113年 - 113 國立臺北大學_碩士班一般入學考試試題_企業管理學系甲組：統計學#146878

1. (10%) This machine is made up of six independent components, and the number represents the failure rate of the component. Find the reliability probability of this machine.

(Note: You are being asked to calculate to four decimal places)

2. (20%) The density function for the triangular distribution is given by

$$f(y) = y + 1 - x \quad \text{for} \quad x - 1 < y \leq x \quad \text{and}$$

$$f(y) = -y + 1 + x \quad \text{for} \quad x < y \leq x + 1$$

Find the mean absolute deviation $E[|Y - x|]$.

3. (20%) Suppose \( X \) and \( Y \) has joint density: \[ f(x, y) = \frac{2}{n(n+1)} \quad \text{for } y = 1, 2, 3, \dots, x, \quad x = 1, 2, 3, \dots, n. \] Find the correlation coefficient of \( x \) and \( y \).

4. (10%) 某研究生建立迴歸模式：\( Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \beta_3 X_{3i} + \varepsilon_i \)，\( \varepsilon_i \) 是誤差，\( i = 1, 2, 3, ..., n \)，\( \varepsilon_i \) 滿足基本假設。估計結果顯示 \( \beta_1 \) 的估計值(b1)在 95% 信賴水準下達顯著(\( H_1: \beta_1 > 0 \), t-value = 2.06)，但 \( X_1 \) 的 VIF (variance inflation factor) 值為 9。請問：如果 VIF 值為 4，那 \( \beta_1 \) 的估計值 95% 信賴水準下是否會轉為不顯著？為什麼？

(1) (10%) 如果北大企業就只有這五家商店，那檢定的虛無假說(null hypothesis)與對立假說(alternative hypothesis)為何？

(2) (10%) 如果北大企業是從旗下商店中隨機抽出這五家商店，那檢定的虛無假說與對立假說為何？

(3) (20%) 請說明(1)與(2)的檢定結果(5%顯著水準)。(F(4, 9) = 3.63; F(5, 9) = 3.48; F(4, 10) = 3.48; F(5, 10) = 3.33)