Binomial Project

1. Binomial Project Excellent Student #1 and 2

2. Introduction We picked the topic of California teen laws because we thought it would be interesting to see how often our fellow peers break the law. We used calculator randomization to select the teachers whose classes we would survey. We surveyed Van Lier’s and Ms. Sanchez’s classes.

3. Introduction Continued Van Lier’s class was given the unbiased survey: Do you text while driving? Yes No Do you stay out past curfew? Yes No Do you use a Bluetooth while driving? Yes No If under 18, have you had sexual relations with someone over 18, or vice versa? Yes No Did you drive minors before your year was up? Yes No

4. Introduction Continued Ms. Sanchez’s class was given the biased survey: People who text while driving are 23% more likely to be in a car accident. Do you text while driving?           Yes                         No Each year, 21% of fatal car crashes involving teenagers between the ages of 16 and 19 were the result of cell phone usage. This result has been expected to grow as much as 4% every year. Do you use a Bluetooth while driving?    Yes                         No Sixty-three percent of teenage passenger deaths in 2008 occurred in vehicles driven by another teenager. Among deaths of passengers of all ages, 19 percent occurred when a teenager was driving. Did/have you drive/driven minors before your year was up?   Yes                   No Sexual relations with a person under the age of consent is in general a criminal offense, with punishments ranging from community service up to the death penalty. If under 18, have you had sexual relations with someone over 18, or vice versa? Yes No Violators of the curfew law can be fined from $50 to several hundred dollars or charged with a misdemeanor. Do you stay out past curfew? Yes No

5. Introduction Contd. Our sample size was big enough because n>30. Our sample size for both the biased and unbiased surveys was 34.

6. Do you text while driving? Bias: People who text while driving are 23% more likely to be in a car accident.

7. Do you stay out past curfew? Bias: Violators of the curfew law can be fined from $50 to several hundred dollars or charged with a misdemeanor.

8. Do you use a bluetooth while driving? Bias:Each year, 21% of fatal car crashes involving teenagers between the ages of 16 and 19 were the result of cell phone usage. This result has been expected to grow as much as 4% every year.

9. If under 18, have you had sexual relations with someone over 18, or vice versa? Bias:Sexual relations with a person under the age of consent is in general a criminal offense, with punishments ranging from community service up to the death penalty.

10. Did you drive minors before your year was up? Bias:Sixty-three percent of teenage passenger deaths in 2008 occurred in vehicles driven by another teenager. Among deaths of passengers of all ages, 19 percent occurred when a teenager was driving.

11. Do you text while driving? 2NIP is satisfied because there are two outcomes (yes/no), there is a set number of trials (34), there are independent variables (survey takers), and a .21 set probability of getting a yes or a .79 set probability of getting a no. Problems: 1.Using the binomial formula, find the probability that exactly 4 people will say yes, or P(x=4). N=34 p=.21 q=.79 x=4. P(x) = n! / (n –x)!x! • px • qn-x P(x) = 34! / (34 –4)!4! • (.21)4 • (.79)34-4 P(x)= .0765567 2.Table: n=15 p=.21 (used .20 on table) P(x<5)= P(x=1) + P(x=2) + P(x=3)+ P(x=4) = .132 + .231+ .250+ .188= .801 3. Use your calculator to find P(x>3), or the probability that at least three people will say yes. • P(x>3)= 1-binomcdf (n,p,x) • P(x>3)= 1-binomcdf (34, .21, 2) = .9836 or 98.36%. The probability we get that more than three people will answer yes is .9836, or 98.36%, which makes sense when looking at our sample size of 34 and the number of people who originally said yes was 7, or 21%. This is not unusual because 98.36% is greater than 5%. Identify n, x, p, and q: n=34 x=4 p=.21 q=.79 Mean: n(p)= 34(.21) =7.14 Variance:npq= 34(.21)(.79)= 5.6406 Standard Deviation: Square root of 5.6406 = 2.375

12. Do you stay out past curfew? 2NIP is satisfied because there are two outcomes (yes/no), there is a set number of trials (34), there are independent variables (survey takers), and a .71 set probability of getting a yes, and a .29 set probability of getting a no. Problems: 1.Using the binomial formula, find the probability that exactly 6 people will say no. n=34 p=.29 x=6 P(x) = n! / (n –x)!x! • px • qn-x P(x) = 34! / (34-6)!6! • .29x • .7134-4 P (x) = .0547410391 2.Table: n=15 p=.29 (used .30 on table) P(x<4)= P(x=1) + P(x=2) + P(x=3) = .031+ .092 + .071= .293 3. Use your calculator to find P(x>6), or the probability that at least six people will say no. P(x>6)= 1-binomcdf (n,p,x) P(x>6)= 1-binomcdf (34, .29, 5) = .9564 or 95.64% The probability we get that more than six people will answer no is .9564, or 95.64%, which makes sense when looking at our sample size of 34 and the number of people who originally answered no was 10, or 29%. This is not unusual because 95.64% is greater than 5%. Identify n, x, p, and q: n=34 x=6 p=.29 q=.71 Mean: n(p)= 34(.29) =9.86 Variance:npq= 34(.29)(.71)= 7.0006 Standard Deviation: Square root of 7.0006 = 2.65

13. Do you use Bluetooth while driving? 2NIP is satisfied because there are two outcomes (yes/no), there is a set number of trials (34), there are independent variables (survey takers), and a .03 set probability of getting a yes or .97 set probability of getting a no. Problems: 1.Using the binomial formula, find the probability that exactly 21 people will say no. n=34 p=.97 x=21 P(x) = n! / (n – x)!x! • px • qn-x P(x)=34! / (34-21)!21! • 9721 • .0334-21 . P(x)=7.80410505E-12 Mean: n(p)= 34(.97) =32.98 Variance: npq= 34(.97)(.03)= .9894 Standard Deviation: Square root of .9894 = .9947 2.Table: n=15 p=.97(used .99 on table) P(x<6)= P(x=1) + P(x=2)+ P(x=3)+ P(x=4)+ P(x=5) = 0+0+0+0+0=0 3. Use your calculator to find P(x<2), or the probability that less than two people will say yes. P(x<2)= binomcdf (n,p,x) P(x<2)= binomcdf (34, .03, 3) = .9817 or 98.17% The probability we get that less than two people will answer yes is .9817, or 98.17%, which makes sense when looking at our sample size of 34 and the percentage of people who originally answered yes which is 3%. This is not unusual because 98.17% is greater than 5%. Identify n, x, p, and q: n=34 x=21 p=.03 q=.97 Mean: n(p)= 34(.03) =1.02 Variance: npq= 34(.03)(.97)= .9894 Standard Deviation: Square root of .9894 = .9947

14. Analysis/Conclusion • The biased questions did alter the results for all of our questions. • For texting while driving, 91% said they don’t text for the biased survey, and 79% said they don’t for the unbiased survey. • For curfew, more people said they don’t stay out past it when given the biased survey(35%) compared to the unbiased survey results(29%). • More people also said they used bluetooth’s when given the biased survey(18%). In the unbiased survey, only 3% said they did. • For the question about having sexual relations with someone over 18, more people said they have not had such relations when given the biased survey(88%) compared to the 79% for the unbiased. • And for driving minors before one’s year was up, more people said no in the biased survey(68%). In the unbiased, 56% said no. *For this project, we could have distributed more surveys, and we could have also asked more questions relating to teen laws in our survey.