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In hypothesis testing, a Type I error occurs when we wrongly reject a true null hypothesis. Imagine you think a coin is biased and test flipping it. Even if the coin is fair, there’s a chance your test suggests it's biased - that's a Type I error. For our Phoenix toll road exercise, rejecting the notion that 30% of residents favor the proposal when exactly 30% do favor it is a Type I error.
To calculate this error, we use the binomial distribution – a statistical test used for counts of outcomes. The formula involves understanding the number of trials (n), usually representing sample size, and the probability of success (p). For our case:

• n = 10 (sample size)
• p = 0.3 (proportion favoring the proposal)

The probability of Type I error (rejecting the null when it’s true) is the chance of 1 or fewer successes in 10 trials. This is computed using: \(P(X \leq 1)\) for a binomial setup, resulting in around 0.1493, or 14.93% chance of making this mistake.

A Type II error occurs when we fail to reject a false null hypothesis. It’s like missing that the coin actually is biased. Using Phoenix residents again, a Type II error implies concluding that the percent favoring toll roads isn't less than 30%, when it actually is lower. Specifically, if only 20% prefer it but we fail to recognize it, that’s a Type II error.For a binomial distribution with n = 10 and p = 0.2, to find this error, we calculate 1 minus the probability of up to 1 success, \(P(X \leq 1)\). Here’s how to break that down:

• Compute \(P(X = 0)\) which gives the chance all 10 disagree (all 0's).
• Then \(P(X = 1)\), which indicates 1 out of 10 are in favor.

Adding those gives us 0.3758. Thus, the Type II error probability is 1 - 0.3758 = 0.6242, or 62.42%.

Understanding the binomial distribution is key to tackling hypothesis testing problems like our exercise. This distribution is used when counting outcomes of binary events (success/failure) across several trials. Common examples include coin tosses or survey responses, where answers fall into two categories.Each event or trial must be independent, such as separate yes/no responses from surveyed residents. The number of trials (n) and the probability of success (p) define your distribution.This case has:

• n = 10 trials (residents surveyed)
• p = 0.3 for Type I, or p = 0.2 for Type II when assessing likelihood.
  

The formula to calculate exact probabilities uses this setup: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) allowing precise probabilities for specific outcomes, foundation to hypothesis testing.

In hypothesis testing, the power of a test reveals how good a test is at finding a true effect or rejecting a false null hypothesis. High power means the test is effective at detecting real, often subtle differences. For our exercise, the power is about accurately recognizing if less than 30% of Phoenix residents support toll roads, assuming actual support is at 20%. An effective test with a low Type II error rate has higher power. This value is computed as 1 minus the Type II error rate. Using our exercise values:

• Type II Error = 0.6242 (from previous calculation),
• Power = 1 - 0.6242 = 0.3758.

Thus, there is a 37.58% power, suggesting our test might not be the best at recognizing the true lower support level.