## Introduction

Depreciation in the value of a car over time is a good example of a linear function with a proportional change in value over the cars age. For research purposes, in this investigation the values of two different makes and models of cars from the past nine years will be compared and explored through multiple mathematical concepts and ideas to determine if a linear function develops.

To gather the data necessary for evaluating the depreciation rate of cars and to determine the linear function of the cars, two makes and models will be selected and prices of these models from the past nine years will be recorded to gather enough relevant data to create accurate linear functions and depreciation rates.

For this investigation, the Toyota Landcruiser Sahara and the Holden Commodore are the cars that have been selected and the prices from 2010 to 2018 will be researched and recorded in data tables.

## Hypothesis:

The hypothesis for this investigation is that after using multiple accurate and reliable mathematical concepts to explore the linearity of the depreciation rate of two makes and models of cars over the span of nine years, a linear function will arise for both makes and models of car.

## Considerations:

### Mathematical concepts

The mathematic concepts that will be used throughout this investigation are:

- Scatterplots – used to identify the relationship between multiple numerical values.
- Regression lines – used to describe how the response variable changes when the explanatory variable changes.
- Residual plots – used to test the linearity assumption by plotting the residuals against the EV
- The correlation coefficient (Pearson’s r) – used to measure the strength of a linear relationship between two or more numerical variables.
- The r-squared value – used to give the percentage of variation in the RV that is explained by the variation in the EV

## Assumptions

The assumptions in this investigation include:

- All cars have the same mileage on them
- All cars have the same type of interior/seat materials
- All cars have working mechanics and are safe to drive
- All cars have the same type of wheels and wheel rims/hubcaps
- All cars have the same amount of wear and tear in their interior and on their exterior
- All prices of the listed cars were accurate at the time of listing and prices have remained static since the car was listed for sale

## Validity of data collection

To ensure all data that was collected for this investigation was valid and reliable, all prices of the car makes and models were double checked with the prices of two or more used car sales websites, the websites used were official listings from car dealerships/ websites and not personal ads on gumtree or eBay. To ensure that the car prices are not fabricated screenshots of each car and its price are included in the appendices of this investigation. The credibility of each website and program used in this investigation was also checked to ensure that no corrupt websites or programs affected the reliability and validity of the data used in the investigation.

## Limitations

Possible limitations of the data that need to be addressed in this investigation include price fluctuations of each car included in the data tables, the possibility of systematic errors and the possibility of technology failing to produce accurate results for mathematical equations.

## Use of technology

The technology in this investigation includes the use of a scientific calculator for all manual calculations, the use of the Excel spreadsheet program for making all the graphs etc shown in the developing a solution section of the paper, the use of a computer for the typing of this investigation and use of the internet to get data on cars and their prices.

The technology used in the investigation will assist in the efficiency and accuracy of the calculations and graphs used to explore the mathematical concepts of the depreciation of car values and the relationship this has with linear function.

## Developing a solution

In this investigation the data for the cars age was changed from the year of the model to how old in years the car actually is. This was changed to aid in the creation of the scatter plots below however, because of this change the graphs ended up having negative values for some things such as the regression lines and the correlation of the data

The graph below illustrates that despite the initial high purchase price of $120,000, the Toyota Landcruiser Sahara depreciates moderately. By nine years of age the vehicles residual value is approximately $52,000. Considering the vehicle is a second-hand off-road vehicle and that the average life span of a family car is approximately 20 years before replacement should be considered, the gradual depreciation rate makes the Sahara fair value for money for a long-term car with a fair amount of re-sale value.

### Toyota Landcruiser Sahara

2010 $52,888

2011 $53,990

2012 $66,990

2013 $74,990

2014 $85,900

2015 $88,492

2016 $97,488

2017 $98,900

2018 $108,888

### The r-squared value for the Sahara

The r-squared value is often portrayed as a percentage and is used to show the percentage of the dependant variable (or the R.V) variation that a linear model explains In the scatterplot above the r-squared value for the Sahara is 0.9768. This is used to explain and represent the differences between the fitted values and data, because a higher r-squared value (the closer to +1 it is) determines the variation of the fitted values and the observed data, it can be assumed that there is a very small amount of variation in the plotted variables and the observed data for the values of this car model over the span of nine years, as the r-squared value shows that approximately 97% of the variation in a variable can be explained by the other.

It is important to note that while the r-squared process is an intuitive model of how the linear model fits into a set of observations, r-squared should always be valued in conjunction with other models of statistics such as residual plots and the correlation coefficient in order to give a clearer overall trend pattern. Multiple observations will help give confidence in the recorded data trends.

### The regression line for the Sahara

A regression line can be calculated using the formula y = a + b x, a regression line is used to describe how the response variable (RV) changes when the explanatory variable (EV) changes, the regression line is often used to predict the y value when there is a given value for x. The regression line for the Toyota Landcruiser Sahara was calculated in excel using the given data in the scatterplot above, the y value for this regression line was given as y = -7220.5x + 117050. It is worth noting that the data for the age of the car that was used to create the scatter plot had to be changed from the year the car model was made to how old the car was in years, because of this change the scatter plot created had a negative association and is also why the regression equation for the regression line is negative.

### Residual plot for the Sahara

Predicted value Predicted value deviation

A residual can be defined as the space between a measured value and a predicted value in a set of data, the residual plot for this investigation was calculated in excel. A residual plot is used to test the linearity assumption by plotting the residuals against the EV, the more random the plots on a residual plot are the more appropriate the data is for a linear model. Because the scatterplot for the Toyota Landcruiser Sahara appears to follow a linear trend and the residual plot for the car has random plots it can be assumed that the data for the depreciation of the Sahara is appropriate for a linear model and that the linear assumption has been attained.

EV RV

9 52,888

8 53,990

7 66,990

6 74,990

5 85,900

4 88,492

3 97,488

2 98,900 Correlation Coefficient

1 108,888 -0.98834

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### Correlation coefficient for the Sahara

The correlation coefficient or Pearson’s r is used to measure the strength of a linear relationship between two or more numerical variables, the correlation coefficient will always lie between either +1 or -1. A value closer to +1 is a positive association and a value closer to -1 is a negative association. The correlation coefficient for the Toyota Landcruiser Sahara was calculated in excel using the =CORREL(A1:A2,B1:B2) function. The correlation coefficient for the Sahara is -0.98834, because the value of the correlation coefficient is so close to the -1 value this shows that the data for the Sahara has an almost perfect negative association.

It is also worth noting that the set of data gathered for the Toyota Landcruiser Sahara has no outliers, this should be distinguished because the presence or absence of outliers in a set of data will greatly affect the outcome of variables such as the r-squared value or the correlation coefficient if outliers are present and left in the data.

The graph below illustrates that despite the moderate purchase price of $28,495, the Holden Commodore depreciates rather quickly. By nine years of age the vehicles residual value is approximately $6,989 over halfway below its original value. Considering the vehicle is a second-hand vehicle and that the average life span of a family car is approximately 20 years before replacement should be considered, the depreciation rate makes the commodore a cheap car to buy for temporary use and little to no re-sale value.

Holden commodore

2010 $6,989

2011 $7,990

2012 $10,000

2013 $12,990

2014 $14,499

2015 $15,800

2016 $18,988

2017 $25,990

2018 $28,495

### The r-squared value for the Commodore

The r-squared value is used to show the percentage of the dependant variable (or the R.V) variation that a linear model explains. In the scatterplot above the r-squared value for the Holden Commodore is 0.9431. This value is used to clarify and represent the differences between the fitted values and data, The r-squared value determines the variation of the fitted values and the observed data, it can be assumed that there is only minimal difference in the plotted variables and the observed data for the values of this car model over the span of nine years because the r-squared value shows that approximately 94% of the variation in a variable can be explained by the other.

It is important to note that while the r-squared process is an intuitive model of how the linear model fits into a set of observations, r-squared should always be valued in conjunction with other models of statistics such as residual plots and the correlation coefficient in order to give a clearer overall trend pattern. Multiple observations will help give confidence in the recorded data trends.

### The regression line for the Commodore

A regression line is calculated using the formula y = a + b x, and is used to describe how the response variable (RV) changes when the explanatory variable (EV) changes, the regression line is often used to predict the y value when there is a given value for x. The regression line for the Holden Commodore was calculated in excel using the data in the scatterplot above, the y value for this regression line was given as y = -2680.2x + 29150. Considering that the data for the age of the car that was used to create the scatter plot had to be changed from the year the car model was made to how old the car was in years, the scatter plot created had a negative association and this is why the regression equation for the regression line is negative.

### Residual plot for the commodore

Predicted value Predicted value deviation

A residual is defined as the space between a measured and a predicted value in a set of data, the residual plot for the Holden Commodore was calculated in excel. A residual plot is used to test the linearity assumption by plotting the residuals against the EV, the more random the plots on a residual plot are the more appropriate the data is for a linear model. Because the scatterplot for the Holden Commodore appears to follow a linear trend and the residual plot for the car appears to have random plots, it can be assumed that the data for the depreciation of the Commodore is appropriate for a linear model and that the linear assumption has been achieved.

### The correlation coefficient for the Commodore

EV RV

9 6,989

8 7,990

7 10,000

6 12,990

5 14,499

4 15,800

3 18,988

2 25,990 Correlation

Coefficient

1 28,495 -0.97114

The correlation coefficient or Pearson’s r is used to measure the strength of a linear relationship between multiple numerical variables, the correlation coefficient will only ever lie between either +1 or -1. A value closer to +1 is a positive association and a value closer to -1 is a negative association. The correlation coefficient for this investigation was calculated in excel using the =CORREL(A1:A2,B1:B2) function. The correlation coefficient for the Holden Commodore is -0.97114. Because the value of the correlation coefficient is close to the -1 value this shows that the data for the Holden Commodore has a very strong negative association.

It is also worth noting that the set of data gathered for the Holden Commodore has no obvious outliers, this should be noted because the presence or absence of outliers in a set of data will greatly affect the outcome of variables such as the r-squared value or the correlation coefficient if outliers are present and left in the data.

If we compare the correlation coefficient (Pearson’s r ) and the r-squared value of both cars it is clear that while both cars have similar values the Sahara has a higher r-squared and correlation coefficient value. Because the Sahara has higher values it also has a stronger negative linear association and less variation between fitted variables and observed data, The Sahara also has a slower depreciation rate than the commodore which makes it a better car for re-sale value. Because the Sahara has stronger variables it also has a stronger linear function.

## Evaluation to verify results

Based on the above facts and my evaluation of the analytical results is that due to the fact that multiple mathematical concepts have been used to compare data on the two vehicles the solutions achieved are accurate and reliable and show that a linear function arises for both vehicles.

The mathematical concepts that have been used in this investigation have been selected because they can all be used for reliable bivariate data and time series analysis. The concepts used in the assignment are:

- Scatterplots
- Regression lines
- Residual plots
- The correlation coefficient (Pearson’s r)
- The r-squared value

The initial assumptions and observations for this investigation are still accurate and have not changed. The strengths of the findings in this investigation are that the data is easily read, and the scatterplots show the relationship between the two variables easily. The limitations of the investigation are the same as mentioned previously.

## Conclusion:

After reviewing the many mathematical results and facts stated in this investigation it is evident that the hypothesis was justified in stating that a linear function would arise for both the Toyota Landcruiser Sahara and the Holden Commodore. Individually the mathematical strategies showed some limitations such as the scatterplots alone are not able to give a full correlation and the linear regression plot is unable to show causation. When each of these individual equations are used in conjunction with each other a strong and conclusive data pattern can be used when evaluating the presence of a linear function through the form of vehicle depreciation.

In conclusion, through the use of mathematical concepts and the inputting of variables, this investigation has been able to definitively track depreciation of two vehicles that were selected and has proven that linear function does arise when one variable is proportional to the change in another variable.