pumpkins in the entire shipment that are A-grade. [Assume that sampling is done without replacement.]

b) The farm that supplies A and B grade pumpkins to the supermarket, sell their C-grade pumpkins to a jam factory, where they are used as bulking agent. Historical records show that the factory can extract an average of 5.27 kg of pumpkin mash per C-grade pumpkin (with a standard deviation of 1.32 kg). A random sample of 100 C-grade pumpkins are subjected to a new experimental mashing process, and the average yield of this sample is 5.40 kg. Determine the probability of getting a sample result at least this high, if the new experimental process had no impact on the quantity of mash produced. Explain whether this p-value provides strong evidence of an improvement in mash output.

c) The farm manager is not sure whether to plant “Queensland Blue” pumpkins or “Tassie Purple” pumpkins. Consumers’ taste preference is one factor that may influence their decision. A random sample of 64 consumers are given a blind test in which they are asked to identify which pumpkin they prefer. 36 identify the Tassie Purple. Using a value of ? = 10%, test the claim that consumers are indifferent between the two types of pumpkin. [Hint: “indifferent” would mean the proportion favouring Tassie Purple would be 50%.]

d) Explain the difference between “statistical significance” and “practical significance”. Feel free to use examples.

Question 8

Public health authorities are concerned about the increased incidence of the so-called “Zombie Bug” (toxoplasmosis gondii), a parasite that can trigger reckless or suicidal behaviour in humans and other animals. They are wondering whether exposure to certain foods or preparation methods might increase the risk of being infected. In one of their studies they wish to investigate whether the way meat is cooked is related to the risk of infection. A random sample of 100 people are tested for the presence of toxoplasmosis gondii, and also asked how they have their steaks prepared. The data appears in the table below.

a) Being careful to state your conclusions, and using a level of significance of ? = 5%, test whether presence of the Zombie Bug is independent of meat preparation.

b) If the null hypothesis is rejected, some people will claim that:
“this proves that the presence of the Zombie Bug causes people to change the way they prepare their meat, in particular that it promotes risky behaviour such as eating meat which is not fully cooked”,
Whilst others will claim that:
“this proves that eating meat which is not fully cooked causes an increased risk of becoming infected by the Zombie Bug, as people eating raw or rare steaks have a higher infection rate.”
Explain if either of these arguments is correct.

Question 9

You are the manager of a very large commercial greenhouse operation, and you need to understand the relationship between soil temperature (S) and days-to-harvest (DTH) for a new variety of broccoli that you will be growing. To investigate this relationship, your gardeners grow a random sample of 92 plants at different soil temperatures (measured in Celsius) and record the time (in days) before each plant is ready for harvest. This data is then subjected Ordinary Least Squares regression using Excel, and the output appears below.

Referring to the printout above:

a) Determine the ordinary least squares regression line of best fit that explains days-to-harvest by soil temperature. Interpret the slope coefficient.

b) Determine and interpret Se, the standard error of the estimate.

c) Determine and interpret the coefficient of determination.

d) Use the OLS line of best fit to predict the time-to-harvest for a broccoli plant grown in soil at 20 Celsius.

e) List the LINE assumptions required for inference in simple linear regression. Then Test using ?= 0.05 whether the apparent relationship between days-to-harvest and soil temperature for this sample of plants can be explained by chance.

f) There are at least five insights that one can get from a scattergram which suggest that a simple bi-variate OLS regression may not be appropriate. List four of these.

g) Further investigation reveals that two different propagation methods were used - some of the broccoli plants were grown from seeds whilst others were grown from transplants. This continued....

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