You are shown a coin that its owner says is fair in the sense that it will produce the same number of heads and tails when flipped a very large number of times.

To verify whether the coin is fair or not, you can perform a hypothesis test called the "Chi-Square test for goodness of fit". This test will help determine if the observed results (number of heads and tails) match the expected results for a fair coin.

Here are the steps to conduct the Chi-Square test for goodness of fit:

1. Define the null hypothesis (H0): Assume that the coin is fair, meaning it has an equal chance of producing a head or a tail.

2. Set the significance level (α): Typically, a significance level of 0.05 (or 5%) is chosen, which means accepting a 5% chance of falsely rejecting the null hypothesis.

3. Gather data: Flip the coin a large number of times, record the outcomes (heads or tails), and count the number of each.

4. Calculate the expected frequencies: Since we assume the coin is fair, we expect an equal number of heads and tails. So, divide the total number of flips by 2 to get the expected count for each outcome.

5. Calculate the test statistic: Use the Chi-Square formula to calculate the test statistic. It measures the difference between the observed and expected results, taking into account the expected frequencies. The formula is:
χ² = Σ((Observed - Expected)² / Expected)

6. Determine the critical value: Based on the significance level and the degrees of freedom, lookup the critical value from the Chi-Square distribution table.

7. Compare the test statistic to the critical value: If the test statistic is greater than the critical value, reject the null hypothesis (H0) and conclude that the coin is not fair. If the test statistic is smaller, fail to reject the null hypothesis and conclude that there is no evidence to suggest that the coin is biased.

Note that the degrees of freedom for this test are given by (n - 1), where n is the number of different categories/outcomes (in this case, heads and tails).

By following this procedure, you can assess the fairness of the coin and determine if it produces an equal number of heads and tails as claimed.