You wish to test whether or not a coin is fair, so you toss it 400 times and obtain 220 heads. Test the null hypothesis that the coin is fair and balanced against the alternative that it is not fair and balanced. Use the 1% significance level. Be sure to identify the null and alternative hypotheses, identify the appropriate distribution and indicate the tail(s) for the critical region, identify and calculate the test statistic, determine the P-value, make a decision and state a conclusion.

To test whether the coin is fair or not, we need to set up the null and alternative hypotheses.

Null hypothesis (H0): The coin is fair and balanced.
Alternative hypothesis (H1): The coin is not fair and balanced.

To determine if the coin is fair, we can use a hypothesis test and calculate the test statistic and the associated p-value.

We will use the binomial distribution to analyze the data since it involves tossing a coin that can yield a head or tail (success or failure) with equal probabilities.

The test statistic we will use in this case is the z-score. To calculate the z-score, we need to know the sample proportion of heads (p̂), expected proportion of heads for a fair coin (p), sample size (n), and standard deviation (σ).

The sample proportion of heads is p̂ = 220/400 = 0.55 (where 220 is the number of heads and 400 is the total number of tosses).

For a fair coin, the expected proportion of heads would be p = 0.5.

The standard deviation can be calculated using the formula σ = sqrt(p(1-p)/n), where p is the expected proportion of heads and n is the sample size.

Using this information, we can calculate the standard deviation:
σ = sqrt(0.5 * (1 - 0.5) / 400) = sqrt(0.25/400) = 0.025

Now, we can calculate the z-score using the formula:
z = (p̂ - p) / σ = (0.55 - 0.5) / 0.025 = 2

The critical region for the test will be determined by the significance level of 1%. Since this is a two-tailed test, we need to split the significance level between the two tails.

To find the critical z-value for a 1% significance level, we divide the significance level by 2 and look up the corresponding z-value in the standard normal distribution table.

At a significance level of 1%, the critical z-value is approximately ±2.576.

Since our calculated z-score of 2 falls within the non-critical region (-2.576 to +2.576), we fail to reject the null hypothesis.

The p-value is the probability of getting a test statistic value as extreme as the one calculated, assuming the null hypothesis is true. In this case, the p-value is the probability of getting 220 or more heads in 400 coin tosses with a fair coin.

To calculate the p-value, we use the standard normal distribution table or a statistical software. The p-value for a z-score of 2 is approximately 0.0228.

Since the p-value (0.0228) is greater than the significance level (0.01), we fail to reject the null hypothesis.

Therefore, based on the data provided, we do not have enough evidence to conclude that the coin is unfair or biased.