Math Exploration : Rationale:
My personal interest in automobile and engines have been strong and passionate since a very young age. I have always wanted to study about cars and I have always been interested in specifications of cars around my environment. Since then, I have always paid very close attention to the new cars in the Indonesian market. Out of the numerous different brands and types of cars seen in my environment, I have realised that the most common type of cars in Indonesia are family cars. This may be due to the expense of owning a car, therefore, families in Indonesia usually buy cars that their family can use on a daily basis with good efficiency.
Families in Indonesia usually buy cars that are efficient in fuel (therefore lower costs) and at the same time have the capabilities and power to support their family’s daily needs. This means that the 2 of the main factors that affect this the most would be the average fuel economy of an automobile (mileage) and also it’s horsepower (bhp). From this, I have decided to find whether or not these two factors are interrelated. Therefore, I have decided to take my passion of cars and use it to find the correlation between the horsepower and the average fuel economy of family cars in Indonesia.
Introduction:
To find the correlation of the average fuel economy and the horsepower of cars in Indonesia, car brands will have to be chosen for data. Indonesia itself has a very vast variety of different cars and car brands. Therefore, from all the car brands to chose from, I have decided to choose the top 6 most common car brands that are found in Indonesia by researching about the share percentage of the Indonesian automobile market. From the car
May 2016 Math Exploration Internal Assessment Pravir Daryanani
brands chosen, I will choose all the family sized cars that are still being sold in the Indonesian automobile market. From the data collected, I will then chart the data and calculate the line of best fit for the data. For specific results and to find whether or not there is a correlation between the factors or not, I have chosen to use Pearson’s correlation coefficient law to investigate the correlation. To find the line of best fit, I have chosen to use the “Least Square Method”.
Shares of the Indonesian automobile market by brand (figure 1.0):
(Indonesian) (figure 1.0)
I was unable to find the most latest car data in Indonesia due to source limitations. Therefore, the most latest data that I could find was 2012’s (first half year) share of the Indonesian automobile market. From the graph above, we can clearly tell the top 6 most common car brands in Indonesia at the given time were Toyota, Daihatsu, Mitsubishi, Suzuki, Nissan and Honda, therefore, I have picked these brands.
I have picked various family cars by the top six brands as shown above (Toyota, Daihatsu, Mitsubishi, Suzuki, Nissan and Honda) and gathered the average fuel economy and the
May 2016 Math Exploration Internal Assessment Pravir Daryanani
horsepower data of the cars. The data I have obtained was from Carbay which is a secondary source because the data that this source collects is from other source’s data itself which conveys that the data obtained is secondary data. I have tabulated the data as shown below (figure 1.1):
Raw Data Table:
(figure 1.1)
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From the tabulated data shown previously, I have then taken all the data and processed it to create a linear regression line from the coordinated data as seen below (figure 1.2). From the coordinated data, the observation for the correlation began with a linear regression line in order to understand the general relationship between the average fuel economy and the horsepower and to identify the appropriate regression line that would fit the data. By using Microsoft Excel, I calculated the linear regression line or line of best fit as shown below (figure 1.2).
(figure 1.2)
To get the equation of the line, the formula of y=mx+b must be used and to solve this formula, a method called the “least squared method” can be used. I have decided to use this method and I have also self-taught this method from websites online. According to the
May 2016 Math Exploration Internal Assessment Pravir Daryanani
least squared method, the following formulae will give the slope of the line of best fit or the m value and also the y-intercept value or the b value in the formula of y=mx+b:
The definitions of the terms are as shown:
–
– – –
→ Mean of all the y-values
→ Mean of all the x-values ∑ → Sum of
X i → The ith term of X
1. To find the m value, I have to find the mean of all the y-values and the mean of all the x values to be applied to the formula as shown previously. To find the mean of the x or y values, I have to sum all the x or y values and then divide the sum by the number of x or y values. After calculating the mean of all the x and y values, I have arrived at the answers shown below:
= 12.36
= 130.71
To attain these values and the other values needed to solve the formula for m, I have created an excel spreadsheet as shown below:
May 2016 Math Exploration Internal Assessment Pravir Daryanani
After obtaining these values, I have applied it to the formula to solve m:
m = 3020.6329 ÷ 468.0561 = -6.4536
After obtaining the m value, I then applied the value to the formula to solve b:
May 2016 Math Exploration Internal Assessment Pravir Daryanani
b = 130.71 – (-6.4536 x 12.36) b = 210.47
Finally, after achieving the m and y values, I applied the values to the formula y=mx+b to find the equation of the line.
Equation of the line: y= -6.4536x +210.47
I have checked if my results for the equation of the line that I have calculated is correct by using Microsoft Excel to calculate my results and find the equation of the line for me. On the graph shown previously (figure 1.2), the equation of the line that was calculated by Microsoft Excel is stated in the graph. I have then compared the result calculated by excel and result that I have calculated and from this, I have found that the equation of the line that I is stated in the graph is the same result that I have achieved. Therefore, this suggests that my result is correct.
To specifically investigate the statistically significant correlation between the average fuel economy and the horsepower of the automobiles shown, I have decided to use Pearson’s correlation coefficient law. According to Pearson’s correlation coefficient law, to find the ‘r’ value (how perfect the correlation is), the formula as seen below must be used:
r=∑((X-M y) (Y-M x) )/√((∑(X-M x) 2 )(∑(Y-M y) 2 ))
Definitions of the terms: ∑: The sum of
X: X Values
Y: Y Values
Mx: Mean of X Values
My: Mean of Y Values
X – Mx & Y – My: Deviation scores
(X – Mx)2 & (Y – My)2: Deviation Squared
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To find the values that will be applied to the formula, I have created a table to attain the values easily as seen below (figure 1.3):
From the values attained, I have applied them to the formula shown previously:
X Values:
∑X → 5359
M x → 5359/41 = 130.707 ∑(X – M x) 2 → = 44178.488
Y Values:
∑Y → 506.75
M y→ 506.75/41 = 12.36 ∑(Y – M y) 2 → = 468.056
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X and Y Values combined: N (number of values of X and Y) = 41
∑(X – M x) (Y – M y) = -3020.633
Applying to the formula for r:
r=∑((X-M y) (Y-M x) )/√((∑(X-M x) 2 )(∑(Y-M y) 2 )) r = -3020.633 / √[(44178.488)(468.056)] = -0.6643
r2 = 0.4413
The r2 value can range between 0-1. The r2 shows how perfect the correlation between the factors chosen are. Moreover, if the r2 value is 1, it means there is a perfect positive correlation between the factors, whereas if the r2 value is 0, it means that there is a perfect negative correlation between the factors. Therefore, from the final r2 value obtained from Pearson’s correlation coefficient law (0.4413), The average fuel economy and the horsepower of the automobiles in Indonesia does not have a very strong correlation whatsoever. Further studies can be done by comparing the ‘r’ value to critical values that confirm whether or not the 2 factors correlate or not.
Conclusion:
After obtaining the results and conducting the exploration, the final results imply that there may be other factors that could affect the average fuel economy or the horsepower of an automobile. However, these results can still be significant if one wants to find out if the average fuel economy of a car could affect the car’s horsepower. There were some limitations in conducting the experiment; the data were only from 6 car brands in indonesia and the car brands chosen were from the Indonesian automobile market shares of 2012 so the car brands used now could be different. Overall, these results can still serve as a basic understanding in
May 2016 Math Exploration Internal Assessment Pravir Daryanani
predicting and understanding the relationship between the average fuel economy and the horsepower of automobiles. As my intention to answer this question is also because I want to find an automobile that is both efficient and powerful enough for the average family in Indonesia, according to my results, there are many automobiles that meet the requirements of an average family in terms of power and mileage. However, I have chosen a few cars that are both efficient and powerful enough to perform as a family car from the cars I have chosen to investigate, these cars are shown below:
– Mitsubishi Outlander (new) (SUV) – I have chosen this automobile as it gives 170 horsepower and can cover about 12.75 kilometres per litre.
– Honda mobilio (MUV) – I have chosen this automobile as it gives a whopping 17.3 kilometres per litre and it has a decent 120 horsepower engine.
– Toyota Corolla Altis (Sedan) – I have chosen this automobile because it gives a decent 13.2 kilometres per litre and a suitable 150 horsepower engine to meet the requirements as a family car.
As seen from the cars chosen above, each car is different in type for each family’s preferred choice for their car. There are 3 types that I have chosen, a Sedan, MUV and a SUV. I have chosen these cars as they are appropriate to be used as a family car in Indonesia with enough power and efficiency to support the family’s daily needs.
Reflection and Evaluation:
The investigation has met my aim shown in my introduction and has given me the answer to my research question: Is there a correlation between the horsepower and average fuel economy of automobiles in Indonesia. As seen previously in my conclusion, I have chosen the 3 cars that I believe to be appropriate for an average family in Indonesia to further answer
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my question and suggest options for a family in Indonesia with the requirements needed in a car as shown previously.
Overall, throughout the investigation, the results that I have achieved has helped me to come to a reasonable answer; there may not be a strong correlation between the horsepower and average fuel economy of a car in Indonesia. There may be some limitations in this answer as this may be due to the fact that there are many other factors that can affect the average fuel economy and horsepower of a car in Indonesia. Take for example, the petroleum used for a car can significantly affect the performance of a car or also, the equipment in the car provided can usually take up power in the car thus resulting in a decrease in the fuel economy of the car. The source used to obtained the information and data needed for my investigation, Carbay, is a secondary source and takes secondary data from other websites. This may result in inaccurate data or false data. The cars that I have chosen may not be 100% appropriate for families in Indonesia as the families may also desire other requirements in a car that the cars I have chosen may not equip. This investigation may however, be important for other people as they might also want to find if there is a correlation between the factors chosen and also if they want to find out what cars in Indonesia have the capabilities of good efficiency and power. Also, if someone else would like to do an investigation on this topic or something similar, they could use my investigation as an example and they may want to further investigate on cars in other countries and use different methods of finding the correlation between the 2 factors for better results. If I could reinvestigate this topic, I would take other factors that can affect the automobile’s power or average fuel economy performance.
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Bibliography and Works cited:
Carbay. Carbay.com, n.d. Web. 15 May 2016. < https://www.carbay.co.id/ >.
"Indonesian Automobiles." Selective Rationality. N.p., n.d. Web. 28 Apr. 2016. < http://selectiverationality.com/indonesian-autos-bad-like-brazil-but-for-different-reasons/ >.
"Line of Best Fit (Least Square Method)." Line of Best Fit (Least Square Method). HotMath, n.d. Web. 19 May 2016. < http://hotmath.com/hotmath_help/topics/line-of-best-fit.html >.