Relationship Between Eye Color and Success in Anatomy. Sam Holladay IB Math Studies Mr. Saputo 4/3/15

Transcription

1 Relationship Between Eye Color and Success in Anatomy Sam Holladay IB Math Studies Mr. Saputo 4/3/15

2 Table of Contents Section A: Introduction.. 2 Section B: Information/Measurement... 3 Section C: Mathematical Processes... 9 Section D: Interpretation of Results..12 Section E: Validity 13 1

3 Introduction Statement of Task My topic for this project is to see if your success in anatomy is dependent on your eye color. I ve developed this topic into a more specific question: does the eye color of a student have any impact on their chances in being successful in Anatomy? I chose this topic because I excel in anatomy seemingly without putting much, if any, extra work into getting the high grades that I do, so I was curious if this could be attributed to some outside variable. Plan In order to collect my data, I have created a survey asking for each student s eye color and grade (in %) in anatomy. While passing this survey out to 5 anatomy classes, I have informed them that the survey is completely anonymous. While not having their names is a loss of validity, it decreases the chance of experimenter bias. I will ask them to give honest answers. I will ask the teacher for a list of the grades in his 3 anatomy classes and cross check the scores with the scores that have been gathered to make sure that the grades given by the students are accurate. For all my calculations I will be using the three basic eye colors blue, green, and brown. After all of my data has been collected and checked for accuracy, I will use this data in 4 mathematical processes (3 simple and 1 complex). The three simple processes that I will be using are finding the mean, standard deviation, and making a box & whisker plot of the data grouped into eye color and as a whole. This will help me to get an understanding of how the data is spread out and the average score associated with each eye color compared to the average score of the entire sample. The box and whisker plot will show the dispersion of my data and will show any outliers that may have occurred. It will also help me find the central tendency much like the mean; however, it will not be affected by any outliers that may occur. These mathematical 2

4 processes will first show me a general trend of the data collected then they will tell whether anatomy and eye color are dependent or not. The complex process that I will be using to interpret the data is a chi squared test to find the dependency of the two variables (eye color and success in Anatomy). My null hypothesis will be that eye color and success in anatomy are independent. If the null hypothesis is rejected, I will compare the calculated value of chi squared to the critical value of chi squared at the 1%, 5%, and 10% significance levels. This will help me to determine and will give more evidence to whether the variables are independent or not independent. Information/Measurement I collected my data by creating a short survey and having all of the students in Anatomy fill it out. I asked them to give accurate and honest information, and informed them that it would be completely anonymous. There are 41 with blue eyes, 28 with green eyes, and 173 with brown eyes giving me an overall sample size of 242 students. The eye color of the students is categorical data, and the scores are discrete data because they were rounded to the nearest 1%. This is the survey that I used: Your current grade in Anatomy in %: Your eye color (circle one): Blue Green Brown Raw Data: Eye Color Score (%) Eye Color Score (%) Eye Color Score (%) Blue 101 Green 95 Brown 100 Blue 99 Green 93 Brown 100 Blue 99 Green 92 Brown 99 Blue 98 Green 92 Brown 98 Blue 96 Green 91 Brown 98 3

5 Blue 96 Green 89 Brown 98 Blue 96 Green 88 Brown 97 Blue 95 Green 88 Brown 97 Blue 94 Green 86 Brown 97 Blue 92 Green 85 Brown 96 Blue 92 Green 82 Brown 95 Blue 90 Green 80 Brown 95 Blue 90 Green 79 Brown 95 Blue 90 Green 78 Brown 95 Blue 89 Green 77 Brown 94 Blue 88 Green 75 Brown 93 Blue 88 Green 75 Brown 93 Blue 88 Green 75 Brown 92 Blue 88 Green 75 Brown 92 Blue 88 Green 74 Brown 91 Blue 87 Green 71 Brown 91 Blue 85 Green 70 Brown 91 Blue 85 Green 69 Brown 91 Blue 84 Green 63 Brown 90 Blue 83 Green 58 Brown 90 Blue 81 Green 56 Brown 90 Blue 81 Green 43 Brown 90 Blue 80 Brown 89 Blue 79 Brown 89 Blue 77 Brown 88 Blue 77 Brown 88 Blue 77 Brown 86 Blue 76 Brown 86 Blue 73 Brown 86 Blue 69 Brown 85 Blue 69 Brown 84 Blue 68 Brown 84 Blue 67 Brown 84 Blue 56 Brown 83 4

6 Blue 56 Brown 83 Brown 82 Brown 82 Brown 82 Brown 81 Brown 80 Brown 80 Brown 79 Brown 79 Brown 79 Brown 78 Brown 76 Brown 76 Brown 76 Brown 76 Brown 76 Brown 76 5

7 Brown 72 Brown 72 Brown 72 Brown 72 Brown 72 Brown 71 Brown 71 Brown 71 Brown 71 Brown 71 6

8 Brown 70 Brown 70 Brown 70 Brown 70 Brown 70 Brown 70 Brown 70 Brown 69 Brown 69 Brown 69 Brown 68 Brown 68 Brown 67 Brown 67 Brown 66 Brown 65 Brown 65 Brown 65 Brown 65 Brown 64 Brown 64 Brown 63 Brown 63 Brown 62 Brown 62 Brown 61 Brown 60 Brown 60 Brown 60 Brown 59 Brown 59 Brown 58 Brown 57 Brown 57 7

9 Brown 56 Brown 56 Brown 55 Brown 55 Brown 54 Brown 53 Brown 53 Brown 52 Brown 52 Brown 52 Brown 51 Brown 50 Brown 50 Brown 50 Brown 49 Brown 49 Brown 49 Brown 48 Brown 48 Brown 48 Brown 45 Brown 45 Brown 45 Brown 45 Brown 45 Brown 43 Brown 42 Brown 40 Brown 37 Brown 34 Brown 33 8

10 Mathematical Processes My simple mathematical processes are the means, standard deviations, and box & whisker plots of the data grouped into their respective categories. I will find mean, standard deviation, and the information for the box and whisker plots (min, Q 1, median, Q 3, max) using a TI 84 graphing calculator. Blue Eyes: Mean = 84.89% Standard Deviation = Green Eyes: Mean = 77.88% Standard Deviation = Brown Eyes: Mean = 72.01% Standard Deviation = With the information provided by the box and whisker plots, the mean, and the standard deviation, it seems as if blue eyed people are more likely to be successful in anatomy. Due 9

11 to these simple processes, I expect to find that eye color and how you do in anatomy are not independent when I perform the Chi Squared test. My complex mathematical process is a Chi Squared test of dependency. In this Chi Squared test, I have compared the observed frequency and the expected frequency. I have done this to decide if I will accept or reject the null hypothesis that eye color and anatomy are independent variables. I will group the grade scores into the letter grade generally associated with them: e.g. A , B 89 80, C 79 70, D/F I combined the categories D (69 60) and F (59 0) because there was not enough data for them individually and that would negatively affect the reliability of the Chi Squared test. Null Hypothesis (H O ) = Eye color and grade in Anatomy are independent. Alternative Hypothesis (H 1 ) = Eye color and grade in Anatomy are not independent. Observed Frequency A B C D/F 69 0 Sum Blue Eyes Green Eyes Brown Eyes Sum Degrees of Freedom: ( 3 1) ( 4 1 ) = 6 I will use the Degrees of Freedom is used to find the critical value of X 2 which is then compared to the calculated value of X 2 to determine if I accept or reject the null hypothesis. 10

12 Expected Frequency A B C D/F 69 0 Sum Blue Eyes Green Eyes Brown Eyes Sum Chi Squared Test: (O E) 2 E f o f e f o f e ( f o f e ) 2 ( f o f e ) 2 f e Total

13 I found the Chi Squared value to equal With the critical value for 6 degrees of freedom at and my X 2 value at , I must reject the null hypothesis because my X 2 value is greater than the critical value. This means that how you do in anatomy is not independent to eye color. Interpretation of Results When I found the results of the mean, I noticed that blue and green eyes had almost equidistant central tendencies as green and brown with blue being the highest and brown being the lowest. However, brown did have the outlier furthest from the rest of the data at 33 which would affect the mean and make it less accurate as a measure of center. Despite the presence of outliers, blue eyes seem to have the best scores, so if i were to create a new project I would test blue eyes specifically to see if how well you do in anatomy is dependent on having blue eyes specifically. I would use brown eyes as a control in a separate test to give myself something to compare blue eyes to. The box and whisker plots showed the range of the dispersion of the data with blue eyes having the highest central tendency green being second and brown having the lowest. This is accurate and can be checked with the standard deviation which reflect a similar trend as the box and whisker plots. The complex mathematical process that I used was a Chi Squared test of independence. The results from the Chi Squared test found the X 2 calculated value to be which is significantly greater than the accepted X 2 critical value at the 5% significance level for 6 degrees of freedom, Thus causing me to reject the null hypothesis that eye color and success in anatomy are independent, and to accept the alternative hypothesis that eye color and success in anatomy are not independent. Through the calculations of the simple processes, it was apparent early on that there was a difference, especially in the mean. This could have been because of the outliers in the data for green and brown eyes, but within the box plots it is evident that the median shows similar evidence as the mean. 12

14 Validity Through my mathematical processes, I came to the conclusion that success in anatomy is dependent on eye color. I used a graphing device calculator on all of my simple mathematical processes and to check my complex mathematical process, so I know that they are all accurate. Despite this, there are some things that could have improved the accuracy of my results. I could have made sure that there were an equal amount of people in each category of eye color, but considering that the national average of blue eyed people is about 30%, that would be near impossible. Having a disproportionate amount of people in each category results in one category being more or less accurate than the others, in this case it is blue and green eyes that are less accurate than brown eyes. I also could have collected more data from other schools and different age groups. This would have increased the accuracy of my results as well as increasing the validity by being more generalizable to more people. In the simple processes of finding mean, standard deviation, and box plots, I found that blue eyes had the highest central tendency. If I were to do another math study, I would test all of the eye colors individually, focusing on blue eyes, to determine if there was a correlation or even a dependence of blue eyes and high grades in anatomy. With the standard deviation and box plots accurately showing the overall dispersion of the data, I was able to determine if the two factors would have an effect on each other early on. Although there are some adjustments that I would make if I were to do it again, my data went into depth and gave accurate results. Overall, I got the results that I expected that I would get, and I am happy with my study and the results that I found. 13