STAT601 P.R. James

Business StatisticsJanuary 30, 2006

Normal Model – Investigative Task

Background:

Is there a slight advantage in a coin toss? Is the standard American quarter biased? Is one choice, heads or tails, better than the other? This is the question I wanted to answer. To do so, I used ten United States of America Quarter Dollars. Five of these quarters were the old pre-1999 quarters and five were the new State quarters. All ten of the quarters were placed in the bottom half of a cut-off plastic water bottle. The corrugated sides of the water bottle provided a lip that helped to further jumble the coins as the coins were dumped to the ground. Prior to dumping the coins, I shook the cup and the coins vigorously for one second while covering the opening. I dropped the coins onto a carpeted surface from a height of one foot. The carpet provided a small amount of cushioning.—only a fraction of the coins bounced noticeably or rolled more than six inches. After the coins stopped moving, I counted the number of coins that were face up or on the heads side, and then logged the number into an MS Excel spreadsheet. I repeated this process one hundred times. These raw data are located in Appendix A.

Results

The results, at least to me, are somewhat surprising. As you can see in the summary chart below, the mean number of coins that land heads-side-up per ten coinsthrown is 5.1 heads with a standard deviation of 1.63. Initially, these results indicate to me that there is an ever so

slight advantage by selecting heads, but as noted with the 95% confidence interval of the mean, we are 95% confident that the next trial will result in 4.8 to 5.4 heads on the throw. These interval limits, once rounded to the nearest integer, equal 5.

The boxplot is shown belowwith the minimum value of 1, maximum value of 9, and the values 4, 5, and 6 for the 25th, 50th, and 75th quartiles, respectively. The inner quartile is 2 compared to the expected value of 2.17 which is 1.33 standard deviations.

This boxplot is symmetrical and thus gives no indication that the distribution is slightly skewed, but the following histogram does show that the data are slightly skewed to the left. The mean is greater than the median, 5.1 versus 5.0.

In addition to the histogram, the Pareto Chart, below, shows where this imbalance is located. There were twice as many occurrences, fourteen to seven, of having seven heads thrown versus three heads thrown. The occurrence of two heads thrown, five times, equals the occurrences of eight heads thrown. Twice, only one heads was thrown while a throw of nine heads happened just once.

As seen in the Probability Plot of Heads Chart below, this experiment resulted in a normal distribution of data. Knowing that the distribution is normal is helpful when comparing the actual results with expected results. Based on a mean of 5.1 and a standard deviation of 1.63, 68.26% of the results should be between 3.47 and 6.73 heads. The experiment contains 66% of the data within the first standard deviation around the mean.

The second standard deviation around the mean, 1.84 to 5.36, contains 97% of the experimental data, which exceeds the expected 95.44%. The third standard deviation around the mean, 0.21 to 9.99, captures 100% of the experimental data. The expected percentage is 99.73%. The concurrence of the expected results and the actual results gives further assurance that the experimental data is correctly classified as normally distributed.

In summary, the data show that a toss of a coin is indeed a fifty-fifty proposition, but I have decided to keep these ten quarters off to the side in case I need to win a coin flip. I will choose heads for now on.

Appendix A: Raw Data

Trial # / # of Heads / Trial # / # of Heads / Trial # / # of Heads / Trial # / # of Heads / Trial # / # of Heads
1 / 5 / 21 / 5 / 41 / 3 / 61 / 2 / 81 / 5
2 / 6 / 22 / 3 / 42 / 5 / 62 / 6 / 82 / 5
3 / 7 / 23 / 4 / 43 / 4 / 63 / 4 / 83 / 5
4 / 5 / 24 / 4 / 44 / 6 / 64 / 3 / 84 / 6
5 / 6 / 25 / 6 / 45 / 6 / 65 / 2 / 85 / 3
6 / 5 / 26 / 2 / 46 / 8 / 66 / 7 / 86 / 1
7 / 5 / 27 / 5 / 47 / 8 / 67 / 6 / 87 / 4
8 / 4 / 28 / 6 / 48 / 4 / 68 / 7 / 88 / 5
9 / 4 / 29 / 5 / 49 / 1 / 69 / 2 / 89 / 8
10 / 6 / 30 / 6 / 50 / 4 / 70 / 5 / 90 / 4
11 / 5 / 31 / 4 / 51 / 6 / 71 / 7 / 91 / 7
12 / 6 / 32 / 4 / 52 / 4 / 72 / 7 / 92 / 9
13 / 7 / 33 / 5 / 53 / 3 / 73 / 6 / 93 / 4
14 / 7 / 34 / 4 / 54 / 7 / 74 / 6 / 94 / 6
15 / 6 / 35 / 5 / 55 / 4 / 75 / 5 / 95 / 7
16 / 4 / 36 / 7 / 56 / 6 / 76 / 7 / 96 / 5
17 / 4 / 37 / 5 / 57 / 6 / 77 / 6 / 97 / 5
18 / 4 / 38 / 3 / 58 / 5 / 78 / 2 / 98 / 4
19 / 8 / 39 / 6 / 59 / 8 / 79 / 3 / 99 / 7
20 / 5 / 40 / 7 / 60 / 4 / 80 / 5 / 100 / 5

Appendix B: Probability Density Function

Normal with mean = 5.1 and standard deviation = 1.63

x f( x ) / 5 0.244290 / 6 0.210146 / 7 0.124073 / 5 0.244290
6 0.210146 / 5 0.244290 / 5 0.244290 / 4 0.194908 / 4 0.194908
6 0.210146 / 5 0.244290 / 6 0.210146 / 7 0.124073 / 7 0.124073
6 0.210146 / 4 0.194908 / 4 0.194908 / 4 0.194908 / 8 0.050278
5 0.244290 / 5 0.244290 / 3 0.106732 / 4 0.194908 / 4 0.194908
6 0.210146 / 2 0.040115 / 5 0.244290 / 6 0.210146 / 5 0.244290
6 0.210146 / 4 0.194908 / 4 0.194908 / 5 0.244290 / 4 0.194908
5 0.244290 / 7 0.124073 / 5 0.244290 / 3 0.106732 / 6 0.210146
7 0.124073 / 3 0.106732 / 5 0.244290 / 4 0.194908 / 6 0.210146
6 0.210146 / 8 0.050278 / 8 0.050278 / 4 0.194908 / 1 0.010348
4 0.194908 / 6 0.210146 / 4 0.194908 / 3 0.106732 / 7 0.124073
4 0.194908 / 6 0.210146 / 6 0.210146 / 5 0.244290 / 8 0.050278
4 0.194908 / 2 0.040115 / 6 0.210146 / 4 0.194908 / 3 0.106732
2 0.040115 / 7 0.124073 / 6 0.210146 / 7 0.124073 / 2 0.040115
5 0.244290 / 7 0.124073 / 7 0.124073 / 6 0.210146 / 6 0.210146
5 0.244290 / 7 0.124073 / 6 0.210146 / 2 0.040115 / 3 0.106732
5 0.244290 / 5 0.244290 / 5 0.244290 / 5 0.244290 / 6 0.210146
3 0.106732 / 1 0.010348 / 4 0.194908 / 5 0.244290 / 8 0.050278
4 0.194908 / 7 0.124073 / 9 0.013984 / 4 0.194908 / 6 0.210146
7 0.124073 / 5 0.244290 / 5 0.244290 / 4 0.194908 / 7 0.124073
5 0.244290