International Baccalaureate, Math AI HL, Internal Assessment (Statistics and Probability), High school final essays of Mathematics

Download International Baccalaureate, Math AI HL, Internal Assessment (Statistics and Probability) and more High school final essays Mathematics in PDF only on Docsity! 1 Mathematical Exploration Does swimming with competitive swimsuit increase a swimmer’s performance? Course: MAI Level: HL Number of pages: 2 INTRODUCTION Through a rigorous analysis of a shark’s skin, scientists managed to attribute its monumental speed to the V-shaped ridges on its skin called dermal denticles. This texture decreases drag and turbulence around the shark’s body allowing the surrounding water to glide more efficiently. In the late 1990s, the world’s leading swimwear company, Speedo, used shark skin as a model for the revolutionary Fastskin competition swimsuit. It was constructed with built in ridges emulating sharkskin and composed of super stretch fabric to help compress the swimmers’ muscles. This resulted in a drag reduction and muscle vibration which allowed swimmers to move through water faster than ever before. Through continuous research and testing, Speedo modified and improved the Fastskin suit by minimizing drag and increasing grip, which optimized overall performance. Today, the majority of the leading swimwear brands, such as Arena, TYR, and Jaked, have launched their own version of the Fastskin suit. This type of swimsuit, known as a “tech suit”, is used during swimming competitions to enhance the performance of competitive swimmers. The suit's compressive properties improve muscle oxygenation, retain the body in a more streamlined position, repel water, and increase flexibility. My interest in this topic stems from a personal experience. From a very young age, my parents wanted me to take swimming lessons. However, at 12 years old, I decided I wanted to join a club and start swimming competitively. In 2017, I participated in my first competition and noticed that I was the only one wearing a regular training swimsuit. I felt ignorant for not knowing why all my teammates were wearing knee length swimsuits and how it differentiated from a regular training swimsuit. A few months later, I was gifted a tech suit and competed with it unaware of the impact it had on my swimming. In 2021, due to an injury, I decided to take some time off swimming, until a recent occurrence. I was cleaning my closet and found my old tech suit and realised I never got an answer to all the questions I had. 5 Afterwards, I will conduct a two sample T-test at a 95% confidence level to observe whether there is a statistically significant difference between the mean swimming time with or without a Fastskin suit. The null and alternate hypothesis for this T-test are as follows: 𝐻! ∶ 𝜇" = 𝜇# 𝐻$ ∶ 𝜇" > 𝜇# • Where 𝜇! refers to the mean time taken to swim 100 meters with a Regular training suit, and 𝜇" refers to the mean time taken to swim 100 meters with a Fastskin suit. However, to perform this statistical test, I must first calculate the sample mean and sample standard deviation of the time taken to swim 100 meters with and without a Fastskin suit. This is done using the following formulae: T" = 1 10&𝑡! "# !$" ; 𝑠 = +∑ (𝑡! − 𝑇0)%"# !$" 9 • Where 𝑡# is the time taken to swim 100 meters during trial 𝑛. CONDUCTING THE TWO SAMPLE T-TEST: 100 METERS In the section below, I present the data that I have collected, the calculations of the sample mean and standard deviation, as well as the results of the two-sample t-test. To simplify the process, I have decided to round all numerical values to 4 significant figures (this will be continued throughout the rest of investigation). 1 - Raw Data for 100-meter swim The following table displays the time taken to swim 100 meters (in seconds) with each type of swimsuit over 10 trials: Type of swimsuit Trial number 1 2 3 4 5 6 7 8 9 10 Regular swimsuit 65 64 66 65 64 66 63 64 63 63 Fastskin Suit 63 64 60 64 65 63 63 63 64 65 6 2 – Sample mean and standard deviation for 100-meter swim With this data, I calculated the average time taken to swim 100 meters over 10 trials with a regular training suit and a Fastskin suit using the previously mentioned formulae: 𝑇0& = 1 10&𝑡! "# !$" = 65 + 64 + 66 + 65 + 64 + 66 + 63 + 64 + 63 + 63 10 = 64.30 𝑠 𝑇0' = 1 10& 𝑡! "# !$" = 63 + 64 + 60 + 64 + 65 + 63 + 63 + 63 + 64 + 65 10 = 63.40 𝑠 As well as the sample standard deviation, again using the previously mentioned formulae: 𝑠" = '∑ ('!()*")#$ !%# , = '(-.(-/.1) &2(-/(-/.1)&2 ⋯ 2(-1(-/.1)&2(-1(-/.1)& , = '$5.$ , = 1.160 s 𝑠# = '∑ ('!()*')#$ !%# , = '(-1(-1./) &2(-/(-1./)&2 ⋯ 2(-/(-1./)&2(-.(-1./)& , = '$6./ , = 1.430 s 3 – Two Sample t-test and p-value Next, as mentioned previously, I will be conducting a t-test to see if there is a statistically significant difference between the mean time taken to swim 100 meters with or without a Fastskin suit. It is noted that I am assuming that the time taken to swim 100 meters follows a normal distribution. Moreover, I will also assume that the standard deviation with a regular training suit and a Fastskin suit is the same, so I will be performing a pooled two sample t-test. To facilitate this next step of my investigation, I have decided to summarise all the information I have collected previously in the table below: Regular Swimsuit Fastskin Suit 𝜇" 𝑠" 𝑛" 𝜇# 𝑠# 𝑛# 64.30 s 1.160 s 10 63.40 s 1.430 10 7 After imputing these values in my calculator, I found that the p- value of this t-test is 0.06975. 4 – Conclusion Since the p-value is greater than the significance level (as 0.06975 > 0.5), there is not enough evidence to reject the null hypothesis. Therefore, I cannot conclude that there is any statistically significant difference between the time taken to swim 100 meters with a regular training suit or a Fastskin suit. DISCUSSION The result of my t-test suggested that the type of swimwear used to swim 100 meters had no impact on my performance. This conclusion was not only disappointing, but intriguing. I wondered why swimmers would spend money on a swimsuit that did not achieve its intended purpose. Hence, I decided to carry out additional research and came across an article by Swimming World Magazine titled “An Explanation of How Tech Suits Benefit Swimmers” stating that wearing a competitive swimsuit can reduce drag by 4.4% to 6.2% and reduce the amount of energy needed to swim by 4.5% to 5.5%1. This made me realise that the Fastskin suit may have a greater benefit when the distance of the swim is increased. To test this hypothesis, I decided to conduct a second t-test to investigate whether swimming a longer distance would lead to a more significant statistical difference between the two swimsuits. To do so, I must collect more data using the exact same methodology as before. The same external variables will be controlled to maintain consistency. The only difference is that I will be swimming 1000 meters 1 Carly McAdam et al, 2023 10 After imputing these values in my calculator and performing t-test, I obtained that p-value for this t-test to be 0.000385. 4 – Conclusion Since the p-value is smaller than the significance level (0.000385 < 0.5), I have sufficient evidence to reject the null hypothesis and accept the alternate hypothesis. Therefore, I can conclude that there is a statistically significant difference in the average time taken to swim 1000 meters with a Fastskin swimsuit compared to a regular swimsuit. EVALUATING THE EFFICIENCY OF THE FASTSKIN SUIT To understand just how significant racing with a tech suit is, I decided to analyse my swimming pattern throughout the 1000-meter swim. To do so, I had an idea to calculate the average speed of swimming throughout the 1000-meter swim. 1 – Raw Data of 1000-meter swim recorded every 100 meters To begin, I arranged the time taken to swim every 100 meters of the 1000-meter swim in the table below: Type of Swimsuit Trial Number Distance (m) 100 200 300 400 500 600 700 800 900 1000 Regular 1 72 151 234 320 407 493 581 663 748 824 2 74 150 230 315 403 492 583 664 745 820 3 75 155 239 326 415 502 588 672 759 836 4 78 155 237 323 412 502 591 674 759 833 5 76 155 239 327 418 506 596 683 771 850 Fastskin 1 75 152 231 312 394 476 560 640 716 791 2 72 145 221 300 381 463 546 625 701 775 3 77 156 236 320 403 486 568 646 723 800 4 76 155 237 322 405 489 572 653 731 806 5 73 148 225 305 387 467 549 627 702 775 11 Then, to investigate my swimming pattern, I needed to find the average (total) time taken to swim 100, 200, …, 1000 meters with both swimsuits. This is done using the formula below: 𝑇0* = 1 5&𝑡!,* ) !$" = 824 + 820 + 836 + 833 + 850 5 = 832.6 𝑠 • Where 𝑡#,: is the total time taken to complete the kth 100 meters of the swim during trial 𝑛, and 𝑇1: is the average total time taken to complete the kth 100 meters of the swim. Eg. Calculating the average time taken to complete the first 100 meter of the 1000-meter swim while wearing the regular swimsuit: 𝑇0" = 1 5&𝑡!," ) !$" = 72 + 74 + 75 + 78 + 76 5 = 75 𝑠 I conducted the process detailed above for all k between 1 and 10 and for both swimsuits and compiled the information into the following table. 2 – Calculating speed over the 1000-meter swim. The data above was used to find the average speed over time using the formula: 𝑣8 = ∆ :;<'=>?@ ∆ ';A@ = $!! ∆ )( • Where 𝑣: is the average speed in the kth 100 meters of the 1000-meter swim, and Δ𝑡: is the time taken to swim the kth 100 meters of the 1000-meter swim, which may be computed Δ𝑇: = 𝑇: − 𝑇:(8 . Type of Swimsuit Distance (m) 100 200 300 400 500 600 700 800 900 1000 Regular 75 153.2 235.8 322.2 411 499 587.8 671.2 756.4 832.6 Fastskin 74.6 151.2 230 311.8 394 476.2 599 638.2 714.6 789.4 12 Eg. Calculating the average speed to swim the first 100 meters with the regular training suit: 𝑣$ = $!! B. = 1.333 m/s The results were inserted in the table below: 3 – Analysis of the data Afterwards, I decided to plot the data from the table above into a speed over distance graph to investigate whether there was an observable difference between my swimming pattern with a Fastskin suit and my swimming pattern with a training suit. This graph may allow us to understand why there is such a significant difference between the time taken to swim 1000 meters with a Fastskin Suit and with a regular training suit. As shown above, my speed when swimming with a regular training suit (orange line) decreased at a higher rate and had a significantly lower minimum speed (which was 6.25% slower). Additionally, when swimming with a Type of Swimsuit Average Speed during the 𝒌𝒕𝒉 𝟏𝟎𝟎 (m/s) 1 2 3 4 5 6 7 8 9 10 Regular 1.333 1.279 1.211 1.157 1.126 1.136 1.126 1.199 1.174 1.312 Fastskin 1.340 1.305 1.269 1.222 1.217 1.217 1.208 1.263 1.309 1.337