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IB Math Studies Internal Assessment<br />What is the Relationship Patterns of Otter Population Over 18 Years? <br />Patsornkarn Vorapharuek <br />School Name: International School of Bangkok<br />November 25, 2010<br />Course IB Math Studies<br />Introduction<br />As human population grows, resources are demanded to support their lifestyle necessities. Animal species become affected by the invasion of humans as they begin losing habitat and food sources. Recently, animal extinction has become an issue well addressed by the public; therefore actions have been taken to increase population of species. Specifically, the Monterey Bay Aquarium has conserved species such as the sea otter in order to prevent the species going into extinction. As public awareness increases, sea otter population should then increase. <br />In order to investigate this hypothesis, data from the Monterey Bay Aquarium Research Center will be looked at from year 1983 to year 2000. Each year, the amount of total sea otters will be counted. Pearson’s correlation coefficient formula will be used to find the r-value, which will determine the strength of the x correlation of the data, therefore, demonstrating if there is a positive, negative or any correlation between the years and amount of otters. The Chi-Square analysis will be used to determine whether the data has significance. <br />This investigation is of a subject matter that is crucial in the changing world today. Demonstration that human action can bring about change to an issue is significant, as it will show that with effort, we can bring about change. <br />Hypothesis<br />The correlation between years to the total amount of otters will be positive. This hypothesis is based upon the increase awareness of the issue of animal and otter extinction; therefore research facilities and the publics are all working against otter extinction, hence the total number of otters should increase throughout the years. <br />Data collection<br />Data is collected from the investigation done by the Monterey Bay Aquarium and its’ alliances. This data is available on the Internet from the years 1983-2000.<br />Table 1: Amount of California Sea Otters Per Year<br />*Years will be shown as following, 1 representing 1983, 2 representing 1984 and so on<br />YearTotal Otters% Change11277-21303+2.0431361+4.4541586+16.551661+4.761752+5.571856+5.981680-9.591941+15.5102101+8.2112239+6.6122359+5.4132377+0.76142278-4.2152229-2.2162114-5.2172090-1.1182371+13.4<br />Table 1: Table 1 demonstrates the years in which data were collected from and the amount of total otters that year. Furthermore, the percent change between each year is demonstrated, it can be shown whether amount of otters each year have increased or decreased. <br />Data Analysis<br />-914400380365Figure 1:<br />Figure 1: Figure 1 demonstrates the data between the years and amount of otter in a scatter gram. <br />Least Square Regression<br />The line of best fit can be produced using the Least Square Regression. This will demonstrate the nature of the data. It will show its strength (strong or weak), whether it is positive or negative and whether it is linear or not. <br />Linear Square Regression Calculations:<br />3886200-4445<br />Figure 2:<br />-114300065828<br />Figure 2 demonstrates the amount of otters per year in relation with the line of least regression. This line demonstrates a linear positive relationship of the data.<br />The Correlation Coefficient: <br />Value of Key Terms Incorporated in the Formula<br />9.517121091920.833457568818091359235<br />R-value calculation: <br />The value calculated demonstrates the correlation between the x and y variables. The value from the number of otters in relation to years calculated equals to 0.8126. value demonstrates whether the data correlates, therefore the values can only be between -1 and 1. -1 would mean the data is a perfect negative correlation and 1 would mean the data is a perfect positive correlation. 0.8126 is not perfectly positive but it is quite a strong positive correlation. This relationship of the two variables demonstrates that there is a relationship between number of otters and the year. In order to determine whether the relationship is dependent though, the Chi-Squared Test must be used.<br />A chart can be used to better describe the relationship strength of the data through the value. <br />Standard Table of Coefficients from Mathematics Studies SL Textbook Published by Haese & Harris Publication<br />ValueStrength of Association0<<0.25Very weak correlation0.25<<0.50Weak correlation0.50<<0.75Moderate correlation0.75<<0.90Strong correlation0.90<<1.0Very strong correlation=1Perfect correlation<br />In the data collected, since the r squared value is 0.8126, this places it at a “strong correlation” in this chart. <br />Chi Squared Calculations<br />The Chi-Squared test measures independence in a set of data. Independence signifies that the x and y value do not have a relationship. The test will check whether the data supports the null hypothesis that states whether the data of amount of otters is dependent or independent to each other. Chi-Squared requires two tables; one will demonstrate the observed values and the other will demonstrate expected values. The observed value table is referred to Table 1 while the expected values will be shown in Table 2. <br />Table 2: Expected Values of Number of Years and Total Otters (Rounded to the Nearest thousandth) <br />YearTotal Otters6.2901271.7106.4221298.5786.7131357.2877.8251582.1758.1991657.8018.6521749.3489.1691853.8318.3071679.6939.5971940.40310.3892100.61111.0732238.92711.6692359.33111.7622378.23811.2802280.72011.0472232.95610.4832119.51710.36924237950762002096.63111.7572377.243<br />Table 2: Demonstrates the expected values calculated from the observed value data. <br />Sample calculations of expected values:<br />In order to calculate the expected value, the value of and are needed. <br />Example for Year 1: the value would be multiplied by the sum of that row (1+1277). The product of that multiplication must then be divided by the value of + (34746)<br />Expected value for year 1 = <br />Expected value for year 1 = 6.290<br />Chi-Square Formula<br />Whereas, is the observed frequency and is the expected frequency. Observed frequency can be found in table 1 and expected frequency can be found in table 2.<br />Table 3: Calculations for the Chi Square value <br /> 16.290-5.2927.984.44926.422-4.4219.553.04536.713-3.7113.792.05447.825-3.8314.631.87058.199-3.2010.231.24868.652-2.657.030.81379.169-2.174.700.51388.307-0.310.090.01199.597-0.600.360.0371010.389-0.390.150.1461111.073-0.070.0054.81 x 1211.6690.330.110.0091311.7621.241.530.1301411.2802.727.400.6571511.0473.9515.631.4151610.4835.5230.442.9031710.3696.6344.004.2411811.7576.2439.003.31512771271.7105.2928.000.02213031298.5784.4219.550.01513611357.2873.7113.790.01015861582.1753.8314.630.00916611657.8013.2010.230.00617521749.3482.657.030.00418561853.8312.174.700.00216801679.6930.310.095.61 x 19411940.4030.600.361.83 x 21012100.6110.390.157.20 x 22392238.9270.070.0052.38 x 23592359.331-0.330.114.64 x 23772378.238-1.241.536.44 x 22782280.72-2.727.400.00322292232.956-3.9615.650.00721142119.517-5.5230.440.01420902096.631-6.6344.000.02123712377.243-6.2440.000.016 26.858191<br />Chi-squared = 26.858191<br />In order to be able to apply the Chi-Squared number and comprehend it, degrees of freedom must be determined. Degrees of freedom is determined by <br />(# Of rows – 1) (Number of columns -1)<br />Therefore, the degree of freedom in this case is 17. <br />With a 5% level of significance, the Chi Square value of 26.858191 accepts the null hypothesis. At 5% significance level with 17 degrees of freedom, the Chi Square value is less than the P-value of 27.587, showing that this data is independent.<br />Based upon Table 3 shown below, the Chi Square value of 26.858191 with 17 degrees of freedom rejects the null hypothesis at 10% significance. This is because the Chi-Square value of 26.858191 is greater than the P-value of 24.769; therefore the data is dependent upon each other. <br />0228600Table 4: Chi-Square Distribution Table<br />Table 4: In this table, the rows represent the degree of freedom. Once the degree of freedom is located, identify where the Chi-Square value belongs on that row. Once that is located, refer to the column. If the Chi-Square value is greater than the P-value, the null hypothesis is rejected (data is dependent) and if it is less than the P-value, the null hypothesis is accepted (data is independent). <br />Conclusion:<br />With reference to the Pearson’s correlation coefficient and the Chi-Square test, the hypothesis of this investigation was supported when; amount of otter increase in total is dependent upon the year. The value of 0.8126 demonstrates that the data has a strong linear relationship. This Chi Squared test rejected the null hypothesis at 10% significance level, therefore demonstrating that the data was dependent upon each other. However, it must be noted that at 5% significance level, the null hypothesis was supported, therefore rejecting the hypothesis. This demonstrates that there is a possibility that this data is inconclusive, the validity section below will demonstrate the possible factors that could have affected the results of this investigation and how manners in which a conclusive result could be achieved. <br />Validity: <br />The result of this investigation occurred because many variables were not controlled; therefore it could contribute to inaccuracy. Firstly, otter population values are values of estimate; there is no insurance that the Monterey Bay Aquarium could have accounted for all otters in the area of investigation. Therefore, actual numbers of otters may be different.<br />Furthermore, data collection could have included a variety of areas, therefore demonstrating that the hypothesis was not only valid for the one area of investigation. It is possible that the otters in this particular were under strict protection law but otters in a nearby area were not, therefore data could vary. Lastly, with more data, such as investigation of more years could aid in clearer understanding. <br />Works Cited<br />"Counting Otters." Oracle Education Foundation. ThinkQuest , 2001. Web.11 Nov. <br /> 2010. <http://library.thinkquest.org/J0111704/mainextcopy/statistics<br /> /statistics.html>.<br />Coad, Mal, et al. Mathematics for International Students Mathematical Studies SL. <br />2004. Adelaide: Haese & Harris Publications, 2009. Print.<br />
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