I spun a penny 175 times and got 70 heads. We wish to find how significant is this evidence against equal probabilities. What is the sample proportion of heads?

Question

Spinning a coin, unlike tossing it, may not give heads and tails equal probabilities. I spun a penny 175 times and got 70 heads. We wish to find how significant is this evidence against equal probabilities.

1. What is the sample proportion of heads?

Answer 1

The sample proportion of heads would be the number of heads x = 70 divided by the sample size n = 175.

This is p-hat = x / n = 70 / 175 = 0.4

To determine how significantly different this is from an assumed equal probability p₀ = 0.5, we can standardize (normalize) by taking the difference (p-hat - p₀) and dividing by the standard error of the proportion, SE(p) = √(p₀q₀/n), where q₀ = 1-p₀ = 0.5.

Calculating SE(p) = √(0.5*0.5/175) = 1/√700 = 0.037796

z = (p-hat - p₀)/SE(p) = -0.1 / 0.037796 = -2.65

From this we can determine the significance of the result, called the p-value.

p-value = P( |z| > 2.65 ) = 2 * P( z > 2.65 ) = 2 * (0.0040) = 0.0081 ± 0.0001 (rounding)

[Probabilities can be found using technology or from a z-score table. Let me know if you need help here.]

This p-value can be interpreted as the likelihood of seeing what we saw, or something more extreme, assuming the population proportion, p₀ = 0.5. The p-value is what's compared to alpha, or the significance level, to determine whether to reject the null hypothesis in a hypothesis test (essentially what we're performing here.)

If I can explain more about hypothesis testing and alpha please let me know. Hope this helps!