Math IA

1. IB Math Studies Internal Assessment:<br />What is the Relationship between SAT Scores and Family Income of the Test Takers around the World?<br />Exam Session: May 2011<br />School name: International School Bangkok<br />Teacher: Mr. Demille <br />Date: December 8th, 2010<br />Course: IB Math Studies<br />Word Count: 1,832<br />Name: Billy Egnehall<br />What is the Relationship between SAT Scores and Family Income of the Test Takers around the World?<br />Introduction<br />The SAT examination is mostly in today’s world of academics, a requirement of getting accepted into collage. Not only is it enough to take the examination but the student has to pass with an average score or above to even have his/her application be considered. Many students around the world recognize this and therefore apply to prep schools for the SAT or their parents send them to a higher educational institution for that purpose. The prep schools such as Princeton are not cheap however as it helps give advice on how to best tackle the SAT examination, neither are higher educational institutions. Also it can be considered a luxury service by some middle class and low class societies in the world to be able to attend either one. This being said, the SAT prep course and higher educational institutions are, as a result, aimed at the high class societies in the world or those who can afford it. If this is true, it would put families with a higher income at an advantage for their children to get accepted into collage compared to families who cannot afford for their children to take the course or school fee and learn the advice of how to pass the SAT examination with a high score. Are the collages which students aim to be accepted into for a better education really based on which families can afford for their children to take the SAT prep course or learn at a higher educational institution? The data collected from Collage Board in year 2007 was analyzed to determine whether there is a relationship between SAT scores and family income of the test takers around the world (Rampell).<br />Statement of Task<br />The main purpose of this investigation is to determine whether there is a relationship between SAT scores and family income of the test takers around the world. The type of data that will be collected is the SAT scores and family income of the two-thirds of test takers who voluntarily reported it to collage board when signing up for the SAT examination worldwide. The SAT scores are used to determine how high of a score the test taker got and family income to determine the possibility to send their children to SAT prep schools or better educational institutions. The data used to generate the data breaks down the average score for ten different income groups of $20,000 range.<br />Plan of investigation<br />I am investigating the relationship of SAT scores and family income of the test takers around the world. I have collected data on SAT scores and family income of the test takers around the world. With the collection of data that I have acquired, a number of mathematical processes were used to analyze the data: a scatter plot of the data, calculation of the least squares regression line and correlation coefficient. I am going to do a χ2 test on the data to show the dependence of SAT scores and family income of the test takers around the world.<br />Mathematical Investigation<br />Collected Data<br />Family income of test takersPercentage of test takerswithin each family income groupCriticalreadingMathWriting ∑Less than $10,0004%4274514231301$10,000–$20,0008%4534724461371$20,000–$30,0006%4544654441363$30,000–$40,0009%4764854661427$40,000–$50,0008%4894964771462$50,000–$60,0008%4975044861487$60,000–$70,0008%5045114931508$70,000–$80,0009%5085164981522$80,000–$100,00014%5205295101559<br />Table 1: Mean SAT scores per section categorized in family income of test taker in 2007<br />More than $100,00026%5445565371637<br />This bottom row, the “More than $100,000” I am going to consider as an outlier therefore excluded in all calculations as it goes from $100,000 up to the millions of dollar of income which is too wide of a range to include into the calculations of this assessment.<br />-68580-55880Graph 1 shows the average SAT score Vs. family income of test taker. As of now, there seems to be very strong positive correlation. It does appear that the SAT scores improve as the family income increases. (Graph was generated through Microsoft Excel)<br />Calculation of the Least Squares Regression<br />The Least Square regression identifies the relationship between the independent variable, x, and the dependent variable, y. It is given by the following formula:<br />y-y= SxySx2 (x-x) where Sxy= xyn- xy and Sx2=x2n-x2<br />Table 2: Values of Least Squares Regression<br />xyxyx21500013011951500022500000025000137134275000625000000350001363477050001225000000450001427642150002025000000550001462804100003025000000650001487966550004225000000750001508113100000562500000085000152212937000072250000009500015591481050009025000000∑ = 495000∑ = 13000∑ = 733350000∑ = 33225000000x = 55000y = 1444.44xy = 79444444.44x2 = 3691666667<br />These are the calculated values used in finding the Least Squares Regression <br />Sxy= xyn- xy<br />Sxy= 7333500009- 79444444.44<br />Sxy= 2038888.893<br />Sx=x2n-x2<br />Sx=332250000009-3025000000<br />Sx=25819.88897<br />y-y= SxySx2 (x-x)<br />y-1444.44444= 2038888.893(25819.88897)2 (x-55000)<br />y= 0.0030583333x+1276.231666<br />Calculation of Pearson’s Correlation Coefficient<br />Pearson’s Correlation Coefficient indicates the strength of the relationship between the two variables (SAT scores and family income of test taker). It is given by the following formula:<br /> r= SxySxSy where Sx= x-x2n, Sy = y-y2n and Sxy is the covariance xyn- xy.<br />Table 3: Values of Pearson’s Correlation Coefficient <br />xyx-x2y-y2150001301160000000020576.308642500013719000000005394.086423500013634000000006633.197531450001427100000000304.3086425500014620308.19753096500014871000000001810.9753097500015084000000004039.3086428500015229000000006014.864198950001559160000000013122.97531∑ = 495000∑ = 13000∑ = 6000000000∑ = 58204.22222x = 55000y = 1444.44<br />These are the calculated values used in finding the Correlation Coefficient. <br />Sx= 25819.88897<br />Sy = 58204.222229<br />Sy = 80.4185041<br />r= 2038888.893(25819.88897)(80.4185041)<br />r=0.9819360378<br />r2=0.9642983824<br />The calculation r2=0.9642983824 suggests that the strength of the association of the data is very strong since 0.90 < r2 < 1. <br />I compared this value of r2 with the standard table of coefficient of determinations which places it in the “very strong” category (Whiffen).<br />14550-174928<br /> r2=0.9642983824<br /> y= 0.0030583333x+1276.231666<br />Graph 2 indicates that there is a strong positive linear correlation. This is also indicated through the value of correlation coefficient, 0.96.(the graph was generated through Microsoft Excel )<br />Calculation of a χ2 test<br />The χ2 test is used to measure whether two classifications or factors from the same sample are independent of each other – if the occurrence of one of them does not affect the occurrence of the other.<br />χ2= fo-fe2fe<br />Observed Values:<br />B1B2TotalA1ABA+BA2CDC+DTotalA+CB+DN<br />Calculations of 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 />Degrees of freedom measure the number of values in the final calculation that are free to vary: <br />Df=rows-1(columns-1)<br />Null (H0) Hypothesis: SAT scores and family income are independent from each other.<br />Alternative (H1) Hypothesis: SAT scores and family income are dependent from each other.<br />Table 4: Observation Values<br />ScoreIncome($)1300-14301431-1561Total15000 – 5500041556000 – 96000-44Total459<br />Table 2 shows the observed values for SAT score Vs. family income. The data pieces have been put into ranges that represent the income of the families of the test takers.<br />Table 5: Calculations for the Expected Values<br />ScoreIncome($)1300-14301300-1430Total15000 – 550004+1(4+0)94+1(1+4)94+156000 – 960004+0(0+4)91+4(0+4)90+4Total4+01+49<br />Table 3 shows the individual calculations for each of the expected values.<br />Table 6: Expected Values<br />ScoreIncome($)1300-14301300-1430Total15000 – 550002.222222.77777556000 – 960001.777772.222224Total459<br />Table 6 shows the expected values retrieved by the calculations in table 4<br />χ2= fo-fe2fe<br />χ2= 4-2.2222222.22222+1-2.7777722.77777+0-1.7777721.77777+4-2.2222222.22222<br />χ2=5.759995408<br />Df=rows-1(columns-1)<br />Df=2-1(2-1)<br />Df=1<br />The χ2 critical value at 5% significance with 1 degree of freedom is 3.841. As the χ2 value is greater than the critical value, 5.760>3.841, the null hypothesis is rejected and SAT score is assumed dependent from family income. <br />Discussion/Validity<br />Limitations<br />Throughout the investigation between the correlation of SAT scores and family income, various limitations may have affected the outcome of the results.<br />One limitation of the data collected could be that it only reflects on the people who filled in the family income section before signing up for the SAT. There is no evidence that the data reflects everyone who has taken the SAT score as there may be people who did not fill that section.<br />Another limitation could be that not everyone in the world decide to take the SAT, people who cannot afford it or take alternative tests are being neglected. Also the data does not confirm of how many SAT takers are being considered. The data can be proved insufficient and inaccurate for those reasons. <br />There is also a limitation in the data as it states income of “$100,000 and above”. That could mean that the data goes on unto family incomes of millions which is not proportionate to the other ranges of family income given. Due to this however, that piece of data was left out in the calculations.<br />Continuing, there might be a limitation to the recording of the data itself as SAT takers are to take a survey where they mention family income when signing up for SAT. This might cause a problem as many SAT takers, mostly in ages 15-17, do not know the actual income of their family therefore wrong data may be entered.<br />Then there could be a limitation to the data due to culture and race. The data does not mention culture and race which might affect the data as there might have been more American surveys who mentioned family income compared to Asian who answered the survey.<br />Another limitation is that the table of expected values in the χ2 test has all values less than 5 which reduces its validity. <br />Adding on to that, there might be a limitation to the amount of data that was collected as 9 pieces of data may not prove to be sufficient enough to reflect the correlation between SAT scores and family income in a world perspective.<br />Lastly, there may be many other factors taking place when considering the correlation between SAT scores and family income such as reasons for having a high family income and IQ of SAT test takers.<br />Conclusion<br />Despite of the previously mentioned limitations, the found χ2 value, 5.760, rejects the null hypothesis that SAT scores are independent from family income and accepts the alternative hypothesis that SAT scores are dependent from family income. Furthermore, the investigation clearly shows that there is a strong and positive correlation between SAT score and family income as it can be an assumed dependence from each other. <br />Work Cited<br />Rampell, Catherine. quot;
SAT Scores and Family Income - NYTimes.com.quot;
The Economy <br />and the Economics of Everyday Life - Economix Blog - NYTimes.com. 28 Aug. <br />2009. Web. 01 Nov. 2010.<http://economix.blogs.nytimes.com /2009/08/27/sat-<br />scores-and-family-income/>.<br />Downey, Joel. quot;
SAT Scores Rise with Family Income.quot;
Cleveland OH Local News, <br />Breaking News, Sports & Weather - Cleveland.com. 10 Apr. 2008. Web. 01 Nov. <br />2010.<http://www.cleveland.com/pdgraphics/index.ssf/2008/04/sat_scores_rise_<br />with_family_in.html>.<br />Whiffen, Glen, John Owen, Robert Haese, Sandra Haese, and Mark Bruce. quot;
Two <br />Variable Statistics.quot;
Mathematics for the International Student: Mathematical <br />Studies SL. By Mal Coad. [S.l.]: Haese And Harris Pub, 2010. 581-82. Print.<br />