'Relationship between SAT Scores and Family Income Essay\r'

'What is the consanguinity mingled with sit put one across ins and Family Income of the Test Takers well-nigh the World? Introduction\r\nThe sit examination is more often than non in today’s world of academics, a requirement of suckting accepted into collage. Not hardly is it enough to take the examination but the school-age child has to arrive at with an average fit or supra to even chip in his/her application be considered. umpteen students nearly the world recognize this and therefore follow up to prep schools for the sit or their p arents send them to a prouder(prenominal) educational institution for that purpose. The prep schools such as Princeton are not cheap further as it helps give advice on how to best assume the sit down examination, neither are higher educational institutions.\r\nAlso it mickle be considered a lavishness service by some middle naval divisionalization and low class societies in the world to be able to attend either one. This universe said, the sit prep conformation 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 indue 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 flight or school fee and learn the advice of how to pass the sit down examination with a high score.\r\n are the collages which students aim to be accepted into for a break off education really based on which families can afford for their children to take the sit prep course or learn at a higher educational institution? The information collected from montage Board in year 2007 was analyzed to pay back whether there is a relationship between sit stacks and family income of the block unwrap takers around the world (Rampell).\r\n debate of Task\r\nThe main purpose of this investigating is to tick whether there is a relationship between sit heaps and family income of the test takers around the world. The type of entropy that will be collected is the sit down tally and family income of the ii-thirds of test takers who voluntarily reported it to collage circuit card 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 hypothesis to send their children to SAT prep schools or part educational institutions. The information used to generate the data breaks down the average score for ten divers(prenominal) income groups of $20,000 work.\r\nPlan of investigation\r\nI am analyse 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 entreaty of data that I have acquired, a physique of mathematical processes were used to analyze the data: a scatter plo t of the data, calculation of the least squares reverse toward the intend line and correlation coefficient. I am passing 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.\r\n mathsematical Investigation\r\n imperturbable Data\r\nFamily income of test takers| Percentage of test takers\r\n deep down for separately one family income group| Critical\r\nreading| Math| Writing| âˆ|\r\nLess than $10,000| 4%| 427| 451| 423| 1301|\r\n$10,000â€$20,000| 8%| 453| 472| 446| 1371|\r\n$20,000â€$30,000| 6%| 454| 465| 444| 1363|\r\n$30,000â€$40,000| 9%| 476| 485| 466| 1427|\r\n$40,000â€$50,000| 8%| 489| 496| 477| 1462|\r\n$50,000â€$60,000| 8%| 497| 504| 486| 1487|\r\n$60,000â€$70,000| 8%| 504| 511| 493| 1508|\r\n$70,000â€$80,000| 9%| 508| 516| 498| 1522|\r\n$80,000â€$100,000| 14%| 520| 529| 510| 1559|\r\nTable 1: Mean SAT scores per section categorized in family income of test taker in 2007\r\n more(prenominal) than $100,000| 26%| 544| 556| 537| 1637|\r\nThis bottom row, the â€Å"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. Graph 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 im register as the family income increases. (Graph was generated through with(predicate) Microsoft transcend)\r\n deliberation of the Least Squares retroversion\r\nThe Least Square regression identifies the relationship between the freelance variable, x, and the dependent variable, y. It is prone by the adjacent formula: y-y= SxySx2 (x-x) where Sxy= xyn- xy and Sx2=x2n-x2\r\nTable 2: determine of Least Squares Regression\r\nx| y| xy| x2|\r\n15000| 1301| 19515000| 225000000|\r\n250 00| 1371| 34275000| 625000000|\r\n35000| 1363| 47705000| 1225000000|\r\n45000| 1427| 64215000| 2025000000|\r\n55000| 1462| 80410000| 3025000000|\r\n65000| 1487| 96655000| 4225000000|\r\n75000| 1508| 113100000| 5625000000|\r\n85000| 1522| 129370000| 7225000000|\r\n95000| 1559| 148105000| 9025000000|\r\n∠= 495000| ∠= 13000| ∠= 733350000| ∠= 33225000000| x = 55000| y =\r\n1444.44| xy = 79444444.44| x2 = 3691666667| These are the calculated determine used in finding the Least Squares Regression Sxy= xyn- xy\r\nSxy= 7333500009- 79444444.44\r\nSxy= 2038888.893\r\nSx=x2n-x2\r\nSx=332250000009-3025000000\r\nSx=25819.88897\r\ny-y= SxySx2 (x-x)\r\ny-1444.44444= 2038888.893(25819.88897)2 (x-55000)\r\ny= 0.0030583333x+1276.231666\r\nCalculation of Pearson’s Correlation Coefficient\r\nPearson’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 formu la: r= SxySxSy where Sx= x-x2n, Sy = y-y2n and Sxy is the covariance xyn- xy.\r\nTable 3: determine of Pearson’s Correlation Coefficient\r\nx| y| x-x2| y-y2|\r\n15000| 1301| 1600000000| 20576.30864|\r\n25000| 1371| 900000000| 5394.08642|\r\n35000| 1363| 400000000| 6633.197531|\r\n45000| 1427| 100000000| 304.308642|\r\n55000| 1462| 0| 308.1975309|\r\n65000| 1487| 100000000| 1810.975309|\r\n75000| 1508| 400000000| 4039.308642|\r\n85000| 1522| 900000000| 6014.864198|\r\n95000| 1559| 1600000000| 13122.97531|\r\n∠= 495000| ∠= 13000| ∠= 6000000000| ∠= 58204.22222| x = 55000| y = 1444.44| | |\r\nThese are the calculated determine used in finding the Correlation Coefficient.\r\nSx= 25819.88897\r\nSy = 58204.222229\r\nSy = 80.4185041\r\nr= 2038888.893(25819.88897)(80.4185041)\r\nr=0.9819360378\r\nr2=0.9642983824\r\nThe calculation r2=0.9642983824 suggests that the strength of the association of the data is very strong since 0.90 < r2 < 1.\r\nI compared this val ue of r2 with the metre table of coefficient of determinations which places it in the â€Å"very strong” home (Whiffen).\r\nr2=0.9642983824\r\ny= 0.0030583333x+1276.231666\r\nGraph 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 ) Calculation of a χ2 test\r\nThe χ2 test is used to measure whether two classifications or factors from the same sample are independent of from each one different †if the occurrence of one of them does not come upon the occurrence of the other. χ2= fo-fe2fe\r\nObserved set:\r\n| B1| B2| wide|\r\nA1| A| B| A+B|\r\nA2| C| D| C+D|\r\n integral| A+C| B+D| N|\r\nCalculations of anticipate Values:\r\n| B1| B2| Total|\r\nA1| A+B(A+C)N| A+B(B+D)N| A+B|\r\nA2| A+C(C+D)N| B+D(C+D)N| C+D|\r\nTotal| A+C| B+D| N|\r\nDegrees of savedom measure the number of determine in the final calculation that are free to vary: Df=rows-1(columns-1)\r\nNull (H0) conjecture: SAT scores and family income are independent from each other. Alternative (H1) Hypothesis: SAT scores and family income are\r\ndependent from each other.\r\nTable 4: Observation Values\r\nScore|\r\nIncome($)| 1300-1430| 1431-1561| Total|\r\n15000 †55000| 4| 1| 5|\r\n56000 †96000| -| 4| 4|\r\nTotal| 4| 5| 9|\r\nTable 2 shows the observed values for SAT score Vs. family income. The data pieces have been put into ranges that agree the income of the families of the test takers. Table 5: Calculations for the Expected Values\r\nScore|\r\nIncome($)| 1300-1430| 1300-1430| Total|\r\n15000 †55000| 4+1(4+0)9| 4+1(1+4)9| 4+1|\r\n56000 †96000| 4+0(0+4)9| 1+4(0+4)9| 0+4|\r\nTotal| 4+0| 1+4| 9|\r\nTable 3 shows the individual calculations for each of the expected values. Table 6: Expected Values\r\nScore|\r\nIncome($)| 1300-1430| 1300-1430| Total|\r\n15000 †55000| 2.22222| 2.77777| 5|\r\n56000 †96000| 1.777 77| 2.22222| 4|\r\nTotal| 4| 5| 9|\r\nTable 6 shows the expected values retrieved by the calculations in table 4 χ2= fo-fe2fe\r\nχ2= 4-2.2222222.22222+1-2.7777722.77777+0-1.7777721.77777+4-2.2222222.22222 χ2=5.759995408\r\nDf=rows-1(columns-1)\r\nDf=2-1(2-1)\r\nDf=1\r\nThe 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 baseless hypothesis is rejected and SAT score is untrue dependent from family income.\r\nDiscussion/Validity\r\nLimitations\r\nThroughout the investigation between the correlation of SAT scores and family income, different limits may have matched the outcome of the results. virtuoso restriction of the data collected could be that it hardly reflects on the community 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.\r\nAnother demarcation could be that not everyone in the world decide to take the SAT, people who cannot afford it or take preference tests are being neglected. Also the data does not confirm of how many a(prenominal) SAT takers are being considered. The data can be proved shy(predicate) and inaccurate for those reasons.\r\nThere is also a limitation in the data as it states income of â€Å"$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.\r\nContinuing, there might be a limitation to the recording of the data itself as SAT takers are to take a survey where they remark family income when signing up for SAT. This might cause a problem as many SAT takers, broadly speaking in ages 15-17, do not know the veritable income of their family therefore wrong data may be ent ered.\r\nThen 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.\r\nAnother limitation is that the table of expected values in the χ2 test has all values less than 5 which reduces its validity. 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.\r\nLastly, 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.\r\n oddment\r\nDespite of the previously mentioned limitations, the found χ2 value, 5.760, rejects the vigor hypothesis that SAT scores are independent fro m 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 fictional dependence from each other.\r\nWork Cited\r\nRampell, Catherine. â€Å"SAT haemorrhoid and Family Income †NYTimes.com.” The Economy and the Economics of Everyday Life †Economix web log †NYTimes.com. 28 Aug. 2009. Web. 01 Nov. 2010.<http://economix.blogs.nytimes.com /2009/08/27/sat- scores-and-family-income/>.\r\nDowney, Joel. â€Å"SAT Scores cram with Family Income.” Cleveland OH Local News,\r\nBreaking News, Sports & Weather †Cleveland.com. 10 Apr. 2008. Web. 01 Nov.\r\n2010.<http://www.cleveland.com/pdgraphics/index.ssf/2008/04/sat_scores_rise_\r\nwith_family_in.html>.\r\nWhiffen, Glen, John Owen, Robert Haese, Sandra Haese, and Mark Bruce. â€Å"Two\r\nVariable Statisti cs.” maths for the International Student: Mathematical\r\nStudies SL. By Mal Coad. [S.l.]: Haese And Harris Pub, 2010. 581-82. Print.\r\n'