4. 研究者想瞭解運動對人體脈搏的影響，故針對 10 位糖尿病患者進行 1 個月的特定減重計畫，並記錄各患者在此減重計畫實施前、後的血壓(收縮壓)如下表：
假設此筆資料的事前分析顯示，減重前後差距變數為常態分配，請問在顯著水準為 1% 的情況下，此筆資料是否提供足夠的證據獲得以下結論：此特定減重計畫會影響糖尿病患者的血壓(收縮壓)？

5. 將高中成績分為三類：優良、普通、不佳，並在各分類中隨機抽取 5 個大學生並且記錄其大學 GPA 如下表： 在顯著水準為 0.05 情形下，請問此筆資料是否提供足夠證據而獲得以下結論：大學生之高中成績會影響其大學成績。

(1)請計算 \( a - g() \) 内的數值。(21%)

(2) \( 95\% \) 信賴水準下，如果 \( \mathrm{A} \times \mathrm{B} \) 的效果不顯著，而 \( \mathrm{A} \) 與 \( \mathrm{B} \) 的 \( \mathrm{F} \) 值都超過臨界值，請問該接著進行何種分析？(9%)

2. 某研究生建立線性迴歸模式為：\( Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \varepsilon_i \)，\( \beta_0 \)、\( \beta_1 \)、\( \beta_2 \) 分別是截距與解釋變數的係數，\( \varepsilon_i \) 是隨機誤差項，\( i = 1, 2, ..., n \)，\( n \) 為樣本大小。依此模式進行資料分析，假設結果滿足線性迴歸的基本假設，\( Y_i \) 的總變異量為 SST，模式的解釋力為 \( R^2 \)。該生想要檢視解釋變數是否有互動效果把 \( X_1 X_2 \) 的乘積項加入模式中，結果發現解釋力增加為 \( R^{2^*} \)。請問該生如何建構 F 統計值 (F-value) 檢定解釋力的增量 (即 \( R^{2^*} - R^2 \)) 是否顯著？(10%) 又該生想要評估解釋變數的影響程度是否不等，請問該生如何建構 F 統計值進行檢定？(10%)*Short Answers. Please show steps in your answers. Answers without proper steps will be deemed incomplete.

3. (10%) In American football, teams take turns in offense and defense. Assume that the only way teams can score is through offense. In the overtime, the team that scores first wins the game and they keep playing until a winner comes through. There is no tie game. Suppose Team A takes the offense first against Team B, and the probability of scoring at offense is \( 50\% \) for either of the teams. What is the winning probability for Team A when they play in the overtime? What is the winning probability of Team B?

4. (10%) When you get accepted by NTPU, you might wish to commute to the campus and live in downtown Taipei City. You will likely need to take the metro and transfer by bus to the school. Suppose the travel time for metro and bus are both normally distributed and independent from each other and it takes you exactly 10 minutes in total for all the walking.Suppose you need to show up in class at 9:10AM on campus, what is the probability for you to be late when you walk out of your door at 8:25AM?

5. (10%) Suppose out of the 250,000 people who came to Taiwan in the first half of 2021, 500 people were COVID-19 carriers. The PCR tests are able to correctly identify the COVID-19 carriers 90 out 100 times on average. On the other hand, PCR tests will show positive results for the non-carriers 5% of the times. How many false positives and false negatives are expected to be identified from the 250,000 people who entered Taiwan? Explain why CDC decided to impose the quarantine (isolation) measures on passengers from abroad even though they had been tested negative for the PCR tests back in 2021?

6. (20%) A manufacturing company produces electrical switches. To test the strength of the switches, robustness testing is carried out on 100 switches to determine how many times before the switch breaks down. The average of number of uses is 1.245 million times and the standard deviation is 0.3 million. The company would like to investigate whether or not the population mean of usage is more than 1.2 million times.A. At the 0.10 level of significance, is there evidence that the population mean of the number of usage is more than 1.2 million? Please form the null hypothesis and alternative hypothesis respectively before performing statistical tests.B. When the true population mean is 1.240 million and the level of significance is set as 0.05, what is the probability of making type I and type II errors respectively?Square root of numbers:

(一)\( y' = \frac{2x - e^x \sin y}{e^x \cos y + 1} \)(8 分)

(二)\( y' + \frac{1}{x}y = 3x^2y^3 \)(8 分)

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