Recommended
More Related Content
Similar to Alvarez Paulo Math IA Draft PDF FINAL
Similar to Alvarez Paulo Math IA Draft PDF FINAL (20)
Alvarez Paulo Math IA Draft PDF FINAL
- 1. INTERNATIONAL BACCALAUREATE MATH STUDIES INTERNAL ASSESSMENT TOPIC: DATA COLLECTION AND STATISTICS Research Question: Is there a relationship between Total Juvenile Crime, Total Students Graduated, and Total Law Enforcement employed, in the United States? Supervisor: Tim Venhuis Candidate: Paulo L. Alvarez Candidate Number: 000046-0008 Word Count: 3256
- 2. 2 Introduction and Statement of Intent With the year 2016 approaching, the US Presidential election comes closer to 146,311,000 Americans who will decide the future of their nation. Amongst the candidates, issues like education and crime are inevitably going to show up. I’ve always taken an interest with these two issues, as they have a significant impact on the development of a nation, and more importantly, its youth. In this vein, could it be possible that a state that has more law enforcement officials employed or more high school graduates, lessen total juvenile crimes reported? Similarly, if a state has less law enforcement officials employed or less high school graduates, will total juvenile crimes reported be greater than states that have higher graduates and law enforcement? This investigation will be geared in addressing these issues using data from The United States of America’s Federal Bureau of Investigation and the National Center for Education Statistics. The USA has been selected as my country of focus because of its reliability in collecting data, relative economic similarity between its states, and its extensive data archiving. The amount of data that will be used in this investigation will be 50, looking at all US states to properly assess the extent of this topic. The overall purpose of this investigation is to see if there exists a relationship between total juvenile crimes, total law enforcement employed, and total high school graduates. The data that will be used in this investigation did not need to be collected through a survey, as it is gathered from the United States of America’s Federal Bureau of Investigation, census site Proximity.com, The US Department of Justice National Report Series for Juvenile Arrests 2012, and the National Center for Education Statistics. The majority of these sources are affiliated with the US National Government, and would be considered credible information, and to that extent credible for this investigation. The data collected from these sources will be processed into two tables; Table One shall detail Law Enforcement and Juvenile Crime according to each 50 US State in 2012. The Second shall detail Education; High School Graduation Rate and Total High School Graduates per each 50 US State in 2012. I have organized these tables in this manner in order to separate the variables that I will test, since I want to observe the relationship between total juvenile crimes, total law enforcement employed, and total high school graduates employed. I have then created 3- column graphs, which cover Total Juvenile Crimes, Total High School Graduates, and Total Law Enforcement Employed in the year 2012. Going back to the tables, all tables include the averages of their respective category and the averages of Total High School Graduates and Total Law Enforcement Employed for my chi-square test. Because my chi-square contingency tables have a degree of freedom of 1 and I’m testing at a 5% significance level, my significance level will be 3.84, and I will use the Yates Correction Continuity Test for both Chi Square Tables.
- 3. 3 In the succeeding pages, two sets of tabulations (in Tables 1 and 2) will be presented for all 50 states of the U.S as samples. At the bottom of these tables, two important measures of central tendency, the mean and median, will be computed for with the help of Microsoft Excel software. In getting the mean, the following formula was utilized: 𝑥 = 𝑥! !" !!! 𝑛 , 𝑤ℎ𝑒𝑟𝑒 𝑥! 𝑖𝑠 𝑎 𝑠𝑎𝑚𝑝𝑙𝑒 𝑎𝑛𝑑 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 Since it was evident from the data that some states like California, Alaska, and Vermont were consistent outliers, the median was also computed as an alternate indicator. The median, regardless of outliers would be a better metric in comparing the variables with. In getting the median for this even-numbered sample size of 50, the following formula was utilized, after arranging the samples from least value to greatest value: 𝑀𝑒𝑑𝑖𝑎𝑛 = 𝑛 2 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 + 𝑛 2 + 1 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 2 , 𝑤ℎ𝑒𝑟𝑒 𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 Substituting 𝑛 = 50: 𝑀𝑒𝑑𝑖𝑎𝑛 = 50 2 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 + 50 2 + 1 𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 2 And then simplifying: 𝑀𝑒𝑑𝑖𝑎𝑛 = 25!! + 26!! 2 With this in mind, the raw data in Table 1 is shown below:
- 4. 4 Table 1: Law Enforcement and Juvenile Crime and US States in 2012 with Averages State Law Enforcement Employed Violent Crime Property Crime Drug Abuse Weapon Possession Total Juvenile Crimes Alabama 12,745 57 698 286 11 1052 Alaska 1,968 246 1485 622 50 2403 Arizona 22,999 152 1109 653 34 1948 Arkansas 9,148 143 1001 328 44 1516 California 117,268 225 669 253 123 1270 Colorado 17,270 111 1108 611 65 1895 Connecticut 10,271 162 599 211 45 1017 Delaware 3,151 389 1245 546 73 2253 Florida 65,683 263 1264 480 56 2063 Georgia 34,769 169 927 302 61 1459 Hawaii 3,720 248 826 880 67 2021 Idaho 4,265 87 1198 549 70 1904 Illinois 45,505 751 1395 1337 291 3774 Indiana 12,032 160 981 387 45 1573 Iowa 7,375 183 1347 403 49 1982 Kansas 9,675 112 809 369 23 1313 Kentucky 9,728 91 562 166 20 839 Louisiana 19,364 445 1385 477 90 2397 Maine 2,826 54 1133 412 26 1625 Maryland 17,956 295 1100 617 102 2114 Massachusetts 19,282 177 305 84 28 594 Michigan 23,165 135 658 274 53 1120 Minnesota 13,476 114 1267 525 47 1953 Mississippi 5,662 63 1004 377 64 1508 Missouri 19,487 187 1258 468 61 1974 Montana 2,405 113 1535 406 15 2069 Nebraska 4,943 115 1711 719 57 2602 Nevada 9,447 243 941 405 40 1629 New Hampshire 3,436 54 650 543 0 1247 New Jersey 37,881 199 523 526 80 1328 New Mexico 6,023 202 1278 644 78 2202 New York 79,358 218 1024 485 56 1783 North Carolina 33,353 162 969 319 138 1588 North Dakota 1,968 89 1343 501 37 1970 Ohio 19,288 100 703 252 43 1098 Oklahoma 12,445 130 958 354 49 1491 Oregon 9,918 133 1215 699 45 2092 Pennsylvania 30,203 303 770 387 90 1550 Rhode Island 3,045 128 735 407 130 1400 South Carolina 15,135 146 911 516 87 1660 South Dakota 2,820 87 1495 1043 60 2685 Tennessee 26,268 281 949 431 85 1746 Texas 72,877 121 785 471 29 1406 Utah 7,042 76 1328 492 85 1981 Vermont 1,677 70 391 239 17 717 Virginia 23,625 74 620 337 41 1072 Washington 14,212 163 1039 399 60 1661 West Virginia 4,475 57 323 138 10 528 Wisconsin 18,638 234 1793 648 143 2818 Wyoming 2,074 51 1264 1122 66 2503 Mean 19,027 171 1,012 482 63 1,728
- 5. 5 Median 12,239 145 1003 450 56 1,661 Table 2: High School Graduation Rate, High School Graduates and Us States in 2012 with Averages State High School Graduation Rate (in Percent) Youth Population (Age 15-‐19) Total High School Graduates (Aged 15-‐19) Alabama 80 343,123 274,498 Alaska 72 51,379 36,993 Arizona 75 460,459 345,344 Arkansas 85 203,600 173,060 California 80 2,813,521 2,250,817 Colorado 77 338,471 260,623 Connecticut 86 250,257 215,221 Delaware 80 64,446 51,557 Florida 76 1,223,857 930,131 Georgia 72 705,508 507,966 Hawaii 82 84,426 69,229 Idaho 83 115,237 95,647 Illinois 83 916,375 760,591 Indiana 87 475,499 413,684 Iowa 90 216,848 195,163 Kansas 86 203,128 174,690 Kentucky 86 295,593 254,210 Louisiana 74 326,087 241,304 Maine 86 88,286 75,926 Maryland 85 404,292 343,648 Massachusetts 85 462,674 393,273 Michigan 77 739,534 569,441 Minnesota 80 367,809 294,247 Mississippi 76 222,938 169,433 Missouri 86 421,368 362,376 Montana 84 66,538 55,892 Nebraska 88 128,796 113,340 Nevada 71 182,317 129,445 New Hampshire 87 93,593 81,426 New Jersey 88 597,591 525,880 New Mexico 70 149,440 104,608 New York 77 1,365,555 1,051,477 North Carolina 83 652,589 541,649 North Dakota 88 47,105 41,452 Ohio 82 823,604 675,355 Oklahoma 85 262,928 223,489 Oregon 69 254,818 175,824 Pennsylvania 86 905,023 778,320 Rhode Island 80 79,688 63,750 South Carolina 78 324,237 252,905 South Dakota 83 57,489 47,716 Tennessee 86 436,141 375,081 Texas 88 1,873,088 1,648,317 Utah 83 220,983 183,416 Vermont 87 46,003 40,023 Virginia 84 547,561 459,951 Washington 76 461,092 350,430 West Virginia 81 120,073 97,259 Wisconsin 88 399,160 351,261 Wyoming 77 38,024 29,278 Mean 82 438,563 357,132
- 6. 6 Median 83 309,915 247,105 Column Graphs 1, 2, and 3: Column Graphs: An advantage to using the column graph for visually organizing my variables is that it highlights states that are either particularly strong or weak in a given variable. These graphs can also be used to make an initial visual judgment regarding, in an attempt at correlation/causation. Lastly, the column graph is useful for my project, as the scope of it takes place in one year, and deals with 50 different subjects/states. Graph 1: Column Graph of Total Juvenile Crimes per State in 2012 Observations: As this investigation will be looking at the effects of High School Graduates and Law Enforcement in a state, it is natural to start off by looking at the Total Juvenile Crimes per State. With regards to total juvenile crimes per state in 2012, Illinois, Wisconsin, South Dakota, Nebraska, and Wyoming make up the top five states with the highest in total crimes reported. While California, Connecticut, Kentucky, Massachusetts, and West Virginia have the lowest. While the investigation factors in all 50 states, these 10 states happen to be the strongest and weakest in regards to crime, thus it could be expected that their law enforcement employed and high school graduates would either be high for low crime and for high crime states.
- 7. 7 Graph 2: Column Graph of Total High School Graduates per State in 2012 Graph 3: Column Graph of Law Enforcement Employed per State in 2012 Observations: With the variables that will be tested with total Juvenile Crimes, law enforcement and total high school graduates are presented visually on graphs 2 and 3 with some disparity. For instance there are states like California, which visually, has the most high
- 8. 8 school graduates and law enforcement employed, yet in regards to crime, isn’t the lowest state. States like Massachusetts and West Virginia are the two lowest states regarding crime, but visually appear to be fairly low with high school graduates and law enforcement employed. A possible explanation for this disparity, and a potential weakness with the data collected, is that the youth population of each state varies in levels. Going back to California, Massachusetts, and West Virginia, California’s youth population is about 2,813,521. Compare that to West Virginia and Massachusetts and their combined youth population of 582,747 is only about 20.7% of California’s. Hence it would be expected that California almost acts like an outlier in that it has a significantly higher youth population than most states, thus yielding higher graduates and law enforcement employed. However, California’s data will not be considered as an outlier since it is a US state, and therefore qualifies as being included in this investigation. So while at a glance these column graphs cannot be used to support correlation/causation of the variables a stronger method to do so would be the Chi- Square test of independence. Chi Square Test For my further process in this investigation, I shall use two Chi-Square tests to determine if Total Juvenile Crimes is independent from Total High School Graduates and Total Law Enforcement Employed. My determiners for the Chi-Square tests are going to be based on the averages of Law Enforcement Employed; 19,027 and Total High School Graduates; 357,132. With the averages I will divide the 50 states with those that are above and including the average, and those that are below the average. A summary of the earlier computations is shown below: Law Enforcement Employed Total High School Graduates (Aged 15-19) Mean 19,027 357,132 Median 12,239 247,105 Next, with regards to Total Juvenile Crime, I have divided the total into violent and non- violent crimes. An example in calculating the total non-violent and violent crimes, I will add the number of Violent Crime and Property Crime reported to make up violent crimes.
- 9. 9 Likewise, I will add the number of Drug abuse and weapons possession reported to make up non-violent crimes. Table 4: Division of Violent Crimes (Bold Red) and Non-Violent Crimes per US State in 2012 State Violent Crime Property Crime Drug Abuse Weapon Possession Alabama 57 698 286 11 Alaska 246 1485 622 50 Arizona 152 1109 653 34 Arkansas 143 1001 328 44 California 225 669 253 123 Colorado 111 1108 611 65 Connecticut 162 599 211 45 Delaware 389 1245 546 73 Florida 263 1264 480 56 Georgia 169 927 302 61 Hawaii 248 826 880 67 Idaho 87 1198 549 70 Illinois 751 1395 1337 291 Indiana 160 981 387 45 Iowa 183 1347 403 49 Kansas 112 809 369 23 Kentucky 91 562 166 20 Louisiana 445 1385 477 90 Maine 54 1133 412 26 Maryland 295 1100 617 102 Massachusetts 177 305 84 28 Michigan 135 658 274 53 Minnesota 114 1267 525 47 Mississippi 63 1004 377 64 Missouri 187 1258 468 61 Montana 113 1535 406 15 Nebraska 115 1711 719 57 Nevada 243 941 405 40 New Hampshire 54 650 543 0 New Jersey 199 523 526 80 New Mexico 202 1278 644 78 New York 218 1024 485 56 North Carolina 162 969 319 138 North Dakota 89 1343 501 37 Ohio 100 703 252 43 Oklahoma 130 958 354 49 Oregon 133 1215 699 45 Pennsylvania 303 770 387 90 Rhode Island 128 735 407 130 South Carolina 146 911 516 87 South Dakota 87 1495 1043 60 Tennessee 281 949 431 85 Texas 121 785 471 29 Utah 76 1328 492 85 Vermont 70 391 239 17 Virginia 74 620 337 41 Washington 163 1039 399 60 West Virginia 57 323 138 10 Wisconsin 234 1793 648 143 Wyoming 51 1264 1122 66
- 10. 10 For the first Chi-Square test that will compare the corresponding means between High School Graduates and Juvenile Crimes, the null hypothesis and alternate hypotheses will be presented: High School Graduates and Juvenile Crimes 𝑯 𝟎: High School Graduates and Juvenile Crimes are independent 𝑯 𝟏 : High School Graduates and Juvenile Crimes are not independent Degrees of Freedom: Using the Degrees of Freedom (df) Formula: 𝒅𝒇 = 𝑟 − 1 𝑐 − 1 , 𝑤ℎ𝑒𝑟𝑒 𝒅𝒇 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝐷𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝐹𝑟𝑒𝑒𝑑𝑜𝑚, 𝒓 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑤𝑠, 𝑎𝑛𝑑 𝒄 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 𝑖𝑛 𝑡ℎ𝑒 𝐶𝑜𝑛𝑡𝑖𝑛𝑔𝑒𝑛𝑐𝑦 𝑇𝑎𝑏𝑙𝑒 𝒅𝒇 = 2 − 1 2 − 1 ∴ 𝒅𝒇 = 𝟏 According to the Degrees of Freedom table, below: The data, therefore, shall be tested at a 5% significance level of 3.84.
- 11. 11 Chi Square Table 1: Average of High School Graduates in 2012 with Violent and Non Violent Crimes Contingency Table Crime Category Total High School Graduates (Aged 15-19) Violent Non-Violent Total ≥ 357,132 17,325 8,073 25,398 < 357,132 41,829 19,166 60,995 Total 59,154 27,239 86,393 Expected Value Table for Average of High School Graduates in 2012 with Violent and Non Violent crimes Crime Category Total High School Graduates (Aged 15-19 Violent Non-Violent Total ≥ 357,132 59,154 × 25,398 86,393 = 17,390 27,239 × 25,398 86,393 = 8,008 25,398 < 357,132 59,154 × 60,995 86,393 = 41,764 27,329 × 60,995 86,393 = 19,231 60,995 Total 59,154 27,239 86,393
- 12. 12 𝒳!"#! ! 𝑓! 𝑓! 𝑓! − 𝑓! ( 𝑓! − 𝑓!)! ( 𝑓! − 𝑓!)! 𝑓! 17,325 17,390 - 65 4,225 0.242 41,829 41,764 65 4,225 0.101 8,073 8,008 65 4,225 0.527 19,166 19,231 -65 4,225 0.219 Total 1.09 ∴ 𝒳!"#! ! = 1.09 Since the 𝒳!"#! ! value of 1.09 is less than the critical value of 3.84, we can reject 𝐻! and accept 𝐻!. Therefore High School Graduates and Juvenile Crimes are independent of each other. Because the contingency table is a 2x2 table with a df of 1, the Yates Correction for Continuity Test must be used. The Yates test was developed by English Statistician Frank Yates, and is meant to account for the upwards bias in a 2x2 contingency table. Yates Correction For Continuity Test Using the Yates Formula: 𝒳!"#$% ! = 𝑓!! 𝑓! − 0.5 ! 𝑓! !
- 13. 13 Therefore in tabular form, the following values were derived: ( 𝑓! − 𝑓!)! 𝑓! ( 𝑓! − 𝑓! − 0.5 ! 𝑓! 0.242 0.239 0.101 0.996 0.527 0.519 0.219 0.216 1.09 1.07 ∴ Since 1.07 < 3.84, we can now accept the 𝐻! and reject 𝐻! to conclude that High School Graduates and Juvenile Crimes are independent. Now that we have tested the total high school graduates with juvenile crimes, a second test will be performed with the second variable with juvenile crimes, the total number of law enforcement employed. Law Enforcement Employed and Juvenile Crimes 𝐻!: Law Enforcement Employed and Juvenile Crimes are independent 𝐻! : Law Enforcement Employed and Juvenile Crimes are not independent
- 14. 14 Chi Square Table 2: Contingency Table of Average Law Enforcement Employed in 2012 with Violent and Non Violent crimes Crime Category Law Enforcement Employed Violent Non-Violent Total ≥ 19,027 19,732 8,895 28,627 < 19,027 38,447 17,806 56,253 Total 58,179 26,701 84,880 Expected Value Table for Law Enforcement Employed in 2012 with Violent and Non Violent crimes Crime Category Law Enforcement Employed Violent Non-Violent Total ≥ 19,027 58,179 × 28,627 84,880 = 19,622 26,701× 28,627 84880 = 9,005 28,627
- 15. 15 < 19,027 58,179 × 56,253 84,880 = 38,557 27,329 × 60,995 84,880 = 17,696 56,253 Total 58,179 26,701 84,880 𝒳!"#! ! 𝑓! 𝑓! 𝑓! − 𝑓! ( 𝑓! − 𝑓!)! ( 𝑓! − 𝑓!)! 𝑓! 19,732 19,622 110 12,100 0.616 38,447 38,557 -110 12,100 0.313 8,895 9,005 -110 12,100 1.34 17,806 17,696 110 12,100 0.683 Total 2.95 ∴ 𝒳!"#! ! = 2.95 Yates Correction For Continuity Test Since the 𝒳!"#! ! value of 2.95 is less than the critical value of 3.84, we can reject 𝐻! and accept 𝐻!. Therefore Law Enforcement and Juvenile Crimes are independent of each other. Similar to the first Chi-Square Table, this contigency table is a 2x2 table and has a df of 1. Hence it must go through the Yates Continuity Test before comparing to the df of 3.84. I used use my Ti-84 graphing calculator and produced the following values:
- 16. 16 ( 𝑓! − 𝑓!)! 𝑓! ( 𝑓! − 𝑓! − 0.5)! 𝑓! 0.2429557217 0.6110615636 0.1011636816 0.3109746609 .5275974026 1.331510272 .2196973636 0.677568377 2.957944161 2.931114874 ∴ 2.931114874 < 3.84 we can now accept the 𝐻! and say that High School Graduates and Juvenile Crimes are independent. Conclusion In exploring the relationship between Total Juvenile Crimes with total high school graduates and total law enforcement employed, I have used two Chi-Square tests then subsequently used the Yates Correction for Continuity test, as my contingency tables are 2x2 and yield a degrees of freedom of 1. I’d then compare the values yielded by the Yates test, and found that for total high school graduates, the sum of ( !!!!!!!.!)! !! = 1.074687763 which is less than the significance level of 3.84 thus the relationship between Total High School Graduates and Total Juvenile Crimes, is independent. For Total Law Enforcement Employed, the sum of ( !!!!!!!.!)! !! = 2.931114874 is less than 3.84, hence the relationship between Total Law Enforcement and Total High School Graduates is independent. Thus, it can be concluded that Total Juvenile Crimes has no relationship with both Total Law Enforcement Employed and the Total High School Graduates in a given US State. In this investigation I had faced some issue with the extent of the data collected and used. For instance the data used came from
- 17. 17 2012, nearly four years have passed since then and the numbers in regards to the variables used may have changed substantially. The reason I had used 2012 as the basis of my investigation, is because no other year beyond 2012 has a complete set of data that I needed, specifically the number of total High School Graduates in a given state. I also acknowledge that the reliability of the data source could come under question, as all of the data used in this investigation are from government sources, and the extent to which the data is true or inflated due to different criteria for all 50 states may be troublesome to the overall data. Lastly, regarding the nature of this issue, the scope used may not be adequate as the investigation only focused on Juvenile crimes. When it may be possible that a student may commit a crime later in their lives.
- 18. 18 Works Cited Page "State Population by Age and Gender: Census 2000, 2010 and Change | Fastest Growing States." State Population by Age and Gender: Census 2000, 2010 and Change | Fastest Growing States. Proximity, 2012. Web. 10 Jan. 2016. United States of America. Department of Justice. Office of Juvenile Justice and Delinquency Prevention. Office of Juvenile Justice and Delinquency Prevention Juvenile Arrests 2012. By Charles Puzzanchera. US Department of Justice, Dec. 2014. Web. United States of America. Federal Bureau of Investigation. Criminal Justice Information Service Division. Full-time Law Enforcement Employees. By CJIS. N.p.: n.p., 2012. FBI Crime in the US. Web. United States of America. Federal Bureau of Investigation. Criminal Justice Information Service Division. Violent Crime. By CJIS. N.p.: n.p., 2013.FBI Crime in the US 2013. Web.