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(三) 由於受查者內部控制之設計及執行通常具有穩定性，因此審計準則公報第 49 號「查核人員對所評估風險之因應」規定，於某些情況下，查核人員可以採用以往控制測試所取得之查核證據作為當期之控制測試查核證據。惟查核人員當期如計劃採用以往查核所取得對特定控制執行有效性之查核證據時，應遵循哪些規範？(18 分)

1. Let $\{Y1, \ldots, Yn\}$ be a random sample from the normal distribution with mean $\mu$ and known variance $\sigma^2$. The hypotheses are $H0: \mu = 50$ vs $Ha: \mu > 50$ and the rejection region $R = \{\hat{y}: \hat{y} > 50 + a\}$. At 0.05 level of significance, the value of $a$ is (5%) and the power of the test for true $\mu = 60$ is . (5%) (若無法算出一個值，請用累積密度函數等方式表達)

2. The probability density functions for X and (Y given X) are given below.$$\begin{array}{l} PX(x) = \frac{\lambda^x e^{-\lambda}}{x!} \text{ where } \lambda > 0 \text{ and } x = 1, 2, \ldots \\ P{Y|X}(y|x) = \binom{x}{y} p^y (1-p)^{x-y} \text{ where } 0 < p < 1 \text{ and } y = 1, 2, \ldots \end{array}$$The probability density function of Y is (5%) and the variance of Y is . (5%)

3. Suppose that each of two investments for a portfolio has a \( 4\% \) chance of a loss of $5 million, a \( 2\% \) chance of a loss of $1 million, and \( 94\% \) chance of a gain of $2 million. Two investments are independent of each other. The 10th percentile for the distribution of this portfolio investment return is __. (5%)

4. Consider a random sample \( \{x_1, \dots, x_n\} \) from the distribution with mean \( \mu \) and known variance \( \sigma^2 \). Define \( \bar{x} = n^{-1} \sum_{i=1}^{i=n} x_i \), \( s_1^2 = (n-2)^{-1} \sum_{i=1}^{i=n} (x_i - \mu)^2 \), and \( s_2^2 = (n-2)^{-1} \sum_{i=1}^{i=n} (x_i - \bar{x})^2 \). If n=60, the distributions of \( (n-2)s_1^2/\sigma^2 \) and \( (n-2)s_2^2/\sigma^2 \) are ________ and ________, respectively. (10%) (Justify your answers)

5. Consider a population of five weights identical in appearance but weighing 1, 3, 5, 7 and 9 ounces. The variance for the population (\( \sigma^2 \)) is ________. (5%) Sampling without replacement from the above population with a sample size of 2 produces ten possible samples. Using the ten sample mean values to calculate the variance of \( \bar{X} \), it is ________. (5%) Why is this value not the same as \( \sigma^2/n \)? ________ (5%)

(1) Conduct a statistical test to determine whether there is a significant difference between the variances in the SAT score of male and female students. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(2) Based on the result of previous question, conduct a statistical test to determine whether the mean SAT score of male students is significantly LOWER than the mean SAT score of female students. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(3) Are the proportions of in-state students the same for male and female students? Report the test statistic and P-value. (5%)

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