(b) \((10\%)\mathrm{H}_0: \mu_x = \mu_y\) against \(\mathrm{H}_1: \mu_x > \mu_y\) with \(\alpha = 0.01\).

(Some critical values $F_\alpha(df_1, df_2)$ and $t_\alpha(df)$, where $\Pr(F > F_\alpha(df_1, df_2)) = \alpha$ and $\Pr(t > t_\alpha(df)) = \alpha$, are given as follows: $$ \begin{array}{l} F_{0.02}(24, 29) = 2.236, F_{0.01}(24, 29) = 2.495, F_{0.02}(29, 24) = 2.306, \\ F_{0.01}(29, 24) = 2.589, F_{0.02}(30, 25) = 2.267, F_{0.01}(30, 25) = 2.538, \\ t_{0.01}(53) = 2.398, t_{0.01}(54) = 2.397, t_{0.01}(55) = 2.396, \\ t_{0.005}(53) = 2.672, t_{0.005}(54) = 2.670, t_{0.005}(55) = 2.668 \end{array} $$ 3. Let $X_1, X_2, \ldots$, and $X_n$ be a random sample from distributions with the following probability density functions. In each case find the maximum likelihood estimator $\hat{\theta}$. (a) \((5\%) f(x; \theta) = \theta x^{\theta - 1}, 0 < x < 1, 0 < \theta < \infty\)

(b) \((5\%) f(x; \theta) = 1 / \theta, 0 < x < \theta < \infty\)

4. (10%) Assume that a stock is picked from a population of stocks at random. The average return of the population of stocks is 20%, and the standard deviation of the population of stocks is 2%. To calculate the maximum of the probability that a stock picked at random from this population, which has a return less than 16% or greater than 24%.

5. In a presidential election campaign, two potential candidates argue who has higher supporting ratio. They tend to incorporate into a candidate group, however, they have inconsistent ideas on who should be the vice-president candidate. Candidate A suggests if the poll demonstrates the difference of supporting rate between two candidate groups is not significant, he will support B to be the presidential candidate; otherwise, A should be the presidential candidate and B the vice. (a) \((7\%)\) If the poll surveys at \(95\%\) confidence level and the sampling error is approximately \(\pm 3\%\), how large of the sample size should be interviewed at least? \(\left\{Z_{0.975} = 1.96\right\}\)

(b) \((7\%)\) The results of the poll: Group (A,B) represents A the presidential candidate and B the vice; Group (B,A) vice versa.

| | Support rate (p) | Sample size (N) | | --- | --- | --- | | Group (A,B) | 48.3% | 2,046 | | Group (B,A) | 46.1% | 2,046 | How do they make the decision? (at 95% confidence level) 6. A bank constructs a computer system for reviewing small-size loans. The following data discloses the cases reviewed by the system: Total loan cases: 54,231 Passed cases by computer review: 49,082 Non-default cases after computer review: 46,880 Default cases after computer review: 2,202 Cases rejected by computer system: 5,149. Then the cases are sent to be reviewed by manually. Non-default cases after manual review: 4,585; default cases after manual review: 564. (a) (8%)By the perspective of the computer system, fill up the following confusion matrix with number of cases.

(b) (4%) What are the ratios of type I error and type II error?

(c) (14%) Calculate "Accuracy ratio", "Precision ratio", "Recall ratio", "False positive ratio", "False negative ratio", "F1-score".

7. (10%)Please derive when n→∞ and small p, binomial distribution will approach to Poisson distribution.

1. We have n activities \( a1, a2, \ldots, an \). Each activity \( ai \) has a start time \( si \) and a finish time \( fi \) where \( 0 \leq si < fi \). If \( ai \) is selected, activity \( ai \) takes place during the half-open time interval \( [si, fi] \). Activities \( ai \) and \( aj \) are compatible if the intervals \( [si, fi] \) and \( [sj, fj] \) do not overlap. In the Activity Selection Problem, we try to select a maximum-size subset of mutually compatible activities. Design an O(nlog n)-time algorithm to solve this problem. (15分)

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