3. A random sample of 100 citizens is selected from a city to investigate the satisfaction rate with municipal services. The decision rule is as follows: if between 55 and 65 people (inclusive) express approval, it is concluded that the satisfaction rate is 60%. Please calculate:

(1) The probability of a Type I error. (6%)

(2) The power of the test when the true proportion is \( p = 0.5 \). (6%)

4. Regarding survey responses, people often hesitate to provide an affirmative answer. Suppose that in a population, 80% would truthfully answer "No" to a question, while 20% would truthfully answer "Yes." However, 70% of the individuals lie about their response. Let $Y$ be the number of people who must be surveyed until the first affirmative ("Yes") response is obtained. Derive the probability distribution of $Y$. (7%)

5. In electronics, current is denoted as $I$, resistance as $R$, and power as $W = I^2 \times R$. Suppose $I$ follows a uniform distribution over the interval $(0, 1)$, and the probability density function of $R$ is $f(r) = \begin{cases} 2r, & 0 \le r \le 1 \\ 0, & \text{otherwise} \end{cases}$ Given that $I$ and $R$ are independent, please find: (1) The probability density function of \(W\). (12%)

(2) The expected value of \( W \), denoted as \( E(W) \). (6%)

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