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Math IA

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Math IA

1. 1. IB Mathematical Studies Internal Assessment:Shoe Size versus Length of Forearms Exam Session: May 2012 School Name: International School Bangkok IB Number: 000307-161 Teacher: Ms. Goghar Date: January 20, 2012 Course: IB Math Studies SL Word Count: 2,338 Name: Jennifer Purgill
2. 2. Purgill 1Introduction Growing up, children always hear all kinds of urban myths about many different topics.Some of these myths include the idea that the length of a person’s arm span is equal to theirexact height, the length of a person’s thumb is always about an inch long no matter who theperson is, the length of your waist from hip to hip is equal to the circumference of your neck, andthat the size of a person’s foot is the exact same as the length of their forearm. As a child whoheard many of these myths while growing up, I was always curious to find out whether or notthey were really true. The myth that was heard most often, though, was the one stating that aperson’s shoe size is extremely close, if not exactly the same, as the length of their forearm. Thismyth will be analyzed using various mathematical processes to test whether it is actually true ornot.Statement of Task The main purpose of this investigation is to deduce whether or not the size of a person’sshoe has a direct correlation with the length of their forearm.Information for the project will becollected from randomly selected students attending the International School of Bangkok. It isnecessary to have a varied group of students tested to ensure that the correlation happens in allpeople, not only a certain race or gender. After data is collected and recorded, a graph will beused as a visual aid to show exactly what the correlation is between the shoe size of a person andtheir forearm length. If the hypothesis is correct, then the results should support the idea that asthe length of a person’s forearm increases, then so does their shoe size. If results are exactenough, it should be proven that the size of a person’s shoe is equal, or very close to equal, totheir forearm length. The main reason I chose this topic for my internal assessment is because I
3. 3. Purgill 2have always heard rumors that the length of people’s forearms and their shoe size are exactly thesame but it will be interesting to see if it can be mathematically proven with proper calculations. Hypothesis: The length of a person’s forearm is directly correlational to the size of their shoe. Ho null hypothesis: The size of a person’s shoe is independent of the length of their forearm. H1 alternative hypothesis: The size of a person’s shoe is not independent of the length of their forearm.Plan of Investigation The data used for this internal assessment was gathered from various students attendingthe International School of Bangkok. The reason data was collected from randomly selectedstudents is to ensure the fact that this occurrence is present in all people, not only people of acertain gender, race, or height. There will be 50 students measured overall, 25 males and 25females. The even number of both males and females measured will ensure that this data doesnot only apply to one gender. They will first have their forearms measured from the point wherethe inside of the elbow ends to the bend of the wrist using centimeters. This information will thenbe recorded on paper. The students will then be asked for their shoe size and that data will berecorded as well. After all 50 students’ information is collected, the shoe sizes will be convertedinto centimeters to see if they are directly correlational, if not exactly equal, to the length of theirforearms. This raw data will be displayed in tables 1 and 2 of the internal assessment.
4. 4. Purgill 3Mathematical Processes/Interpretation of ResultsCollected DataTable 1: Table displaying50 students’ shoes sizes in centimeters and their forearm lengthsStudent Size of Shoe (cm) Forearm Length (cm)1 25.7 25.52 22.8 22.73 25.4 25.54 24.1 24.05 24.1 24.36 24.8 24.77 23.5 23.48 25.1 25.09 24.8 24.610 22.8 23.011 26.0 25.812 25.4 25.513 24.8 25.014 23.5 23.515 24.5 24.516 23.5 23.417 25.4 25.518 23.5 23.419 24.1 24.020 24.1 24.321 24.8 25.022 22.8 23.023 24.1 24.024 23.5 23.025 23.8 24.026 27.6 27.527 26.7 26.528 27.0 27.229 26.3 26.230 26.3 26.131 27.9 28.032 27.9 27.733 28.3 28.234 26.7 26.535 28.3 28.136 28.3 28.537 27.0 27.238 27.6 27.539 27.6 27.5
5. 5. Purgill 440 27.3 26.941 26.7 26.542 26.3 26.243 27.0 27.044 27.9 28.045 27.3 27.246 27.6 27.847 27.6 27.948 26.3 26.549 27.0 27.450 26.7 26.5Sum 1288.1 1287.2Average 25.8 25.7The averages of the x and y columns displayed above were calculated by first finding the sum ofall 50 data points per column then diving that total by 50. A sample calculation to show findingthe average of x for this is shown below:The average of the y column was found using the same equation but substitutes y in place of x:The modes of the shoe size for the 50 students (x) are 23.5, 24.1, and 27.6The mode of the forearm length for the 50 students (y) is 26.5
6. 6. Purgill 5A t-test was performed to check if the difference between a person’s shoe size and their forearmlength are statistically significantly different in the case of women.Figure 1: Results of t-test for all 50 students comparing shoe size and forearm lengthIt was found that the size difference between a student’s shoe size and the length of their forearmis not statistically significant. This means that the size of a student’s shoe and the length of theirforearm are very similar if not exactly the same in most of the measurements presented.Figure 2: Graph displaying the comparison between students’ shoe size compared to thecorresponding students’ forearm lengthsIt can be seen in the graph above that as the size of the students’ shoes increase, the lengths oftheir forearms increase as well. There is a high positive correlation that can be seen between thetwo variables. A line of best fit was used to analyze the data, and it is stated that the correlationbetween the two variables is 0.9976. The linear fit for this graph was calculated using theformula:
7. 7. Purgill 6Table 2: Table displaying chi-squared information where is the students’ shoe sizes (cm) andis the length of the respective student’s forearm 22.8 22.7 517.6 519.8 515.3 9.00 9.00 22.8 23.0 524.4 519.8 529.0 9.00 7.29 22.8 23.0 524.4 519.8 529.0 9.00 7.29 23.5 23.0 540.5 552.3 529.0 5.29 7.29 23.5 23.4 549.9 552.3 547.6 5.29 5.29 23.5 23.4 549.9 552.3 547.6 5.29 5.29 23.5 23.4 549.9 552.3 547.6 5.29 5.29 23.5 23.5 552.3 552.3 552.3 5.29 4.84 23.8 24.0 571.2 566.4 576.0 4.00 2.89 24.1 24.0 578.4 580.8 576.0 2.89 2.89 24.1 24.0 578.4 580.8 576.0 2.89 2.89 24.1 24.0 578.4 580.8 576.0 2.89 2.89 24.1 24.3 585.6 580.8 590.5 2.89 1.96 24.1 24.3 585.6 580.8 590.5 2.89 1.96 24.5 24.5 600.3 600.3 600.3 1.69 1.44 24.8 24.6 610.1 615.1 605.2 1.00 1.21 24.8 24.7 612.6 615.1 610.1 1.00 1.00 24.8 25.0 620.0 615.1 625.0 1.00 0.49 24.8 25.0 620.0 615.1 625.0 1.00 0.49 25.1 25.0 627.5 630.0 625.0 0.49 0.49 25.4 25.5 647.7 645.2 650.3 0.16 0.04 25.4 25.5 647.7 645.2 650.3 0.16 0.04 25.4 25.5 647.7 645.2 650.3 0.16 0.04 25.7 25.5 655.4 660.5 650.3 0.01 0.04 26.0 25.8 670.8 676.0 665.5 0.04 0.01 26.3 26.1 686.4 691.7 681.2 0.25 0.16 26.3 26.2 689.1 691.7 686.4 0.25 0.25 26.3 26.2 689.1 691.7 686.4 0.25 0.25 26.3 26.5 697.0 691.7 702.3 0.25 0.64 26.7 26.5 707.6 712.9 702.3 0.81 0.64 26.7 26.5 707.6 712.9 702.3 0.81 0.64 26.7 26.5 707.6 712.9 702.3 0.81 0.64 26.7 26.5 707.6 712.9 702.3 0.81 0.64 27.0 26.9 726.3 729.0 723.6 1.44 1.44 27.0 27.0 729.0 729.0 729.0 1.44 1.69 27.0 27.2 734.4 729.0 739.8 1.44 2.25 27.0 27.2 734.4 729.0 739.8 1.44 2.25 27.3 27.2 742.6 745.3 739.8 2.25 2.25 27.3 27.4 748.1 745.3 750.8 2.25 2.89 27.6 27.5 759.0 761.8 756.3 3.24 3.24 27.6 27.5 759.0 761.8 756.3 3.24 3.24 27.6 27.5 759.0 761.8 756.3 3.24 3.24
8. 8. Purgill 7 27.6 27.7 764.5 761.8 767.3 3.24 4.00 27.6 27.8 767.3 761.8 772.8 3.24 4.41 27.9 27.9 778.4 778.4 778.4 4.41 4.84 27.9 28.0 781.2 778.4 784.0 4.41 5.29 27.9 28.0 781.2 778.4 784.0 4.41 5.29 28.3 28.1 795.2 800.9 789.6 6.25 5.76 28.3 28.2 798.1 800.9 795.2 6.25 6.25 28.3 28.5 806.6 800.9 812.3 6.25 7.84Sum 1288.1 1287.2 33302.6 33326 33280.5 141.3 142.4Average 25.8 25.7 666.1 666.5 665.6 2.83 2.85The average for the x column displayed in the table above was found by using the formula:This same formula was used to find the averages for the rest of the columns as well, butsubstitute for what data the column contains and finding the sum of the respective columnrather than the column of x.The correlation coefficient ) will be used to test is there is a correlation between students’ shoesizes and their forearm length. The formula for this test is:In this formula, Sxyis the covariance.
9. 9. Purgill 8In this formula for the correlation coefficient ( ), the work for is shown above. In theformula, the work to find and are shown below:When the data from the table is inserted into the calculator, the correlation coefficient (r) for thedata is equal to 0.994.This means that there is a positive direct correlation between the length ofthe students’ forearms and their shoe sizes. Because r=0.994 this means that r2=0.988, whichindicates that there is a strongcorrelation between the length of students’ forearm lengths and thesize of shoe they wear.Chi-squared is a test used to determine whether two factors are independent or dependent of eachother. When using chi-squared, a table of observed and expected data is shown and calculated tocheck if there is a significant difference between the two factors. The formula for this is:
10. 10. Purgill 9Observed values:Table 3: Sample calculation for chi-squared table of observed values A1 A2 Total B1 A B A+B B2 C D C+D Total A+C B+D nExpected Values:Table 4: Sample calculation for chi-squared table of expected values A1 A2 Total B1 A+B B2 C+D Total A+C B+D nTable 5: Table displaying the relationship between the shoe size (cm) and length of the forearms(cm) of the students Size of Shoe (cm)Length of Forearm 22.8-26.0 26.1-28.3 Total(cm) 22.7-25.7 24 0 24 25.8-28.5 1 25 26 Total 25 25 50
11. 11. Purgill 10Figure 3: Graph displaying shoe size (cm) versus forearm length (cm) for the 50 studentsIt can be seen in the graph shown above that it is not mathematically possible to have at least fivedata points for each section of the chi-squared table (see table 5) due to the data points being invery close proximity to each other. This supports the alternate hypothesis that a person’s shoesize is not independent of their forearm length.Degrees of freedom measure the number of values in the final calculation that are free to vary.The formula for degrees of freedom is:
12. 12. Purgill 11Discussion/ValidityLimitationsThroughout the investigation, there have been a few limitations present which may affect thereliability of the outcomes found. One limitation of the experiment is that all of the data wascollected from students of the International School of Bangkok. Because of this, the participantsdo not represent a random sample but instead an opportunity sample and there was not a verywide variety of types of participants although the students were from many different cultures andbackgrounds. All of the students measured were in a similar age group (ages 14 to 18) and it ispossible that not all of the students had grown to their full potential (i.e. their arms still haveroom to grow and or their shoe size is not as large as it will be when they have stopped growing).Another limitation to the data was that all measurements were done using simple rulers andtherefore the numbers recorded may not have been 100% accurate as well as the fact that it isdifficult to determine the exact location where a person’s forearm begins and ends. Because ofthis, some of the measurements may have been off by a few millimeters or centimeters for thecolumn of forearm length in both the males and females measured.
13. 13. Purgill 12ConclusionDespite the limitations mentioned in the previous section, the data overall supports thealternative hypothesis and rejects the null hypothesis that the size of a person’s shoe isindependent of the forearm length of the person. This means that in fact a person’s shoe size isdependent on a person’s forearm length. Furthermore, the data not only supports the alternativehypothesis, but it also shows that there is not an extreme significance between a person’s shoesize and the length of their forearm. The data is also not biased towards one gender or race overanother because there was a wide variety of people with different backgrounds and ancestrymeasured as well as equal amounts of both genders tested. Because of this and the results foundit can be determined that the shoe size is in fact directly correlational to the length of a person’sforearm.