# Application Of Mathematical And Statistical Methods Based On Given Data Assessment Answer

Refer to your subject guide [pp. 5, 19] for marks allocation and general report requirements.

This assessment involves obtaining data from ABS data sets, applying mathematical and statistical methods to the data, and providing business advice based on your calculated results.

Relevant data and/or results may be included in your report using small data tables, as an acceptable method of presentation. Where relevant, show all workings for calculations.

All submissions must comply with the requirements listed in the Course Handbook, with all citations & referencing complying with the Harvard AGPS referencing style.

Students should note the word limit for this report. Responses must be within 10% +/- of the word limit. The reference list does not form part of the word count.

Please include your word count on your title/cover page.

Tony Lau is considering opening a business, either near his home in Sydney, or on the Central Coast where he would like to semi-retire within a few years. Tony has asked you to analyse ABS and other data, to assist him with making his decision. Tony has asked you to answer three questions.

Part 1  Employment and Incomes

1. Tony wants to know how many people living in each region are of working age [15-64 years] and whether this is growing over time. Tony asks that you include graphs showing this population change over recent years
2. Tony also asks you to use regression [least-squares] to predict the working age population in 2023 for both regions. Comment on the reliability of your prediction.
3. Tony is also interested in the average [mean] employee income earned by people working in these two regions, and asks that you include graphs showing any changes over recent years.
4. Finally, Tony is interested in how many employees are in each region. He asks you to provide the mean and median number of employees, for 2013-2017, for each region, and to also report the range of employee numbers for each region for those five years

Part 2  Business Ownership and Wealth Indicators

Tony is convinced that business owners earn more money than employees, and that he can predict future business growth by measuring certain measurements of wealth. Tony feels this will help his to decide on the best location for his business.

1. Tony asks you to calculate the relationship [correlation] between the number of people in each region that own a business, and the median total income for individuals in that region [excluding Government pensions]. Tony also asks you to calculate the relationship between mean capital gains and the number of business owners. Tony wants you to report these correlations for each region.
2. Include your comments on the results, and the significance (if any) for Tony.

Part 3  Product Choice

Part of Tony’s business idea involves selling low profile sports sedan tyres for expensive cars such as Mercedes and BMW. Tony intends to stock either the Triomphe brand, from France, or the Palio brand, from Italy. Tony has negotiated exclusive arrangements with both brands, so whichever brand he chooses he will be the only outlet in the Sydney geographic basin. Palio has offered Tony a profit margin 8% higher than Triomphe. For both brands, Tony will offer customers a free replacement under warranty, if a tyre lasts less than two years. ABS and RMS data reveals that residents of the Central Coast and North Sydney regions travel 10,000km per year on average.

The tyre manufacturers have provided Tony with the following production data:

• Triomphe: 28,000km mean tyre life; standard deviation 2500km.
• Palio: 28,000km mean tyre life; standard deviation 6500km.
• The data from each brand forms an approximate normal curve.

Required:

1. For each brand, Tony asks you to calculate:
1. the probability that a tyre will last more than 32,000km.
2. the probability that a tyre will last more than 26,000km but less than 31,000km.
3. the likely proportion of tyres Tony will need to replace for free under his warranty.
2. Based on part i. advise Tony on which brand you think he should sell. Justify your choice.
3. Tony is keen on Palio’s higher profit margin, but very worried about their quality control and variability, so he tested a random sample of 60 Palio tyres, finding a sample mean tyre life of 22,164km. Construct a 99% confidence interval

# Part-1

1. Working age population between age 15-64 years:

Central Coast:

The above line chart indicates growth in working age population between 2013 and 2018 in Central Coast Area. It can be seen that the line chart indicates an upward sloping trend in the population over a period of time as seen by the rising blue line in the graph. This means that the working age population between 15-64 years age group is increasing over time. The number was 201,626 in 2013 and increased to 207,306 in 2018.

North Sydney:

The above line chart indicates growth in working age population between 2013 and 2018 in North Sydney Area. It can be seen that the line chart indicates an upward sloping trend in the population over a period of time as seen by the rising blue line in the graph. This means that the working age population between 15-64 years age group is increasing over time. The number was 51,411 in 2013 and increased to 53,647 in 2018.

2. Working Population in 2023

For Central Coast, for 2023:

In order to predict the working population, linear regression was performed with output as follows:

According to above, the linear regression equation is: Working population (y) = -2,184,393+1185.286*year (x)

Hence, for x = 2023,

Working Population (y) = -2184393+1185.286*Year

Working Population = -2184393+1185.286*2023 = 213,440.5

For North Sydney, for 2023:

In order to predict the working population, linear regression was performed with output as follows:

According to above, the linear regression equation is: Working population (y) = -892,785+469*year (x)

Hence, for x = 2023,

Working population (y) = -892785 + 469*2023 = 56,002

The regression equation has been formulated for the working population data for both the regions and based on that the prediction/forecast for 2023 has been made. Therefore, the prediction is reliable.

3. Mean Employee Income

The following graphs indicate the mean employee income in Central Coast Area and North Sydney Area for the period of 2013 till 2017:

From the above graphs, it is clear that for Central Coast, the average employee income is increasing over the years. It was \$49,706 in 2013 and reached \$55,344 in 2017. However, for North Sydney, the average employee income is growing till 2016 but after that, it has become almost stagnant. It was \$88,196 in 2013, \$100,437 in 2016 and there was negligible increase to \$100,491 in 2017.

4. Number of employees:

The following screenshots indicate the number of employees in each of the regions, along with the average and median number of employees also:

Central Coast:

North Sydney:

It can be seen that the minimum and maximum number of employees for Central Coast is 146,556 and 161,585, respectively. The average number of employees during the period of 2013 to 2017 is 154,469.6. The median number of employees during the period of 2013 to 2017 is 156,330.

It can be seen that the minimum and maximum number of employees for North Sydney is 40,069 and 45,658, respectively. The average number of employees during the period of 2013 to 2017 is 43,210.8. The median number of employees during the period of 2013 to 2017 is 44,001.

Part-2

Correlation Tables

North Sydney:

 Own unincorporated business income earners (no.) Median total income (excl. Government pensions and allowances) (\$) Gross Capital Gains reported by taxpayers - Mean (\$) 2011 2013 5887 66254 50547 2014 6096 68493 62812 2015 6240 70603 73534 2016 6482 72915 80363 2017 6692 73016 94683 2018 Correlation between No of business owners and Median Total Income 0.97 Correlation between no of business owners and mean gross capital gain 0.99

The correlation between the number of business owners and mean total income excluding government pensions and allowances is very high i.e. 0.97. Similarly, the correlation between total number of business owners and mean gross capital gain is 0.99 which is also very high. Almost perfect positive correlation is seen for both.

Central Coast:

 Own unincorporated business income earners (no.) Median total income (excl. Government pensions and allowances) (\$) Gross Capital Gains reported by taxpayers - Mean (\$) 2011 2013 22168 41433 19131 2014 22305 42501 25416 2015 22786 43676 40112 2016 23371 44951 33431 2017 23895 45608 40040 2018 Correlation between No of business owners and Median Total Income 0.98 Correlation between no of business owners and mean gross capital gain 0.79

The correlation between the number of business owners and mean total income excluding government pensions and allowances is very high i.e. 0.98. Similarly, the correlation between total number of business owners and mean gross capital gain is 0.79 which is also very high.

The correlation between the number of business owners and mean total income excluding government pensions and allowances is very high for both the regions whereas the correlation between total number of business owners and mean gross capital gain is high for North Sydney as compared to Central Coast.

Part-3:

 Triomphe Palio the probability that a tyre will last more than 32,000km. 0.0548 0.2676 the probability that a tyre will last more than 26,000km but less than 31,000km. 0.3364 0.2616 the likely proportion of tyres Tony will need to replace for free under his warranty. Prob that tyre will last less than 20000kms (for 2 years) 0.00069 0.10935

Probabilities have been calculated by computing the Z scores and then finding the probabilities at various Z scores.

We can see that a very small i.e. almost negligible chance for need to change the tyre in case of Triomphe whereas there is almost 10% change of failure and need to change in case of Palio.

Since the profit margin for Palio is 8% higher than Triomphe but the failure rate is 10% therefore, Tony should proceed with Triomphe brand tyres and sell them more. For Palio he will be facing a loss of 2% because of the excess failure rate.

For 99% confidence interval, Z score = 2.58

X(bar) = Sample mean = 22164

N =60

Assuming the standard deviation for the sample to be 5145 (calculated w.r.t. 6500 for 28000kms).

 Upper 22164+(2.58*5145)/sqrt(60) 23877.7 Lower 22164-(2.58*5145)/sqrt(60) 20450.3

At 99%, the confidence interval is 20450 to 23878kms.