Relationship between SAT Scores and Family Income Essay

What is the Relationship between sit stimulates and Family Income of the Test Takers nigh the World? IntroductionThe sit down mental examing is mostly in todays world of academics, a requirement of getting authorized 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 action be considered. Many students around the world recognize this and on that pointfore apply to training schools for the sit down or their pargonnts send them to a mellower educational institution for that offer. The prep schools such as Princeton ar non cheap however as it helps give advice on how to best tackle the SAT examination, nevery are higher educational institutions.Also it evoke be considered a luxury service by slightly 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 tippytoe 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 re everyy based on which families can afford for their children to take the SAT prep course or learn at a higher educational institution? The information collected from Collage Board in year 2007 was try outd to determine whether there is a affinity between SAT slews and family income of the test takers around the world (Rampell).Statement of TaskThe main purpose of this investigation is to determine whether there is a relationship between SAT lashings and family income of the test takers around the world. The type of info that will be collected is the SAT oodles and family income of the two-thirds of test takers who voluntarily describe it to collage board when signing up for the SAT examination worldwide. The SAT scores are use 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.Plan of investigationI 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 2 test on the data to show the colony of SAT scores and family income of the test takers around the world.Mathematical InvestigationCollected DataFamily income of test takers region of test takerswithin each family income group Criticalreading Math Writing Less than $10,000 4% 427 451 423 1301$10,000$20,000 8% 453 472 446 1371$20,000$30,000 6% 454 465 444 1363$30,000$40,000 9% 476 485 466 1427$40,000$50,000 8% 489 496 477 1462$50,000$60,000 8% 497 504 486 1487$60,000$70,000 8% 504 511 493 1508$70,000$80,000 9% 508 516 498 1522$80,000$100,000 14% 520 529 510 1559Table 1 Mean SAT scores per section categorized in family income of test taker in 2007More than $100,000 26% 544 556 537 1637This 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. Graph 1 shows the average SAT score Vs. family income of test taker. As of now, there seems to be very strong absolute correlation. It does appear that the SAT scores improve as the family income increases. (Graph was generated through Microsoft Excel)Calculation of the least(prenominal) Squares RegressionThe Least Square regression identifies the relationship between the independent variable, x, and the dependent variable, y. It is abandoned by the following polity y-y= SxySx2 (x-x) where Sxy= xyn- xy and Sx2=x2n-x2Table 2 Values of Least Squares Regressionx y xy x215000 1301 19515000 22500000025000 1371 34275000 62500000035000 1363 47705000 122500000045000 1427 64215000 202500000055000 1462 80410000 302500000065000 1487 96655000 422500000075000 1508 113100000 562500000085000 1522 129370000 722500000095000 1559 148105000 9025000000 = 495000 = 13000 = 733350000 = 33225000000 x = 55000 y =1444.44 xy = 79444444.44 x2 = 3691666667 These are the calculated surveys used in decision the Least Squares Regression Sxy= xyn- xySxy= 7333500009- 79444444.44Sxy= 2038888.893Sx=x2n-x2 Sx=332250000009-3025000000Sx=25819.88897y-y= SxySx2 (x-x)y-1444.44444= 2038888.893(25819.88897)2 (x-55000)y= 0.0030583333x+1276.231666Calculation of Pearsons Correlation CoefficientPearsons 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 r= SxySxSy where Sx= x-x2n, Sy = y-y2n and Sxy is the covariance xyn- xy.Table 3 Values of Pearsons Correlation Coefficientx y x-x2 y-y215000 1301 1600000000 20576.3086425000 1371 900000000 5394.0864235000 1363 400000000 6633.19753145000 1427 100000000 304.30864255000 1462 0 308.197530965000 1487 100000000 1810.97530975000 1508 400000000 4039.30864285000 1522 900000000 6014.86419895000 1559 1600000000 13122.97531 = 495000 = 13000 = 6000000000 = 58204.22222 x = 55000 y = 1444.44 These are the calculated set used in finding the Correlation Coefficient.Sx= 25819.88897Sy = 58204.222229Sy = 80.4185041r= 2038888.893(25819.88 897)(80.4185041)r=0.9819360378r2=0.9642983824The calculation r2=0.9642983824 suggests that the strength of the association of the data is very strong since 0.90 r2 1.I compared this value of r2 with the standard table of coefficient of determinations which authoritys it in the very strong category (Whiffen).r2=0.9642983824y= 0.0030583333x+1276.231666Graph 2 indicates that there is a strong positive linear correlation. This is excessively indicated through the value of correlation coefficient, 0.96.(the represent was generated through Microsoft Excel ) Calculation of a 2 testThe 2 test is used to value 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. 2= fo-fe2feObserved Values B1 B2 TotalA1 A B A+BA2 C D C+DTotal A+C B+D NCalculations of Expected Values B1 B2 TotalA1 A+B(A+C)N A+B(B+D)N A+BA2 A+C(C+D)N B+D(C+D)N C+DTotal A+C B+D NDegrees of freedom whole tone the number of values in the final calculation that are free to vary Df=rows-1(columns-1)Null (H0) possible action SAT scores and family income are independent from each other. Alternative (H1) Hypothesis SAT scores and family income aredependent from each other.Table 4 Observation ValuesScoreIncome($) 1300-1430 1431-1561 Total15000 55000 4 1 556000 96000 - 4 4Total 4 5 9Table 2 shows the sight 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. Table 5 Calculations for the Expected ValuesScoreIncome($) 1300-1430 1300-1430 Total15000 55000 4+1(4+0)9 4+1(1+4)9 4+156000 96000 4+0(0+4)9 1+4(0+4)9 0+4Total 4+0 1+4 9Table 3 shows the individual calculations for each of the expected values. Table 6 Expected ValuesScoreIncome($) 1300-1430 1300-1430 Total15000 55000 2.22222 2.77777 556000 96000 1.77777 2.22222 4Total 4 5 9Table 6 shows the expected values retrieved by the calculations in tabl e 4 2= fo-fe2fe2= 4-2.2222222.22222+1-2.7777722.77777+0-1.7777721.77777+4-2.2222222.22222 2=5.759995408Df=rows-1(columns-1)Df=2-1(2-1)Df=1The 2 critical value at 5% significance with 1 degree of freedom is 3.841. As the 2 value is great than the critical value, 5.7603.841, the null hypothesis is rejected and SAT score is assumed dependent from family income.Discussion/ rigourousnessLimitationsThroughout the investigation between the correlation of SAT scores and family income, various limitations may have affected the outcome of the results. One limitation of the data collected could be that it only reflects on the people who fill 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.another(prenominal) 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 n eglected. Also the data does not confirm of how some SAT takers are being considered. The data can be proved insufficient and inaccurate for those reasons.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.Continuing, there readiness 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 superpower 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.Then there could be a limitation to the data due to socialization 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 famil y income compared to Asian who answered the survey.Another limitation is that the table of expected values in the 2 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.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.ConclusionDespite of the previously mentioned limitations, the found 2 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 intelligibly 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.Work CitedRampell, Catherine. SAT Scores and Family Income NYTimes.com. The Economy and the Economics of commonplace Life Economix Blog NYTimes.com. 28 Aug. 2009. Web. 01 Nov. 2010..Downey, Joel. SAT Scores Rise with Family Income. Cleveland OH Local News,Breaking News, Sports & Weather Cleveland.com. 10 Apr. 2008. Web. 01 Nov.2010..Whiffen, Glen, John Owen, Robert Haese, Sandra Haese, and Mark Bruce. TwoVariable Statistics. Mathematics for the International Student MathematicalStudies SL. By Mal Coad. S.l. Haese And Harris Pub, 2010. 581-82. Print.