四、研究顯示台灣少女(14~18歲)的平均智商為118，標準差為10，假設該研究的隨機抽樣樣本數為400，請計算台灣少女平均智商的.95信賴區間( 15分)；另外，研究者可以根據此信賴區間，聲稱該信賴區間有.95的機率會包含母群參數嗎？如果不是，為什麼？(10分)

1. Good morning, everyone. My name is snowball. I am a polar bear. I am two years old. This is my twin brother, Fluffy. My brother and I are from Canada. We are very tall and big. And we are good swimmers. (10%)

2. Peter needs a healthy diet. He learns some dos and don’ts from his doctor. First, he should drink enough water and eat some meat every day. Also, he should eat a lot of fresh fruit and vegetables. Finally, he can’t eat too much candy or junk food. Now, every evening, Peter eats rice, meat, and vegetables for dinner. He is healthy now, his family is happy about that. (10%)

3. Our teacher used Google Maps to show London to us. He had us go along the Thames River because a lot of famous places are along it. We started with the London Eye. We really wanted to ride on it and enjoy a bird’s eye view of the city. After that, we walked south along the river, and we saw Big Ben on our right. After having this online trip, I can’t wait to see these beautiful sights in person. (10%)

4. News magazines, like newspapers, tell about important people and facts in the world. Doctors may read special medicine magazines to know the lastest medicine. There are also magazines for farmers, which give them new ideas about how to grow beautiful plants, vegetables, and rice. People who like sports can read sports magazines and know more about their favorite teams or players. (10%)

5. Maybe we’ll have Guide Robots. They help the blind people and lead the way for them, because they have a map in their computers. Sometimes they will be as smart as guide dogs. Family Robots go shopping for everything we need, and they also do all the housework. Office Robots will finish all the paper works. Maybe we’ll have Police Robots, too. They help the policemen to stop illegal things. (10%)

6. Weather satellites help scientists monitor the weather patterns on Earth. Information from weather satellites is invaluable for making weather forecasts and warning of potentially dangerous weather conditions such as hurricanes and tornadoes. Weather satellites travel in what is called low earth orbit. That means they travel between 320 and 800 kilometers above the Earth. (10%)

7. In spite of their advantages, motor vehicles have many critics because of their harmful effects on human lives and on air, water, land, and wildlife. The automobile may be the most destructive machine ever invented. Though we tend to deny it, riding in cars is one of the most dangerous things we do in our daily lives. (10%)

8. Extraction is the “pulling out” of a substance from one phase by another phase. The term is usually used to describe removal of a desired compound from a solid or liquid mixture by a solvent. In laboratory, the extraction of an organic compound from one liquid phase by another liquid, a technique called liquid-liquid extraction, or simply “extraction”, is common manipulation. (15%)

9. By definition a catalyst is a substance that enhances the rate of a chemical reaction but is not permanently altered by the reaction. Catalysts perform this feat because they decrease the activation energy required for a chemical reaction. In other words, catalysts provide an alternative reaction pathway that requires less energy. (15%)

令 X為一隨機變數，其機率密度函數為 \[ f(x) = \begin{cases} (x+1)/2, & \text{if } -1 < x < 1, \\ 0, & \text{otherwise.} \end{cases} \]試求隨機變數 \( Y = X^2 \) 的機率密度函數。(10 分)

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