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To derive a confidence interval for \( p \) using Chebyshev's inequality, let's follow a step-by-step process. Given that \( X_1, X_2, \dots, X_n \) are independent and identically distributed (i.i.d.) Bernoulli random variables with parameter \( p \) (i.e., \( X_i \sim \text{Bernoulli}(p) \)), we can proceed as follows: ### 1. Mean and Variance of Bernoulli Distribution The expected value and variance of each \( X_i \) are: - \( E(X_i) = p \) - \( \text{Var}(X_i) = p(1 - p) \) ### 2. Sample Mean Define the sample mean \( \bar{X} \) as the average of the \( n \) observations: \[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \] By the law of large __________ _________ ___ ___ ___ ____ ______.
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