Heads Up! A Coin Flip Experiment

Heads Up!A Coin Flip Experiment

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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





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, -0.57) which is 0.57

Significance Test: Dimes

Although some conditions are not met, we will still



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







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.

Heads Up! A Coin Flip Experiment

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