Step 1: Understanding the Concept:
A Type I error occurs when the null hypothesis (\(H_0\)) is rejected despite being true. The probability of committing a Type I error is denoted by \(\alpha\), which is also the significance level or size of the test.
Step 2: Key Formula or Approach:
\(\alpha = P(\text{Reject } H_0 | H_0 \text{ is true})\).The experiment is modeled by a binomial distribution. Let X represent the count of heads in 5 tosses, so \(X \sim \text{Bin}(n, p)\).The rejection criterion is \(X \ge 3\). We need to determine \(P(X \ge 3)\) assuming \(H_0\) is valid, i.e., \(p = 1/2\).
Step 3: Detailed Explanation:
Given \(H_0\), X follows a binomial distribution with parameters \(n=5\) and \(p=1/2\).
The probability mass function (PMF) is defined as \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\).
\[ P(X=k) = \binom{5}{k} \left(\frac{1}{2}\right)^k \left(\frac{1}{2}\right)^{5-k} = \binom{5}{k} \left(\frac{1}{2}\right)^5 = \frac{\binom{5}{k}}{32} \]
The Type I error probability is the probability of rejecting \(H_0\), i.e., \(P(X \ge 3)\).
\[ \alpha = P(X \ge 3) = P(X=3) + P(X=4) + P(X=5) \]Calculating each probability:
- \( P(X=3) = \frac{\binom{5}{3}}{32} = \frac{10}{32} \)
- \( P(X=4) = \frac{\binom{5}{4}}{32} = \frac{5}{32} \)
- \( P(X=5) = \frac{\binom{5}{5}}{32} = \frac{1}{32} \)
Summing the probabilities:\[ \alpha = \frac{10}{32} + \frac{5}{32} + \frac{1}{32} = \frac{16}{32} = \frac{1}{2} \]
Step 4: Final Answer:
The probability of a Type I error is \( \frac{1}{2} \).