Binomial Project

1. Binomial Project Sports & Nutrition By Excellent Students #1 and #2

2. Introduction - For our project, we decided to have a survey about sports and nutrition because we wanted to see how many students do take care of their body. - We surveyed 2 classes for this project: Lockhart and Rais. We numbered all the teachers on the list and then used random-int on the calculator to get 5 numbers (5 teachers). From those five, we were given Lockhart and Rais to survey. - Our sample size was big enough because we had a sample size of 30 which is the accepted size.

3. unbiased Survey • Do you work out daily?Yes or No • Do you eat healthy?Yes or No • Do you play a sport?Yes or No • How much do you weigh?___ • Do you sleep around 7-9 hours a night?Yes or No • Do you feel tired or weak often?Yes or No • Do you get more than 10 hours of exercise/ week? Yes or No • Do you find yourself losing weight?Yes or No • Do you think your weight is affected by your and exercise and eating choice?Yes or No • Do you think sports or exercising in general is a necessity?Yes or no

4. check 2NIP! 1. Do you work out Daily?Yes or No • Do you eat healthy?Yes or No • Do you play a sport?Yes or No • How much do you weigh?___ • Do you sleep around 7-9 hours a night?Yes or No • Do you feel tired or weak often?Yes or No • Do you get more than 10 hours of exercise/ week? • Do you find yourself losing weight?Yes or No • Do you think your weight is affected by your and exercise and eating choice?Yes or No • Do you think sports or exercising in general is a necessity?Yes or no Numbers 1, 2, 3, 5, 6, 7, 8, 9, 10 all follow 2NIP. For each of those questions there are: 2 total outcomes (yes or no), there is a set number of trials (60: 30 biased, 30 unbiased), there were independent survey-takers, and there was a set probability (yes or no- .5) Number 4 does not follow 2NIP because there are not 2 outcomes. A weight is any number, meaning multiple outcomes.

5. biased Survey • Do you work out daily and get the exercise your body needs in order to stay healthy?Yes or no • Are you eating healthy and treating your body the way you should be?Yes or No • Do you play a sport rather than just sitting at home being lazy and gaining weight?Yes or No • How much do you weigh?___ If you don’t like your weight It’s probably because you’re not exercising and eating right. • Are you sleeping 7-9 hours and keeping your body nurtured? Yes or No • Do you feel tired and weak often due to unhealthy eating and lack of sleep? Yes or No • Are you giving your body the exercise it needs?Yes or No • Do you see yourself losing weight at all if you are working out daily?Yes or No • Do you believe your weight is affected by your exercise and eating habits considering they really affect your weight?Yes or No • Do you think sports or exercise is a necessity to keep a strong steady body?Yes or No

6. check 2NIP! • Do you work out daily and get the exercise your body needs in order to stay healthy?Yes or no • Are you eating healthy and treating your body the way you should be?Yes or No • Do you play a sport rather than just sitting at home being lazy and gaining weight?Yes or No • How much do you weigh?___ If you don’t like your weight It’s probably because you’re not exercising and eating right. • Are you sleeping 7-9 hours and keeping your body nurtured? Yes or No • Do you feel tired and weak often due to unhealthy eating and lack of sleep? Yes or No • Are you giving your body the exercise it needs?Yes or No • Do you see yourself losing weight at all if you are working out daily?Yes or No • Do you believe your weight is affected by your exercise and eating habits considering they really affect your weight?Yes or No • Do you think sports or exercise is a necessity to keep a strong steady body? Yes or No Numbers 1, 2, 3, 5, 6, 7, 8, 9, 10 all follow 2NIP. For each of those questions there are: 2 total outcomes (yes or no), there is a set number of trials (60: 30 biased, 30 unbiased), there were independent survey-takers, and there was a set probability (yes or no- .5) Number 4 does not follow 2NIP because there are not 2 outcomes. A weight is any number, meaning multiple outcomes.

7. Unbiased: Do you work out daily? Biased: Do you work out daily and get the exercise your body needs in order to stay healthy? unbiased biased

8. 2. Unbiased: Do you eat healthy? Biased: Are you eating healthy and treating your body the way you should be? unbiased biased

9. 3. Unbiased: Do you play a sport? Biased: Do you play a sport, rather than sitting at home being lazy and gaining weight? unbiased biased

10. 4. Unbiased: How much do you weigh? Biased: How much do you weigh? If you don’t like your weight it’s probably because you’re not exercising and eating right. unbiased biased

11. 5. Unbiased: Do you sleep 7-9 hours/night? Biased: Are you sleeping 7-9 hours and keeping your body nurtured? unbiased biased

12. 6. Unbiased: Do you feel tired or weak often? Biased: Do you feel tired or weak often due to unhealthy eating habits and lack of sleep? unbiased biased

13. 7. Unbiased: Do you get more than 10 hours of exercise/ week? Biased: Are you giving your body the exercise it needs? unbiased biased

14. 8. Unbiased: Do you find yourself loosing weight? Biased: Do you see yourself loosing weight at all if you are working out daily? unbiased biased no yes

15. 9. Unbiased: Do you think your weight is affected by your exercise and eating choice? Biased: Do you believe your weight is affected by your exercise and eating habits considering they really affect your weight? unbiased biased yes no

16. 10. Unbiased: Do you think that sports or exercising in general is a necessity? Biased: Do you think sports or exercising in general is necessary to keep a stable body? unbiased biased yes no

17. Analysis The graphs and tables shown in the slides previous are good representations of bias answers. The biased questions seemed to be answered less truthfully than the non-bias surveys and the results from the two surveys differed. It seemed as though the bias surveys were answered differently than the non-bias ones, it was as if people felt our bias and felt as if they were a bad person for answering a question a certain way.

18. Important Notations • n = Sample Size • P = Probability of “Yes” or “No” • q = 1 - P • x = # of people who said Yes

19. 1. Binomial Formula Unbiased Survey. #1. Do you work out daily? Yes or No n= 30 p=.53 x=9 q=.47 P(x=9)=n!/(n-x)!x! * p^x * q^n-x P(x=9)=30!/(30-9)!9! * .53^30 * .47^30-9 = P(x=9)= .0061 Table Unbiased Survey. #1. Do you work out daily? Yes or No n=15 p=.47 x=7 q=.53 P(x=6)=.153 Software Unbiased Survey. #1. Do you work out daily? Yes or No n= 30 p=.53 x<8 q=.47 P(x<8) = 1-binomcdf(30,.53,7) = .9991 Not unusual because .9991 is not less than .05

20. 2. Binomial Formula Unbiased Survey. #2. Do you eat healthy? Yes or No n= 30 p=.83 x=25 q=.17 P(x=25)= n!/(n-x)!x! * p^x * q^n-x P(x=25)=30!/(30-25)!25! * .83^30 * .17^30-25 = P(x=25)=.1919 Table Unbiased Survey. #2. Do you eat healthy? Yes or No n= 15 p=.87 x=13 q=.13 P(x=6)= .267 Software Unbiased Survey. #2. Do you eat healthy? Yes or No n= 30 p=.83 x<8=25 q=.17 P(x<8)= 1-binomcdf(30,.83,7) = 1 Not unusual because 1 is not less than .05

21. 3. Binomial Formula Unbiased Survey. #1. Do you work out daily? Yes or No n= 30 p=.53 x=9 q=.47 P(x=9)=n!/(n-x)!x! * p^x * q^n-x P(x=9)=30!/(30-9)!9! * .53^30 * .47^30-9 = P(x=9)= .0061 Table Unbiased Survey. #1. Do you work out daily? Yes or No n=15 p=.47 x=7 q=.53 P(x=6)=.153 Software Unbiased Survey. #1. Do you work out daily? Yes or No n= 30 p=.53 x<8 q=.47 P(x<8) = 1-binomcdf(30,.53,7) = .9991 Not unusual because .9991 is not less than .05

22. 1.mean=np=(30)(.53)=15.9 variance=npq=(30)(.53)(.47)= 7.473 standard deviation=root of variance=2.7337 2.mean=np=(30)(.83)=24.9 10.mean=np=(30)(.83)=24.9 variance=npq=(30)(.83)(.17)=4.233 variance=npq=(30)(.83)(.17)=423.3 Standard deviation=root of variance=2.0574standard deviation=root of variance=20.5743 3.mean=np=(30)(.37)=11.1 variance=npq=(30)(.37)(.63)=6.993 Standard deviation=root of variance=2.6444 5.mean=np=(30)(.67)=20.1 variance=npq=(30)(.67)(.33)=6.633 Standard deviation=root of variance=2.5755 6.mean=np=(30)(.6)=18 variance=npq=(30)(.6)(.4)=7.2 Standard deviation=root of variance=2.6833 7.mean=np=(30)(.33)=9.9 variance=npq=(30)(.33)(.67)=6.633 Standard deviation=root of variance=2.5755 8.mean=np=(30)(.43)=12.9 variance=npq=(30)(.34)(.57)=7.353 Standard deviation=root of variance=2.7116 9.mean=np=(30)(.67)=20.1 variance=npq=(30)(.67)(.33)=6.633 Standard deviation=root of variance=2.5755 Unbiased mean, variance & standard deviation

23. 1.mean=np=(30)(.6)=18 variance=npq=(30)(.6)(.4)= 7.2 standard deviation=root of variance=2.6833 2.mean=np=(30)(.6)=18 10.mean=np=(30)(.6)=18 variance=npq=(30)(.6)(.4)=7.2 variance=npq=(30)(.6)(.4)=7.2 Standard deviation=root of variance=2.6833standard deviation=root of variance=2.6833 3.mean=np=(30)(.67)=20.1 variance=npq=(30)(.67)(.33)=6.633 Standard deviation=root of variance=2.5755 5.mean=np=(30)(.63)=18.9 variance=npq=(30)(.63)(.37)=6.993 Standard deviation=root of variance=2.6444 6.mean=np=(30)(.13)=3.9 variance=npq=(30)(.13)(.87)=3.393 Standard deviation=root of variance=1.8420 7.mean=np=(30)(.83)=24.9 variance=npq=(30)(.83)(.17)=4.233 Standard deviation=root of variance=2.0574 8.mean=np=(30)(.37)=11.1 variance=npq=(30)(.37)(.63)=6.993 Standard deviation=root of variance=2.6444 9.mean=np=(30)(.57)=17.1 variance=npq=(30)(.57)(.43)=7.353 Standard deviation=root of variance=2.7116 Biased mean, variance & standard deviation

24. Conclusion Through this project, we are able to conclude that the answers given on the bias surveys were much different than the ones from the normal surveys although the questions were the same. Some survey takers were able to catch on to the bias while others were not. From our data, we are able to conclude that the majority of the students we surveyed claim to be healthy and fit. Most students on both surveys answered “Yes” when asked if they exercised weekly and “Yes” that they are eating healthy foods. The data shows that in the bias data people were more inclined to answer the questions in favor of more healthy choice because of the bias way the question was worded.

25. If we were to repeat this project... * We would asked less questions so that we could have focused more on the topic and made sure that the questions for that topic were relevant and made sense. * We also probably would choose to focus more on Sports next time, instead of focusing on sports and nutrition. *Also, we would ask a question in our survey whether or not the survey-taker is a boy or a girl considering it affects some of the other questions. * Finally, we would survey the same class with both surveys to see if the bias truly affected the answers to the questions.