Heads Up! A Coin Flip Experiment By

About I’ve always wondered: Do coins follow the theoretical proportion of P(head)= 0.5? Because the heads/tails of coins are different, this might skew the probability. I wanted to test different coins and see which ones were the closest to the theoretical probability.

Data Collection The first row is the number of heads/tails in a trial of sample size n=15. The second row is the number of heads/tails in a trial of sample size n=30. The third row is the number of heads/tails in a trial of sample size n=50.

Sample Proportions of Heads [p-hat] P-hatPennyNickelDimeQuarter

Confidence Intervals for Proportions The formula: p ± z* √{p(1 − p)/n} where p is p- hat, or the sample proportion of heads. Z* is the critical value that can be found in Table B n is the sample size The confidence interval is usually in interval format: (a,b)

95% Confidence Intervals P-hat/ Confidence Intervals PennyNickelDimeQuarter (0.28,0.79) 0.33 (0.09,0.57) 0.40 (0.15,0.65) 0.47 (0.21,0.72) (0.39,0.74) 0.40 (0.22,0.58) 0.50 (0.32,0.68) 0.57 (0.39,0.74) (0.40,0.68) 0.46 (0.32,0.60) 0.50 (0.36,0.64) 0.48 (0.34,0.62)

Significance Tests for Proportions Conditions: np>10, n(p-1)>10, and population >10n Hypotheses: H 0 : p=0.5 [this is the proportion of heads] H a : p≠0.5 Test Statistic: P-value: 2*normalcdf(Z,999) if Z>0 or 2*normalcdf(-999,Z) if Z<0

Significance Test: Pennies Although some conditions are not met, we will still conduct the significance tests and proceed “with caution”: Sample Size 15: Test Statistic: 0.26 P-value: 2*normalcdf(0.26,999) which is 0.80 Sample Size 30: Test Statistic: 0.73 P-value: 2*normalcdf(0.73,999) which is 0.47 Sample Size 50: Test Statistic: 0.57 P-value: 2*normalcdf(0.57,999) which is 0.57

Significance Test: Nickels Although conditions are not met, we will still conduct the significance tests and proceed “with caution”: Sample Size 15: Test Statistic: P-value: 2*normalcdf(-999,-1.29) which is 0.2 Sample Size 30: Test Statistic: -1.1 P-value: 2*normalcdf(-999,-1.1) which is 0.27 Sample Size 50: Test Statistic: P-value: 2*normalcdf(-999, -0.57) which is 0.57

Significance Test: Dimes Although some conditions are not met, we will still conduct the significance tests and proceed “with caution”: Sample Size 15: Test Statistic: P-value: 2*normalcdf(-999,-0.77) which is 0.44 Sample Size 30: Test Statistic: 0 P-value: 2*normalcdf(0,999) which is 0.99 Sample Size 50: Test Statistic: 0 P-value: 2*normalcdf(0,999) which is 0.99

Significance Test: Quarters Although some conditions are not met, we will still conduct the significance tests and proceed “with caution”: Sample Size 15: Test Statistic: P-value: 2*normalcdf(-999,-0.26) which is 0.80 Sample Size 30: Test Statistic: 0.73 P-value: 2*normalcdf(0.73,999) which is 0.47 Sample Size 50: Test Statistic: P-value: 2*normalcdf(-999,-0.28) which is 0.78

P-values P-hat/ Confidence Intervals/ P-value PennyNickelDimeQuarter (0.28,0.79) (0.09,0.57) (0.15,0.65) (0.21,0.72) (0.39,0.74) (0.22,0.58) (0.32,0.68) (0.39,0.74) (0.40,0.68) (0.32,0.60) (0.36,0.64) (0.34,0.62) 0.34

Conclusions Pennies: All of the P-values were too high to be significant so we would not reject the null hypothesis. Nickels: All of the P-values were too high to be significant so we would not reject the null hypothesis. Dimes: All of the P-values were too high to be significant so we would not reject the null hypothesis. Quarters: All of the P-values were too high to be significant so we would not reject the null hypothesis. This means that even though both sides of a coin may be different weights, it is not statistically significant enough to alter the probability of heads/tails.

Other Notes As sample size increased: Pennies- The p-value has no pattern Nickels- The p-value increased Dimes- The p-value increased Quarters- The p-value decreased I would have expected that the p-value increases with an increase in sample size due to the Central Limit Theorem [as sample size increases, p-hat gets closer to 0.5]

Things to Remember Since I only flipped coins a maximum of 50 times, there still will be inaccuracy because the sample size is not that big. Also, some conditions for the smaller sample sizes were not met in order to do the significance tests.