example 1

This is only for reference.  The level achieved for this kind of essay would be in the range of 2 to 3.

 

Internal Exploration

IB Mathematics Standard Level

 

  

Topic

Ways in which one can model the relationship between car crashes and the age of the driver, using linear regression, verifying with the normal distribution.

 

 

 

 

 

 

 

 

 

Rationale

I have decided to center this mathematical exploration around the topic of age and car accidents. For this assessment, I will be finding the relationship between age, gender, and car crashes in the UK. Basically, how is age affecting the possibility of having an accident? As a car enthusiast myself, I realized while I was doing my driver’s license and I went to the highway for the first time, it is going really fast and I can get into a car accident really easily. Since I am very interested in cars and I know how fast they can go I know that they can also be very dangerous if you go too fast when you are inexperienced. I want to investigate the relationship between the different age groups and car accidents are. I am very interested in it because I love cars and it can be a warning for younger and older people to drive safer and more responsibly.

Introduction

Road traffic collisions are a major global concern. They account for more than 1.2 million deaths worldwide each year and an even larger number of serious injuries (World Health Organization,

2015). Obtaining a better understanding of the factors that contribute to driver crash risk is critical for the development of effective road safety policies and initiatives. The car accident in the Member States of the European Union claims about 25.600 lives and leave more than 1,4 million people injured in 2016 (IRTAD Annual Road Safety Report). Many measures have been taken to reduce traffic accidents. After a pilot phase of about 3 years, the Council endorsed the “Report from the Commission on the progress with the project and its future prospects” (European Commission).

Statement of Task The objective of this investigation is to determine how age is affecting the chance of causing a car accident, in order to find out what age it is more likely to cause an accident. The type of data will be collected from the website “National safety”. The data of these statistics will be used to determine what age is more likely to cause an accident and verify with Normal Probability Distribution.

Plan of Investigation:

I investigate and model the relationship between age, gender and car crashes. I will use a correlation and regression equation. Furthermore, I use the mean and standard deviation to compare with a normal distribution.

Data, Materials, and Methods

The youngest and oldest drivers were the fewest in number in the population. Specifically, drivers aged 17–20 years were 87% fewer in number, and drivers aged 70 years and above were 46% fewer in number than drivers aged 40–49 years who were representing the largest driver age group in the population (National Safety Council). For both genders, fewer trips were made at night than during the daytime or evening. Only 8% of trips were taken by men and 5% of trips were taken by women which were during the night (National Safety Council). 

Number of total car crashes by age, gender group in the UK in the EU, year 2016                  (National Safety Council)

Crash Counts
Males	Day	Evening	Night	Total	Females	Day	Evening	Night	Total
17-20	2,801	938	2,251	5,990	17-20	949	342	568	1,859
21-30	4,101	1.347	2,616	6,718	21-30	2,069	539	610	3,218
31-40	4,702	1,218	1,637	7,557	31-40	2,228	426	324	2,978
41-50	4,251	948	1,112	6,311	41-50	1,831	333	215	2,379
51-60	3,086	591	578	4,255	51-60	1,079	182	108	1,369
61-70	1,682	286	234	2,202	61-70	553	79	40	672
71-80	1.188	155	101	257	71-80	474	50	22	546

Table 1: This table shows how the different age groups, gender groups, and daytime who were included in the car crashes in the year 2016. [14,959 drivers on average]

1. Calculations: Male car crashes

Males	Day	Evening	Night	Total
17-20	2,801	938	2,251	5,990
21-30	4,101	1.347	2,616	6,718
31-40	4,702	1,218	1,637	7,557
41-50	4,251	948	1,112	6,311
51-60	3,086	591	578	4,255
61-70	1,682	286	234	2,202
71-80	1.188	155	101	257

Males car crashes Regression curve

On the x-axis the age and on the y-axis total number of car crashes taken. [TI Smartview 84Plus family]

Linear Regression Equation:
y=ax+b, where  a= -105.475 and b=9502.08

Therefore, y= – 105.475x+9502.08

And the correlation coefficient is -0.85 which reveals that there is a strong negative relation between drivers’ age and car crashes. As the drivers’ age is increasing the number of car crashes is decreasing. The car crash rates involvement by driver age and gender were calculated based on the estimated number of licensed drivers. However, the youngest age group had a high relative risk of being involved in a fatal and non-fatal accident compared to the 61- to 70-year-old group.

For example, Let us consider the driver’s age group of  25, 35 and 45  years.

The number of car crashes by each age group can be found by

y = – 105.475x+9502.08               Substituting x=25, 35 and 45, we get

y  = – 105.475(25)+ 9502.08             y  = – 105.475(35)+ 9502.08      y  = – 105.475(45)+ 9502.08

= 6865.205                                        = 5810.455                                 = 4755.705

The percentage of crashes in a year, therefore

y = 6865.205/14959 = 0.459          y = 5810.45/14959 = 0.388           y = 4755.705/14959 = 0.318

= 45.9%                                                                     = 38.8%                                       = 31.8%

The risk of car crash for the age group of 25 years drivers is more than other age groups.

2. Calculations: Female car crashes

Females	Day	Evening	Night	Total
17-20	949	342	568	1,859
21-30	2,069	539	610	3,218
31-40	2,228	426	324	2,978
41-50	1,831	333	215	2,379
51-60	1,079	182	108	1,369
61-70	553	79	40	672
71-80	474	50	22	546

Females car crashes Regression curves

On the x-axis the age and on the y-axis total number of car crashes taken.[TI Smartview 84Plus ]

Linear Regression Equation:
y=ax+b, where  a = -38 and b=3570.143

Therefore, y= – 38x+3570.143

And the correlation coefficient is -0.77 which reveals that there is a strong negative relation between drivers’ age and car crashes. The 30–39 year age group had the highest frequency of daytime crashes, while the 21–29 year age group had the highest crash frequency during the evening and night hours.

For example, Let us consider the driver’s age group of  25, 35 and 45 years.

The average number of car crashes by this person can be found by

y= – 38x+3570.143                              y= – 38x+3570.143                  y= – 38x+3570.143

y= – 38(25)+3570.143                        y= – 38(35)+3570.143               y= – 38(45)+3570.143

= 2620.143                                       = 2240.143                                 = 1860.143

The percentage of crashes in a year, therefore

y = 2620.143/14959 = 0.175          y = 2240.143/14959 = 0.149    y = 1860.143/14959 = 0.124

= 17.5%                                                                     = 14.9%                                       = 12.4%

The risk of car crash for the age group of 25 years drivers is more than other age groups.

3.Total Males and Female car crashes Comparison

Age Group	Males	Females	Total
17-20	5990	1859	7849
21-30	6,718	3218	9936
31-40	7,557	2978	10535
41-50	6,311	2379	8690
51-60	4,255	1369	5624
61-70	2202	672	2874
71-80	257	546	803

Male and Females car crashes Regression curves

On the x-axis the age and on the y-axis total number of car crashes taken.[TI Smartview 84Plus ]

Analysis: The male and female car crash regressions graphs comparison above clearly indicate that the rate of male car crashes is decreasing faster than female car crashes. The number of car accidents decreased slowly with age at 70+. Men and women showed similar trends but with lower frequency for women. The risk of car crashes for teens was higher than that of drivers aged 70-79. Clearly, there is more risk of crashing in males between the age limit 30-39 and in females 21-30.

Evaluation and Data Analysis

The rates of car accident rates are usually used to find out the risk of different groups of drivers. Here I was applying a crash risk method, for which the number of crashes is corresponding in size to the amount of driving exposure. I also found from the comparison graph of male and female car crashes that normal distribution crash rates were reduced steeply from age 17–20 years through age 71-80 years. Older drivers had reduced crash involvement and death risks when using the crash risk. Finally, nighttime driving among teen drivers accounted for a smaller proportion of their crash risk than implied by crash rates. Young males are reported to have higher crash rates at night compared to daytime. In both male and female drives of the age group of 31-40 have a high chance of a car crash.

Conclusion

Road accidents are a big global problem. A wealth of research into road safety has evaluated driver characteristics such as age and gender, which are associated with an increased risk of accidents.

I also found out about the nonfatal crash risks among different groups. In addition to crash involvement, driver’s age and gender have also been shown to affect the severity of crash outcomes. This research applied a new approach to model crash risks based on data that bears a linear relation with crashes. This work has important practical implications for improving road safety initiatives. This is an important comparison for identifying the risk of driving conditions.

 

Bibliography/References

1. “IRTAD Annual Road Safety Report 2019” Accessed 20 October 2019 Web https://www.nrso.ntua.gr/irtad-annual-road-safety-report-2019/
2. “European Commission”’ Web 8 October 2019. <https://ec.europa.eu/info/statistics_en>
3. “CARE” Community database on Accidents on the Roads in Europe, Accessed 20 October 2019,  https://www.irfnet.ch/files-upload/member_area-pdf-files/RS_WG/CARE-and-the-European-Road-Safety-Observatory.pdf
4. “Austrian Road Safety Board” Web Accessed 10 November 2019. https://www.kfv.at/
5. “World Health Organization”, 2015
6. “National Safety Council” accessed on 5 December 2019 https://www.nsc.org/
7. “All Graphs” by me. TI Smartview CE for the TI 84 family
8. Normal Distribution accessed 25 Jan 2020 https://homepage.divms.uiowa.edu/~mbognar/applets/normal.html