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6. Consider the linear population model with intercept:$y_t = \alpha + \beta x_t + \varepsilon_t$The variance structure is known as below: $E(\varepsilon_t^2) = \sigma^2 h_t^2,$Which regression may obtain efficiency $\hat{\beta}$ estimator? (A) Regress \( y_{t} \) on 1, \( x_{t} \) (B) Regress \( y_{t} / h_{t} \) on 1/ht ,xt/ht(C) Regress \( y_{t} / h_{t}^{2} \) on 1/h2t ,xt/h2t(D) Regress \( y_{t} \) on 1/ht ,xt/ht(E) Regress \( y_{t} / h_{t} \) on 1, \( x_{t} \)

5. Consider the linear model with intercept: $y_t = \alpha + \beta x_t + u_t$ If the $A = \sum_{i=1}^{n} (x_i - \bar{x})^2$ is positive definite and $E(u_i \mid x_i) = 0, \forall t = 1, 2, \dots, T$, which statement is true under sample size $T \to \infty$. (A) \( \hat{\beta} \) is consistent, but we cannot judge whether \( \hat{\beta} \) is efficiency (B) \( \hat{\beta} \) is unbiased and efficiency (C) \( \hat{\beta} \) is consistent and efficiency (D) \( \hat{\beta} \) is, but we cannot judge whether \( \hat{\beta} \) is efficiency (E) \( \hat{\beta} \) is unbiased and consistent.

4. Consider the simple regression $y_i = \alpha + \beta x_i + \varepsilon_i$And suppose population parameter $\beta = 2$ and least square estimator $\hat{\beta} = 2.3$. Now consider the regression $(x_i - y_i) = \alpha^* + \beta^* x_i + u_i.$What is the value of \( \hat{\beta} \)? (A) 1 (B)-1 (C)-1.3 (D)-0.3 (E) 0.3

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43. 李先生 75 歲，因呼吸急促到急診，胸部 X 光顯示左側肋膜有積液，下列肋膜液分析結果 何者不屬於滲出液 ( exudate ) 的特性？ (A)Pleural fluid pH < 7.30 (B)Pleural fluid : serum protein ratio < 0.5 (C)Pleural fluid : serum LDH ratio > 0.6 (D)Pleural fluid LDH > 0.67 upper limit of normal for serum LDH

73.下列胸水檢查結果何者較不像是 exudate？ (A) pleural fluid protein level ＞ 3.0 g/dL(B) pleural fluid: serum protein ratio ＞ 0.5(C) pleural fluid lactic dehydrogenase ＜ 100 IU/L(D) pH 值 ＜ 7.20

1 “As a result, climate change is likely to deepen existing global inequalities.” Millions of people in East Africa are suffering from severe drought, while violent storms continue to damage parts of Southeast Asia. Coffee farmers in Central America are also facing declining yields due to rising temperatures. (A) Poorer nations are generally less able to cope with such environmental changes and are often located in regions where climate conditions are more extreme. (B) As global temperatures rise, these countries are expected to bear the greatest burden. (C) Experts estimate that by 2030, billions of dollars will be required annually for adaptation efforts. (D) In addition, higher borrowing costs make it even more difficult for these nations to respond effectively. (E) This may lead to a worsening cycle of vulnerability and financial strain

2. "Meanwhile, it has also diverted attention from other ongoing global threats." The COVID-19 pandemic has shown how a global risk can quickly become a widespread crisis, affecting both public health and economic stability. (A) In the short term, it has intensified problems such as job losses and declining social trust. (B) At the same time, long-term challenges like climate change and technological inequality remain unresolved. (C) Although vaccines were developed rapidly, the crisis revealed the limits of reactive responses. (D) Greater preparation and international cooperation might have reduced the damage. (E) This experience highlights both the risks the world faces and the opportunity to build a more resilient future.

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11. 設 t 為實數，直線 \(L_t\)：\((3\cos t)x + (5\sin t)y = 15\)。若 f(t) 表示原點 (0,0) 到直線 \(L_t\) 的距離，求 f(t) 的最小值為____。