heads up! a coin flip experiment by. about i’ve always wondered: do coins follow the theoretical...
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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
8 7 10 5 9 6 8 7
17 13 18 12 15 15 13 17
27 23 23 27 25 25 26 24
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-hat Penny Nickel Dime Quarter
15 0.53 0.33 0.40 0.47
30 0.57 0.40 0.50 0.57
50 0.54 0.46 0.50 0.48
Confidence Intervals for ProportionsThe 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
Penny Nickel Dime Quarter
15 0.53(0.28,0.79)
0.33(0.09,0.57)
0.40(0.15,0.65)
0.47(0.21,0.72)
30 0.57(0.39,0.74)
0.40(0.22,0.58)
0.50(0.32,0.68)
0.57(0.39,0.74)
50 0.54(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 ProportionsConditions:
np>10, n(p-1)>10, and population >10n
Hypotheses:
H0: p=0.5 [this is the proportion of heads]
Ha: p≠0.5
Test Statistic:
P-value: 2*normalcdf(Z,999) if Z>0 or 2*normalcdf(-999,Z) if Z<0
Significance Test: PenniesAlthough 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: -1.29
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: -0.57
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: -0.77
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: -0.26
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: -0.28
P-value: 2*normalcdf(-999,-0.28) which is 0.78
P-values
P-hat/Confidence Intervals/P-value
Penny Nickel Dime Quarter
15 0.53(0.28,0.79)
0.80
0.33(0.09,0.57)
0.20
0.40(0.15,0.65)
0.44
0.47(0.21,0.72)
0.80
30 0.57(0.39,0.74)
0.47
0.40(0.22,0.58)
0.27
0.50(0.32,0.68)
0.99
0.57(0.39,0.74)
0.47
50 0.54(0.40,0.68)
0.57
0.46(0.32,0.60)
0.57
0.50(0.36,0.64)
0.99
0.48(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 patternNickels- The p-value increasedDimes- The p-value increasedQuarters- The p-value decreasedI 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.