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Discussion and Results
The original intention was to examine the data using a Chi-squared test to determine if the side of a penny that was up at the maximum of its trajectory was independent of its initial condition. After taking our photos and determining what side faced up in each, we began calculating the Chi-squared statistic for each by hand using a TI-89 calculator; however, we soon realized that our data set was far too small to be of any statistical significance. In order for a Chi-squared test to be valid, each box in the table must have a value greater than or equal to five. The calculations are shown below. Observed values are blue and calculated expected values are red. For each box the expected values were calculated by multiplying the sum of the values in the column by the sum of the values in the row and dividing the product by the sum of all values in the chart.
| Heads Force One | Heads Force Two | Tails Force One | Tails Force Two | |
| Heads at Peak | 3 3.235 | 4 2.588 | 2 2.588 | 2 2.588 |
| Tails at Peak | 2 1.76 | 0 1.412 | 2 1.412 | 2 1.412 |
H0: Initial side and side in mid air are independent.
HA: Initial side and side in mid air are not independent.
χ² = Σ (Expected - Observed)²/Expected α = .05 df = (4-1)(2-1) = 3
χ² = 2.9981
Critical χ² = 7.814 FAIL TO REJECT Null Hypothesis (H0)
Failing to reject the null hypothesis means that the data indicates that the initial side and the side the penny is is mid air are independent.
In an attempt to increase the value of each cell, we lumped all of the data with an initial state of heads together and likewise for tails
| Heads Initially | Tails Initially | |
| Heads at Peak | 7 5.8235 | 4 5.176 |
| Tails at Peak | 2 3.176 | 4 2.8235 |
H0: Initial side and side in mid air are independent.
HA: Initial side and side in mid air are not independent.
χ² = 4.1697 α = .05 df = (2-1)(2-1) = 1
Critical χ² = 3.841 REJECT Null Hypothesis
Reject the null hypothesis means that the data indicates that the initial side and the side the penny is is mid air are not independent.
When we combined the forces we reached a different conclusion than when they were separated. This further supports the notion that we did not have enough data for our experiment to have any statistical significance.