P-Value for a Fair Coin

Determining the P-Value for a Fair Coin

To assess the fairness of a coin, you can perform a hypothesis test to determine if the observed outcomes deviate significantly from what would be expected under the assumption of a fair coin. Here's how you can approach this problem:

---

1. Define the Hypotheses

• Null Hypothesis ($H_0$): The coin is fair, meaning the probability of landing heads ($p$) is 0.5.

  $H_0: p = 0.5$

• Alternative Hypothesis ($H_a$): The coin is not fair, meaning the probability of landing heads is not equal to 0.5.

  $H_a: p \neq 0.5$

---

2. Choose a Significance Level

Select a significance level ($\alpha$), which is the threshold for determining whether the observed data is statistically significant. Common choices are 0.05 or 0.01.

---

3. Conduct the Experiment

Flip the coin $n$ times and record the number of heads ($x$). The larger the sample size, the more reliable your test results will be.

---

4. Calculate the Test Statistic

The number of heads follows a binomial distribution:

$x \sim \text{Binomial}(n, p)$

For large $n$, you can approximate the binomial distribution with a normal distribution and calculate the z-score:

$z = \frac{x - n \cdot 0.5}{\sqrt{n \cdot 0.5 \cdot (1 - 0.5)}} = \frac{x - n \cdot 0.5}{\sqrt{n}/2}$

---

5. Compute the P-Value

For a two-tailed test, calculate the p-value as:

$p\text{-value} = 2 \cdot P(Z > |z|)$

Where $Z$ is the standard normal distribution. This p-value represents the probability of observing a test statistic as extreme as the one calculated, under the null hypothesis.

---

6. Make a Decision

• If $p\text{-value} \leq \alpha$, reject the null hypothesis. This indicates that the observed data provides sufficient evidence to conclude that the coin is not fair.
• If $p\text{-value} > \alpha$, fail to reject the null hypothesis. This means there isn't enough evidence to suggest the coin is biased.

---

Example

Suppose you flip a coin 100 times and observe 60 heads.

1. Calculate the z-score:

   $z = \frac{60 - 50}{\sqrt{100}/2} = \frac{10}{5} = 2$

2. Determine the p-value:

   For $z = 2$, the p-value for a two-tailed test is approximately 0.0456.

3. Compare to $\alpha = 0.05$:

   Since 0.0456 < 0.05, reject the null hypothesis. The coin is likely biased.

This detailed explanation outlines the process of calculating the p-value to assess the fairness of a coin using hypothesis testing.