2. 嘉義大學農場生產的雞蛋的寬度分布情況如下：

請問此數據是否支持下列假說：雞蛋的寬度大致呈常態分布為 N(16.5, 1)？ (10 %)

3. 一新醫學療法宣稱，其在延長癌症末期病人的壽命方面，較傳統療法更具效果。根據醫學期刊紀錄，在傳統療法下，病人平均延長 4.2 年的壽命，而根據對 80 位病人接受新療法所得到的結果，其樣本平均值與標準差分別為 4.5 年和 1.1 年。在顯著水準=0.05 下，檢定這個宣稱是否正確？(10 %)

(a) Find the normal equations. (5 %)

(b) Prove that the matrices H and are symmetric, idempotent. (5 %)

(c) Prove that H is orthogonal to (5 %)

(b) Calculate the E(Y). (5 %)

1. 貨幣政策在穩定經濟體系扮演重要角色，然而我們必須先了解貨幣供給如何在中央銀行、存款機構與存款戶間進行，請以數學推導與詳細說明貨幣乘數。(25%)

2. 請以經濟模型詳細說明實質景氣循環之內涵。(25%)

3. Migration is a central issue for the 2024 US presidential election, with debates focusing on its economic, social, and political impacts. Suppose that there are two countries, Country 1 and Country 2, each with two types of labor: high-skilled (H) and low-skilled (L). Both countries have the same amount of physical capital, high-skilled labor and low-skilled labor but differ in how labor inputs contribute to production. The production functions for the two countries are:$Y_1 = A K_1^\alpha L_1^\beta H_1^{1-\alpha-\beta},$$Y_2 = A K_2^\alpha (L_2 + \eta H_2)^{1-\alpha}.$where $K_1$ and $K_2$ are physical capital in Country 1 and Country 2, $\eta > 1$ is the relative productivity of high-skilled labor to low-skilled labor in Country 2, and $0 < \alpha < 1$, $0 < \beta < 1$. Assume all markets are competitive. Answer the following questions.(a) Derive the marginal product of high-skill and low-skill labor, express the high-to-low skilled wage ratio in terms of the parameters, and discuss whether high-skilled labor and low-skilled labor are complements or substitutes in production for both Country 1 and Country 2. (10%)

(i) How does the influx of illegal low-skilled migrants affect the high-to-low-skilled wage ratio in each country? Which group of workers (high-skilled or low-skilled, or both) is more adversely affected in each country?

(ii) Based on your analysis, which country's workers are more likely to oppose the influx of illegal low-skilled migrants? Explain with economic reasoning.

114年 - 114 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144147

113年 - 113 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144169

112年 - 112 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#144168

111年 - 111 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#145446

110年 - 110 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146105

110年 - 110 國立臺灣大學_碩士班招生考試_應用數學科學研究所數理統計組:機率統計#102196

109年 - 109 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#145663

108年 - 108 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146113

107年 - 107 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146100

106年 - 106 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146505