Math Internal Assessment
- 1. IB Math Studies Internal Assessment:<br />Body Length and shoe Size<br />Name: Jan Rojthana<br />Date: November 2010<br />Course: IB Math Studies<br />Period: 1<br />Introduction:<br />Shoeprint length and shoe size is positively correlated and increase linearly. However, shoe size can only give an estimation of height. Difficulties include variation in the style of the shoe and heel that influences the variability of the imprint left by shoes of the same size.<br />Shoe size is usually proportional to height. In general, shorter people have smaller feet, and taller people have larger feet. These measurements are not always exact, but statistically, height can be predicted by shoe size. Therefore, I would suppose that if I ask random people for their shoe size, the smaller people will not exactly have smaller shoe size while the taller will not exactly have larger shoe size.<br />Statement of Task:<br />In the investigation, I am going to collect and analyze data on the height and shoe size of 20 women and 20 men in International School Bangkok by choosing from their random heights. The aim of the experiment is to find how the height affects the shoe size or a correlation between two variables. Research question states that does a relationship between the two variables, the height and shoe size in a group of women and men exist, or do they vary independently of each other? The data collected is analysis to study the relationship between the human shoe sizes and the individual’s height.<br /> <br />Hypothesis:<br />As the height of a person increases, the shoe size should also get bigger to balance the height. Therefore, there should be a relationship between height and shoe size.<br />The Measurement:<br />The type of data that will be collected is the height and shoe size of men and women. The best-fit line was drawn through the scatter plot (shoe size versus height) to determine if a correlation between foot size and height was observed.<br />Data Collection:<br />Male Students#of peopleShoe size (US)Height (cm)18.017228.017338.516549.017049.016869.517079.5170810.0168910.01731010.01731110.01731210.01741310.01751410.51831511.01831611.51881712.01751812.01871913.01812013.0185Average:10.2175<br />Female Students#of peopleShoe size (US)Height (cm)15.015525.015736.015746.515156.515367.015777.016387.016897.0170107.5148118.0163128.0173138.0175148.5163158.5165168.5172179.0155189.0164199.01632010.0175Average:7.6162<br />Table2: The table shows the collection of height of random men and their shoe size.Table1: The table shows the collection of height of random women and their shoe size.<br />Data Analysis:<br />Height versus Shoe Size (Female)<br />Graph 1: This graph shows the shoe size’s relationship to the height of female students. The lines of the connecting the two variables shows the positive correlation.<br />Height versus Shoe Size (Male)<br />Graph 2: This graph shows the shoe size’s relationship to the height of male students. The lines of the connecting the two variables shows the positive correlation.<br />Standard Deviation Calculations: <br />The standard deviation for the height of female: <br />Sx= x2n- x2<br />Sx= 2624420- 8.12<br />Sx= 35.31<br />The standard deviation for the shoe size of female: <br />Sy= y2n- y2<br />Sy= 57.7620- 0.382<br />Sy= 1.66<br />The standard deviation for the height of male: <br />Sx= x2n- x2<br />Sx= 3062520- 8.752<br />Sx= 38.14<br />The standard deviation for the shoe size of male: <br />Sy= y2n- y2<br />Sy= 104.0420- 0.512<br />Sy= 2.22<br />Least Squares Regression:<br />The least Squares Regression for female students<br />y-y= SxySx2 (x-x) Where Sxy =xyn- x y.<br />Sxy= 1231.220- (8.1)(0.38)<br />Sxy=58.482<br />Then,<br />y-y= SxySx2 (x-x)<br />y-0.38= 58.4821246.80 (x-8.1)<br />y-0.38=0.046906 (x-8.1)<br />y=0.05x-6.4×10-5<br />The least Squares Regression for male students<br />y-y= SxySx2 (x-x) Where Sxy =xyn- x y.<br />Sxy= 178520- (8.75)(0.51)<br />Sxy=84.79<br />Then,<br />y-y= SxySx2 (x-x)<br />y-0.51= 84.791454.66 (x-8.75)<br />y-0.51=0.058289 (x-8.75)<br />y=0.058289x-2.875×10-5<br />Pearson’s Correlation Coefficient: Pearson’s Correlation Coefficient for female student:<br />r= SxySxSY<br />r= 58.48235.31(1.66)<br />r= 0.998<br />r2=0.996 <br />Pearson’s Correlation Coefficient for male student:<br />r= SxySxSY<br />r= 84.7938.14(2.22)<br />r= 1.001<br />r2=1.003 <br />Data Analysis:<br />Best fit line of Height versus Shoe Size (Female)<br />Graph 3: This graph shows that there are some weak positive correlations of female’s heights and shoe size.<br />Best fit line of Height versus Shoe Size (Male)<br />Graph 4: This graph shows that there are some weak positive correlations of male’s heights and shoe size.<br />Chi- Square Test: <br />Observe Values: <br />B1B2TotalA1ABA+BA2CDC+DTotalA+CB+DN<br />Expected Values: <br />B1B2TotalA1a+b(a+c)Na+b(b+d)NA+BA2a+c(c+d)Nb+d(c+d)NC+DTotalA+CB+DN<br />Observe Values: <br />Below 154155-164165-174175-184Total5-7.53520108-10.5051542411-13.500066Table 3: The first column indicates the shoe size of both male and female while the top row indicates the height of both male and female. Total310171040<br />Expected Values: <br />Below 154155-164165-174175-184Total5-7.50.752.54.252.5108-10.51.8610.262411-13.50.451.52.551.56Total310171040<br />Table 4: This table shows the expect value indicates to table 3. <br />Chi- square test: <br />ℵ2= ( observed value-expected value)2expected value<br />ℵ2= 3<br />Degrees of freedom: <br />(R-1) (C-1) = (4-1) (3-1) = 6<br />Therefore,<br />Null Hypothesis H0<br /> The height and the shoe size are independent<br />Null Hypothesis H1<br />The height and the shoe size are dependent <br />Conclusion:<br />After done the investigation that is supported by the graph and the chi-square test, it proves that there is no definite relationship but the idea of it goes the taller the person, the larger the shoe size. This is because the taller the person it, the bigger the base needs to be, which is called “balance”. This is because if someone takes the average size feet for everyone, and averaged the shoe size for each height, and then graphed average shoe size against height, it would have a pretty linear graph. However, if someone just took two random people, and compared height and shoe size, the results could be pretty much anything which tells that the supporting of the evidence is really important. Therefore, the hypothesis which states that as the height of a person increases, the shoe size should also get bigger to balance the height. Therefore, there should be a relationship between height and shoe size, is accurate. <br />Limitation:<br />The limitation in this investigation is that there might be some inaccuracy information during the collecting which indicate to the wrong data which might gives the wrong provident of this investigation. <br />Work Sites:<br />http://www.clt.uwa.edu.au/__data/page/112506/fsp09_anthropometry.pdf<br />http://www.answerbag.com/q_view/9981<br />